A Finite Volume MHD Code in Spherical Coordinates for Background Solar Wind

Abstract

A second-order Godunov-type finite volume method (FVM) to advance the equations of single-fluid solar wind plasma magnetohydrodynamics (MHD) in time has been implemented into a numerical code. This code operates on a three-dimensional (3D) spherical shell with both non-staggered and staggered grids on the overlapping grid system with hexahedral cells of quadrilateral frustum type. By merging geometrical factors in spherical coordinates into the reformulation of fluxes, flux evaluation is made easy to achieve, and thus many numerical schemes with the total variation diminishing (TVD) slope limiters and approximate Roe solvers intended for Cartesian case can follow in the present context of spherical grid described here. At the same time, alternative strategies to ensure a solenoidal magnetic field, such as projection Poisson (PP) solver, hyperbolic divergence cleaning (HDC) method derived from generalized Lagrange multiplier (GLM) formulation of MHD system and constrained transport (CT) method, are employed. In this chapter, an FVM is described exemplarily on a six-component composite grid system by using a minmod limiter for oscillation control. Additionally, an implicit dual time-stepping technique is demonstrated to model the steady state solar wind ambient. Being of second order in space and time, this model is written in FORTRAN language with Message Passing Interface (MPI) parallelization, and validated in modeling the large-scale structure of solar wind from the Sun to Earth process (hereafter called Sun-to-Earth Process MHD model, also STEP-MHD model for brief). To demonstrate the suitability of our code for the simulation of solar wind ambient from the Sun to Earth, selected results from Carrington rotations (CR) during different solar activity phases are presented to show its capability of producing structured solar wind in agreement with observations.

Appendix: Discretizations, Reconstructions and MPP Flux-corrected Method

This appendix presents the used numerical techniques in some details for the discretized formulation of the solar wind MHD governing equations.

The STEP code advances the variables at the centroid point $(\overline{r}_i, \overline{\theta }_j, \phi _k)$ by constructing the semi-discretization for the solar wind MHD governing Eqs. (3.2) in a cell with geometrical center (i, j, k):

$$\begin{aligned} \begin{aligned} \frac{d\overline{\mathbf {U}}}{dt}=&-\frac{3\left( r_{ip}^2\mathbf {F}_{r,ip,j,k}^n-r_{im}^2\mathbf {F}_{r,im,j,k}^n\right) }{\left( r_{ip}^3-r_{im}^3\right) }- \frac{\sin \theta _{jp}\mathbf {F}_{\theta , i,jp,k}^n-\sin \theta _{jm}\mathbf {F}_{,\theta , i,jm,k}^n}{r_i^\theta \left( \cos \theta _{jm}-\cos \theta _{jp}\right) }\\&-\frac{\varDelta \theta (j)}{r_i^\phi \left( \cos \theta _{jm}-\cos \theta _{jp} \right) }\frac{\mathbf {F}_{\phi , i,j,kp}^n-\mathbf {F}_{\phi , i,j,km}^n}{\varDelta \phi (k)} +\overline{\mathbf {S}}_{i,j,k} \end{aligned} \end{aligned}$$

(A.1)

where $\mathbf {U}=\left( \rho , \rho v_{r}, \rho v_{\theta }, \rho v_{\phi }r\sin \theta , e, B_r, B_\theta , B_\phi \right) ^T$ or $\left( \rho , \rho v_{r}, \rho v_{\theta }, \rho v_{\phi }r\sin \theta , e, B_r, B_\theta ,\right. \left. B_\phi , \psi \right) ^T$ with primitive variables $\mathbf {W}=\left( \rho , v_{r}, v_{\theta }, v_{\phi }, p, B_r, B_\theta , B_\phi \right) ^T$ or $\left( \rho , v_{r}, v_{\theta }, v_{\phi }, p,\right. \left. B_r, B_\theta , B_\phi , \psi \right) ^T$.

The STEP code uses Roe and HLL solvers with PP, HDC and CT methods to keep the magnetic field divergence constraint. Flux formulations are displayed below.

3.1.1 Roe Type Flux with Powell’s Eight-wave Formulation

According to the papers [71, 72, 74, 141], the Roe flux form is given as follows:

(A.2)

(A.3)

(A.4)

(A.5)

(A.6)

(A.7)

Be care of the subscripts. For instance, the subscript $r\jmath lm$ ($r\jmath lp$) represents the r direction, $\jmath$th component, the arithmetic average at the interface $l=r_{im}$ ($r_{ip}$ ) of the variable between the left and right sides. $\widetilde{\mathbf {U}}_{[i,j,k]}$ denotes a second-order accurate linear reconstruction [219] on cell with geometrical center (i, j, k):

$$\begin{aligned} \begin{aligned}&\widetilde{\mathbf {U}}_{[i,j,k]}=\overline{\mathbf U}_{i,j,k}+(\widetilde{\partial _r\mathbf U})_{i,j,k}(r-\overline{r}_i)+ (\widetilde{\partial _\theta \mathbf U})_{i,j,k}(\theta -\overline{\theta }_j)+(\widetilde{\partial _\phi \mathbf U})_{i,j,k}(\phi -\phi _k)\\&(\widetilde{\partial _r\mathbf U})_{i,j,k}=\mathrm{minmod} \left( \frac{\overline{\mathbf U}_{i+1,j,k}-\overline{\mathbf U}_{i,j,k}}{\overline{r}_{i+1}-\overline{r}_i}, \frac{\overline{\mathbf U}_{i,j,k}-\overline{\mathbf U}_{i-1,j,k}}{\overline{r}_{i}-\overline{r}_{i-1}} \right) \\&(\widetilde{\partial _\theta \mathbf U})_{i,j,k}=\mathrm{minmod} \left( \frac{\overline{\mathbf U}_{i,j+1,k}-\overline{\mathbf U}_{i,j,k}}{\overline{\theta }_{j+1}- \overline{\theta }_j}, \ \frac{\overline{\mathbf U}_{i,j,k}-\overline{\mathbf U}_{i,j-1,k}}{\overline{\theta }_{j}-\overline{\theta }_{j-1}}\right) \\&(\widetilde{\partial _\phi \mathbf U})_{i,j,k}=\mathrm{minmod} \left( \frac{\overline{\mathbf U}_{i,j,k+1}-\overline{\mathbf U}_{i,j,k}}{\phi _{k+1}-\phi _k}, \ \frac{\overline{\mathbf U}_{i,j,k}-\overline{\mathbf U}_{i,j,k-1}}{\phi _k-\phi _{k-1}} \right) \\ \end{aligned} \end{aligned}$$

(A.8)

For the Powell’s eight wave formulation of MHD equations [141], the Jacobian matrices $\mathbf {A}=\frac{\partial \mathbf {F}_r}{\partial \mathbf {W}}$, $\mathbf {B}=\frac{\partial \mathbf {F}_\theta }{\partial \mathbf {W}}$, and $\mathbf {C}=\frac{\partial \mathbf {F}_\phi }{\partial \mathbf {W}}$ corresponding to the primitive variables in r, $\theta$, and $\phi$ directions are:

$$\begin{aligned} \mathbf {A}=\left( \begin{array}{cccccccc} v_r&{}\rho &{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{}v_r&{}0&{}0&{}\frac{1}{\rho }&{}0&{}\frac{B_\theta }{\mu \rho }&{}\frac{B_\phi }{\mu \rho }\\ 0&{}0&{}v_r&{}0&{}0&{}0&{}-\frac{B_r}{\mu \rho }&{}0\\ 0&{}0&{}0&{}v_r&{}0&{}0&{}0&{}-\frac{B_r}{\mu \rho }\\ 0&{}\gamma p&{}0&{}0&{}v_r&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}v_r&{}0&{}0\\ 0&{}B_\theta &{}-B_r&{}0&{}0&{}0&{}v_r&{}0\\ 0&{}B_\phi &{}0&{}-B_r&{}0&{}0&{}0&{}v_r \end{array} \right) \end{aligned}$$

$$\begin{aligned} \mathbf {B}=\left( \begin{array}{cccccccc} v_{\theta }&{}0&{}\rho &{}0&{}0&{}0&{}0&{}0\\ 0&{}v_{\theta }&{}0&{}0&{}0&{}-\frac{B_{\theta }}{\mu \rho }&{}0&{}0\\ 0&{}0&{}v_{\theta }&{}0&{}\frac{1}{\rho }&{}\frac{B_{r}}{\mu \rho }&{}0&{}\frac{B_\phi }{\mu \rho }\\ 0&{}0&{}0&{}v_{\theta }&{}0&{}0&{}0&{}-\frac{B_{\theta }}{\mu \rho }\\ 0&{}0&{}\gamma p&{}0&{}v_{\theta }&{}0&{}0&{}0\\ 0&{}-B_{\theta }&{}B_{r}&{}0&{}0&{}v_{\theta }&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}v_{\theta }&{}0\\ 0&{}0&{}B_\phi &{}-B_{\theta }&{}0&{}0&{}0&{}v_{\theta } \end{array} \right) \end{aligned}$$

$$\begin{aligned} \mathbf {C}=\left( \begin{array}{cccccccc} v_\phi &{}0&{}0&{}\rho &{}0&{}0&{}0&{}0\\ 0&{}v_\phi &{}0&{}0&{}0&{}-\frac{B_\phi }{\mu \rho }&{}0&{}0\\ 0&{}0&{}v_\phi &{}0&{}0&{}0&{}-\frac{B_\phi }{\mu \rho }&{}0\\ 0&{}0&{}0&{}v_\phi &{}\frac{1}{\rho }&{}\frac{B_r}{\mu \rho }&{}\frac{B_\theta }{\mu \rho }&{}0\\ 0&{}0&{}0&{}\gamma p&{}v_\phi &{}0&{}0&{}0\\ 0&{}-B_\phi &{}0&{}B_r&{}0&{}v_\phi &{}0&{}0\\ 0&{}0&{}-B_\phi &{}B_\theta &{}0&{}0&{}v_\phi &{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}v_\phi \end{array} \right) \end{aligned}$$

In fact, the Jacobian of each direction may be transferred to each other.

$$\begin{aligned} \mathbf {T}_{AB}\left( \mathbf {A}\left( \mathbf {T}_{AB}\mathbf {W}\right) \right) \mathbf {T}_{AB}^{-1}= \mathbf {T}_{AB} \left( \begin{array}{cccccccc} v_{\theta }&{}\rho &{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{}v_{\theta }&{}0&{}0&{}\frac{1}{\rho }&{}0&{}\frac{B_{r}}{\mu \rho }&{}\frac{B_\phi }{\mu \rho }\\ 0&{}0&{}v_{\theta }&{}0&{}0&{}0&{}-\frac{B_{\theta }}{\mu \rho }&{}0\\ 0&{}0&{}0&{}v_{\theta }&{}0&{}0&{}0&{}-\frac{B_{\theta }}{\mu \rho }\\ 0&{}\gamma p&{}0&{}0&{}v_{\theta }&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}v_{\theta }&{}0&{}0\\ 0&{}B_{r}&{}-B_{\theta }&{}0&{}0&{}0&{}v_{\theta }&{}0\\ 0&{}B_\phi &{}0&{}-B_{\theta }&{}0&{}0&{}0&{}v_{\theta } \end{array} \right) \mathbf {T}_{AB}^{-1}=\mathbf {B}\left( \mathbf {W}\right) \end{aligned}$$

where

$$\begin{aligned} \mathbf {T}_{AB}=\mathbf {T}_{AB}^{-1}= \left( \begin{array}{cccccccc} 1&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{}0&{}1&{}0&{}0&{}0&{}0&{}0\\ 0&{}1&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}1&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}1&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}1&{}0\\ 0&{}0&{}0&{}0&{}0&{}1&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}1 \end{array} \right) \end{aligned}$$

Similarly, one can also obtain the Jacobian matrix $\mathbf {C}\left( \mathbf {W}\right) =\mathbf {T}_{AC}\left( \mathbf {A}\left( \mathbf {T}_{AC}\mathbf {W}\right) \right) \mathbf {T}_{AC}^{-1}$, and

$$\begin{aligned} \mathbf {T}_{AC}=\mathbf {T}_{AC}^{-1}= \left( \begin{array}{cccccccc} 1&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}1&{}0&{}0&{}0&{}0\\ 0&{}0&{}1&{}0&{}0&{}0&{}0&{}0\\ 0&{}1&{}0&{}0&{}0&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}1&{}0&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}1\\ 0&{}0&{}0&{}0&{}0&{}0&{}1&{}0\\ 0&{}0&{}0&{}0&{}0&{}1&{}0&{}0 \end{array} \right) \end{aligned}$$

The corresponding eigenvalues and eigenvectors of $\mathbf {B}\left( \mathbf {W}\right)$ and $\mathbf {C}\left( \mathbf {W}\right)$ can be generated as follows:

$$ \lambda _{\theta \jmath }=\lambda _{r\jmath }\left( \mathbf {T}_{AB}\mathbf {W}\right) ,\mathbf {r}_{\theta \jmath }=\mathbf {T}_{AB}\left( \mathbf {r}_{r \jmath }\left( \mathbf {T}_{AB}\mathbf {W}\right) \right) ,\mathbf {l}_{\theta \jmath }=\left( \mathbf {l}_{r \jmath }\left( \mathbf {T}_{AB}\mathbf {W}\right) \right) \mathbf {T}^{-1}_{AB} $$

$$ \lambda _{\phi \jmath }=\lambda _{r\jmath }\left( \mathbf {T}_{AC}\mathbf {W}\right) ,\mathbf {r}_{\phi \jmath }=\mathbf {T}_{AC}\left( \mathbf {r}_{r \jmath }\left( \mathbf {T}_{AC}\mathbf {W}\right) \right) ,\mathbf {l}_{\phi \jmath }=\left( \mathbf {l}_{r \jmath }\left( \mathbf {T}_{AC}\mathbf {W}\right) \right) \mathbf {T}^{-1}_{AC} $$

For simplicity, here is only listed the eigensystem corresponding to r direction.

$$\begin{aligned} \lambda _{r\jmath lm}=\lambda _{r\jmath }\left( \mathbf {U}_{lm}\right) =\lambda _{r\jmath }\left( \frac{1}{2}\left( \widetilde{\mathbf {U}}_{[i-1,j,k]}(r_{im},\theta _j^r,\phi _k)+ \widetilde{\mathbf {U}}_{[i,j,k]}(r_{im},\theta _j^r,\phi _k)\right) \right) \end{aligned}$$

(A.9)

The corresponding eigenvectors $\mathbf {L}_{r\jmath lm}$ and $\mathbf {R}_{r\jmath lm}$ are as follows:

$$\begin{aligned} \begin{aligned}&\mathbf {L}_{r\jmath lm}=\left( \mathbf {l}_{r\jmath }\frac{\partial \mathbf {W}}{\partial \mathbf {U}}\right) \left( \mathbf {U}_{lm}\right) =\left( \mathbf {l}_{r\jmath }\frac{\partial \mathbf {W}}{\partial \mathbf {U}}\right) \left( \frac{1}{2}\left( \widetilde{\mathbf {U}}_{[i-1,j,k]}(r_{im},\theta _j^r,\phi _k)+ \widetilde{\mathbf {U}}_{[i,j,k]}(r_{im},\theta _j^r,\phi _k)\right) \right) \\&\mathbf {R}_{r\jmath lm}=\left( \frac{\partial \mathbf {U}}{\partial \mathbf {W}}\mathbf {r}_{r\jmath }\right) \left( \mathbf {U}_{lm}\right) =\left( \frac{\partial \mathbf {U}}{\partial \mathbf {W}}\mathbf {r}_{r\jmath }\right) \left( \frac{1}{2}\left( \widetilde{\mathbf {U}}_{[i-1,j,k]}(r_{im},\theta _j^r,\phi _k)+ \widetilde{\mathbf {U}}_{[i,j,k]}(r_{im},\theta _j^r,\phi _k)\right) \right) \end{aligned} \end{aligned}$$

(A.10)

$$\begin{aligned} \frac{\partial \mathbf {W}}{\partial \mathbf {U}}=\left( \begin{array}{cccccccc} 1&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ -\frac{v_r}{\rho }&{}\frac{1}{\rho }&{}0&{}0&{}0&{}0&{}0&{}0\\ -\frac{v_{\theta }}{\rho }&{}0&{}\frac{1}{\rho }&{}0&{}0&{}0&{}0&{}0\\ -\frac{v_{\phi }}{\rho }&{}0&{}0&{}\frac{1}{\rho r \sin \theta }&{}0&{}0&{}0&{}0\\ \alpha \zeta &{}-v_r\zeta &{}-v_{\theta }\zeta &{}-\frac{v_{\phi }\zeta }{r \sin \theta }&{}\zeta &{}-\frac{B_{r}\zeta }{\mu }&{} -\frac{B_{\theta }\zeta }{\mu }&{}-\frac{B_{\phi }\zeta }{\mu } \\ 0&{}0&{}0&{}0&{}0&{}1&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}1&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}1 \end{array}\right) \end{aligned}$$

(A.11)

$$\begin{aligned} \frac{\partial \mathbf {U}}{\partial \mathbf {W}}=\left( \begin{array}{cccccccc} 1&{}0&{}0&{}0&{}0&{}0&{}0&{}0\\ v_r&{}\rho &{}0&{}0&{}0&{}0&{}0&{}0\\ v_{\theta }&{}0&{}\rho &{}0&{}0&{}0&{}0&{}0\\ v_{\phi }r \sin \theta &{}0&{}0&{}\rho r \sin \theta &{}0&{}0&{}0&{}0\\ \alpha &{}\rho v_r&{}\rho v_{\theta }&{}\rho v_{\phi }&{}\frac{1}{\zeta }&{}\frac{B_{r }}{\mu } &{} \frac{B_{\theta }}{\mu }&{}\frac{B_{\phi }}{\mu }\\ 0&{}0&{}0&{}0&{}0&{}1&{}0&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}1&{}0\\ 0&{}0&{}0&{}0&{}0&{}0&{}0&{}1 \end{array}\right) \end{aligned}$$

(A.12)

where $\alpha =\frac{1}{2}(v_r^2+v_{\theta }^2+v_{\phi }^2)$ and $\zeta =\gamma -1$.

For convenience, denote ${c}_{fr}, {c}_{sr}, {c}_{f\theta }, {c}_{s\theta }, {c}_{f\phi }, {c}_{s\phi }$ the fast and slow magnetosonic speeds in $(r,\theta ,\phi )$ directions

$$\begin{aligned} \begin{aligned}&c_{fr}=\frac{1}{\sqrt{2}}\sqrt{c_s^2+c_A^2+\left( \left( c_s^2+c_A^2\right) ^2-4c_s^2\frac{B_r^2}{\mu \rho }\right) ^{\frac{1}{2}}}, c_{sr}=\frac{1}{\sqrt{2}}\sqrt{c_s^2+c_A^2-\left( \left( c_s^2+c_A^2\right) ^2-4c_s^2\frac{B_r^2}{\mu \rho }\right) ^{\frac{1}{2}}} \\&c_{f\theta }=\frac{1}{\sqrt{2}}\sqrt{c_s^2+c_A^2+\left( \left( c_s^2+c_A^2\right) ^2-4c_s^2\frac{B_\theta ^2}{\mu \rho }\right) ^{\frac{1}{2}}}, c_{s\theta }=\frac{1}{\sqrt{2}}\sqrt{c_s^2+c_A^2-\left( \left( c_s^2+c_A^2\right) ^2-4c_s^2\frac{B_\theta ^2}{\mu \rho }\right) ^{\frac{1}{2}}} \\&c_{f\phi }=\frac{1}{\sqrt{2}}\sqrt{c_s^2+c_A^2+\left( \left( c_s^2+c_A^2\right) ^2-4c_s^2\frac{B_\phi ^2}{\mu \rho }\right) ^{\frac{1}{2}}}, c_{s\phi }=\frac{1}{\sqrt{2}}\sqrt{c_s^2+c_A^2-\left( \left( c_s^2+c_A^2\right) ^2-4c_s^2\frac{B_\phi ^2}{\mu \rho }\right) ^{\frac{1}{2}}} \end{aligned} \end{aligned}$$

where $c_s=\sqrt{\frac{\gamma p}{\rho }}$ and $c_A=\sqrt{ \frac{B_r^2+B_\theta ^2+B_\phi ^2}{\mu \rho } }$ are the sound and total Alfvénic speeds, respectively. $c_{ax}=\frac{B_x}{\sqrt{\mu \rho }}$ is the Alfvénic speed in x-direction with $x=r, \theta , \phi$.

The eigenvectors associated with those eigenvalues are:

$$ \lambda _{r1}=v_r-c_{fr} $$

$$ \mathbf {r}_{r1}=(\rho \alpha _f,-c_{fr}\alpha _f, c_{sr} \alpha _s\beta _\theta M, c_{sr} \alpha _s\beta _{\phi } M, \alpha _f\gamma p, 0, \alpha _s\sqrt{\mu \rho }c_s \beta _\theta , \alpha _s\sqrt{\mu \rho }c_s \beta _{\phi })^T $$

$$ \mathbf {l}_{r1}=(0,-\frac{c_{fr} \alpha _f}{2c_s^2}, \frac{c_{sr} \alpha _s\beta _{\theta }M}{2c_s^2},\frac{c_{sr}\alpha _s\beta _{\phi }M}{2c_s^2}, \frac{\alpha _f}{2\rho c_s^2}, 0, \frac{\alpha _s\beta _{\theta }}{2\sqrt{\mu \rho } c_s},\frac{\alpha _s\beta _{\phi }}{2\sqrt{\mu \rho } c_s}) $$

$$ \lambda _{r2}=v_r-|c_{ar}| $$

$$ \mathbf {r}_{r2}=(0, 0, -\frac{\beta _{\phi }}{\sqrt{2}}, \frac{\beta _{\theta }}{\sqrt{2}}, 0, 0, -\beta _{\phi }M\sqrt{\frac{\mu \rho }{2}}, \beta _{\theta }M\sqrt{\frac{\mu \rho }{2}})^T $$

$$ \mathbf {l}_{r2}=(0, 0, -\frac{\beta _{\phi }}{\sqrt{2}}, \frac{\beta _{\theta }}{\sqrt{2}}, 0, 0, -\frac{\beta _{\phi }M}{\sqrt{2\mu \rho }}, \frac{\beta _{\theta }M}{\sqrt{2\mu \rho }}) $$

$$ \lambda _{r3}=v_r-c_{sr} $$

$$ \mathbf {r}_{r3}=(\rho \alpha _s, -c_{rs} \alpha _s, -c_{fr} \alpha _f\beta _{\theta } M, -c_{fr} \alpha _f \beta _{\phi } M, \alpha _s \gamma p, 0, -\alpha _f \sqrt{\mu \rho }c_s \beta _{\theta }, -\alpha _f\sqrt{\mu \rho }c_s \beta _{\phi })^T $$

$$ \mathbf {l}_{r3}=(0, -\frac{c_{sr} \alpha _s}{2c_s^2}, -\frac{c_{fr} \alpha _f \beta _{\theta } M}{2c_s^2}, -\frac{c_{fr} \alpha _f \beta _{\phi } M}{2\alpha ^2}, \frac{\alpha _s}{2\rho c_s^2}, 0, -\frac{\alpha _f \beta _{\theta }}{2\sqrt{\mu \rho }c_s}, -\frac{\alpha _f\beta _{\phi }}{2\sqrt{\mu \rho }c_s}) $$

$$ \lambda _{r4}=v_r, \ \ \mathbf {r}_{r4}=(1,0,0,0,0,0,0,0)^T,\ \ \mathbf {l}_{r4}=(1, 0, 0, 0, -\frac{1}{c_s^2}, 0, 0, 0) $$

$$ \lambda _{r5}=v_r,\ \ \mathbf {r}_{r5}=(0, 0, 0, 0, 0, 1, 0, 0)^T, \ \ \mathbf {l}_{r5}=(0, 0, 0, 0, 0, 1, 0, 0) $$

$$ \lambda _{r6}=v_r+c_{sr} $$

$$ \mathbf {r}_{r6}=(\rho \alpha _s, c_{sr}\alpha _s, c_{fr}\alpha _f\beta _{\theta } M, c_{fr} \alpha _f\beta _{\phi } M, \alpha _s\gamma p, 0, -\alpha _f\sqrt{\mu \rho }c_s\beta _\theta , -\alpha _f\sqrt{\mu \rho }c_s\beta _{\phi })^T $$

$$ \mathbf {l}_{r6}=(0, \frac{c_{sr} \alpha _s}{2c_s^2}, \frac{c_{fr}\alpha _f\beta _\theta M}{2c_s^2}, \frac{c_{fr} \alpha _f\beta _{\phi } M}{2c_s^2}, \frac{\alpha _s}{2\rho c_s^2}, 0, -\frac{\alpha _f\beta _\theta }{2\sqrt{\mu \rho }c_s}, -\frac{\alpha _f\beta _{\phi }}{2\sqrt{\mu \rho }c_s}) $$

$$ \lambda _{r7}=v_r+|c_{ar}| $$

$$ \mathbf {r}_{r7}=(0, 0, -\frac{\beta _{\phi }}{\sqrt{2}}, \frac{\beta _{\theta }}{\sqrt{2}}, 0, 0, \beta _{\phi }M\sqrt{\frac{\mu \rho }{2}}, -\beta _{\theta }M\sqrt{\frac{\mu \rho }{2}})^T $$

$$ \mathbf {l}_{r7}=(0, 0, -\frac{\beta _{\phi }}{\sqrt{2}}, \frac{\beta _{\theta }}{\sqrt{2}}, 0, 0, \frac{\beta _{\phi }M}{\sqrt{2\mu \rho }}, -\frac{\beta _{\theta }M}{\sqrt{2\mu \rho }}) $$

$$ \lambda _{r8}=v_r+c_{fr} $$

$$ \mathbf {r}_{r8}=(\rho \alpha _f, c_{fr}\alpha _f, -c_{sr} \alpha _s \beta _{\theta } M, -c_{sr} \alpha _s\beta _{\phi } M, \alpha _f\gamma p, 0, \alpha _s\sqrt{\mu \rho }c_s\beta _{\theta }, \alpha _s\sqrt{\mu \rho } c_s \beta _{\phi })^T $$

$$ \mathbf {l}_{r8}=(0,\frac{c_{fr}\alpha _f}{2c_s^2}, -\frac{c_{sr} \alpha _s\beta _{\theta }M}{2c_s^2}, -\frac{c_{sr}\alpha _s\beta _{\phi }M}{2c_s^2},\frac{\alpha _f}{2\rho c_s^2}, 0, \frac{\alpha _s\beta _{\theta }}{2\sqrt{\mu \rho }c_s}, \frac{\alpha _s\beta _{\phi }}{2\sqrt{\mu \rho }c_s}) $$

where

$$ M=sign(B_r), \alpha _f^2=\frac{c_s^2-c_{sr}^2}{c_{fr}^2-c_{sr}^2}, \alpha _s^2=\frac{c_{fr}^2-c_s^2}{c_{fr}^2-c_{sr}^2}, \beta _{\theta }=\frac{B_{\theta }}{\sqrt{B_{\theta }^2+B_{\phi }^2}}, \beta _{\phi }=\frac{B_{\phi }}{\sqrt{B_{\theta }^2+B_{\phi }^2}} $$

The eigenvectors given above are orthogonal, and, since $\alpha _f$, $\alpha _s$, $\beta _\theta$, and $\beta _\phi$ all lie between zero and one, the eigenvectors are all well-formed, once these four parameters are defined. The only difficulties in defining these occur when $B_{\theta }^2+B_{\phi }^2=0$, in which case $\beta _\theta$ and $\beta _\phi$ are ill-defined, and when $B_{\theta }^2+B_{\phi }^2=0$, and $B_r^2=\mu \rho c_s^2$, in which case $\alpha _f$ and $\alpha _s$ are ill-defined. The first case is fairly trivial; $\beta _\theta$ and $\beta _\phi$ represent direction cosines for the tangential component of the $\mathbf {B}$-field, and in the case of a zero component, it is only important to choose so that $\beta _{\theta }^2+\beta _{\phi }^2=1$ [141] or $\beta _{\theta }=\beta _{\phi }=\frac{1}{\sqrt{2}}$ [16, 158]. An approach for the case in which $\alpha _f$ and $\alpha _s$ are ill-defined has been outlined in detail by Roe and Balsara [150]. Indeed, they have shown in Roe and Balsara [150] that multiple solutions in this case for the linearized Riemann problem does not affect the value for the interface flux.

In case of $B_\theta ^2+B_\phi ^2=0$ (say, $B_{\theta }^2+B_{\phi }^2<10^{-10}$), one can set $\beta _{\theta }=\beta _{\phi }=\frac{1}{\sqrt{2}}$ as done by Brio and Wu [16]. The second case of $|c_{ar}|= c_s$ and $B_\theta ^2+B_\phi ^2=0$ corresponds to the magnetosonic case, in which $c_{fr}^2=c_{sr}^2=c_s^2$, and $\alpha _f^2+\alpha _s^2=1$. In order to remove the singularities in eigenvectors, in case of $|c_{ar}|- c_s=0, B_\theta ^2+B_\phi ^2=0$ (say, $\left| |c_{ar}|- c_s\right| <10^{-10}$ and $B_{\theta }^2+B_{\phi }^2<10^{-10}$), one can follow the formulation of Case V suggested by Roe and Balsara [150] or directly set $\alpha _f^2=\alpha _s^2=\frac{1}{2}$ [158]. Thus, one will have well-defined orthogonal eigenvectors.

3.1.2 Roe Type Flux in GLM Form

Using the GLM [35], the divergence constraint is coupled with the conservation laws by introducing a new variable $\psi$. Now, the coupling part can be written in the following form

$$\begin{aligned} \begin{aligned}&\frac{\partial \mathbf {B}}{\partial t}+\nabla \cdot \left( \mathbf {vB}-\mathbf {Bv}+\psi I\right) =0 \\&\frac{\partial \psi }{\partial t}+\nabla \cdot \left( c_h^2\mathbf {B}\right) =-\frac{c_h^2}{c_p^2}\psi \end{aligned} \end{aligned}$$

(A.13)

Thus, the governing equations involve the ninth variable in $\mathbf {U}=\left( \rho , \rho v_{r}, \rho v_{\theta }, \rho v_{\phi }r\right. \left. \sin \theta , e, B_r, B_\theta , B_\phi , \psi \right) ^T$, with primitive variables $\mathbf {W}=\left( \rho , v_{r}, v_{\theta }, v_{\phi }, p, B_r, B_\theta ,\right. \left. B_\phi , \psi \right) ^T$. In the GLM approach, divergence errors are propagated to the domain boundaries at finite speed $c_h$ and damped at a rate given by $c^2_h/c^2_p$. The parameter $c_h$ is the constant advection velocity of $\psi$. In order to avoid bringing additional constraint on the time step, it is evaluated as the fastest eigenvalue in the whole computation domain $c_h=\max _{i,j,k}\left( |v_r|+c_{fr},|v_\theta |+c_{f\theta },|v_\phi |+c_{f\phi }\right)$. The choice of constant $c_p$ will be given below. In the GLM formulation, computing the flux across the interface in each direction can follow from decoupling the equations of normal magnetic field and $\psi$ from the other seven equations [35, 121, 122, 170]. That is, when solving a one dimensional Riemann problem at a zone interface (say the r direction), these additional waves are decoupled from the remaining ones and are described by the following $2\times 2$ linear hyperbolic system without the source term:

$$\begin{aligned} \left\{ \begin{array}{l} \frac{\partial B_{r}}{\partial t}+\frac{1}{r^2}\frac{\partial r^2\psi }{\partial r}=0\\ \frac{\partial \psi }{\partial t}+\frac{1}{r^2}\frac{\partial }{\partial r}\left( c_h^2 r^2 B_{r}\right) =0 \end{array} \right. \end{aligned}$$

(A.14)

For a generic pair of left and right input states $(B_{rL}, \psi _L)$ and $(B_{rR}, \psi _R)$, the Godunov flux at the interface l of the system (A.14) can be computed exactly [35, 121, 122, 170] as

$$\begin{aligned} {F}_{r,B_{r}}=\psi _l=\frac{\psi _L+\psi _R}{2}-\frac{c_h(B_{rR}-B_{rL})}{2} \end{aligned}$$

(A.15)

$$\begin{aligned} {F}_{r,\psi }=c_h^2B_{rl}=c_h^2\left( \frac{B_{rL}+B_{rR}}{2}-\frac{\psi _R-\psi _L}{2c_h}\right) \end{aligned}$$

(A.16)

This allows us to carry out the solution of the $2\times 2$ linear Riemann problem separately before using any standard 7-wave Riemann solver for the one-dimensional MHD equations. Similarly, one can obtain the flux corresponding to the $\theta$ and $\phi$ directions. Specifically, the flux of the magnetic field and $\psi$ components at the cell interfaces are as follows:

$$\begin{aligned} {F}_{r,B_{r}, im,j,k}^n=~&\frac{1}{2}\left[ \widetilde{\psi }_{[i-1,j,k]}(r_{im},\theta _j^r,\phi _k)+\widetilde{\psi }_{[i,j,k]}(r_{im},\theta _j^r,\phi _k) \right] + \nonumber \\&\frac{1}{2}c_h\left( \widetilde{B}_{r[i-1,j,k]}(r_{im},\theta _j^r,\phi _k)-\widetilde{B}_{r[i,j,k]}(r_{im},\theta _j^r,\phi _k)\right) \nonumber \\ {F}_{r,\psi , im,j,k}^n=~&c_h^2\left\{ \frac{1}{2}\left( \widetilde{B}_{r[i-1,j,k]}(r_{im},\theta _j^r,\phi _k)+\widetilde{B}_{r[i,j,k]}(r_{im},\theta _j^r,\phi _k)\right) +\right. \nonumber \\&\left. \frac{1}{2c_h}\left( \widetilde{\psi }_{[i-1,j,k]}(r_{im},\theta _j^r,\phi _k)-\widetilde{\psi }_{[i,j,k]}(r_{im},\theta _j^r,\phi _k) \right) \right\} \nonumber \\ {F}_{\theta ,B_{\theta }, i,jm,k}^n=~&\frac{1}{2}\left[ \widetilde{\psi }_{[i,j-1,k]}(r_i^\theta ,\theta _{jm},\phi _k)+\widetilde{\psi }_{[i,j,k]}(r_i^\theta ,\theta _{jm},\phi _k) \right] + \nonumber \\&\frac{1}{2}c_h\left( \widetilde{B}_{\theta [i,j-1,k]}(r_i^\theta ,\theta _{jm},\phi _k)-\widetilde{B}_{\theta [i,j,k]}(r_i^\theta ,\theta _{jm},\phi _k)\right) \nonumber \\ {F}_{\theta ,\psi ,i,jm,k}^n=~&c_h^2\left\{ \frac{1}{2}\left( \widetilde{B}_{\theta [i,j-1,k]}(r_i^\theta ,\theta _{jm},\phi _k)+\widetilde{B}_{\theta [i,j,k]}(r_i^\theta ,\theta _{jm},\phi _k)\right) +\right. \nonumber \\&\left. \frac{1}{2c_h}\left( \widetilde{\psi }_{[i,j-1,k]}(r_i^\theta ,\theta _{jm},\phi _k)-\widetilde{\psi }_{[i,j,k]}(r_i^\theta ,\theta _{jm},\phi _k) \right) \right\} \nonumber \\ {F}_{\phi ,B_{\phi }, i,j,km}^n=~&\frac{1}{2}\left\{ \widetilde{\psi }_{[i,j,k-1]}(r_i^\phi ,\theta _j^\phi ,\phi _{km})+ \widetilde{\psi }_{[i,j,k]}(r_i^\phi ,\theta _j^\phi ,\phi _{km}) \right\} \nonumber \\&+\frac{1}{2}c_h\left( \widetilde{B}_{\phi [i,j,k-1]}(r_i^\phi ,\theta _j^\phi ,\phi _{km})-\widetilde{B}_{\phi [i,j,k]}(r_i^\phi ,\theta _j^\phi ,\phi _{km})\right) \nonumber \\ {F}_{\phi ,\psi ,i,j,km}^n=~&c_h^2\left\{ \frac{1}{2}\left( \widetilde{B}_{\phi [i,j,k-1]}(r_i^\phi ,\theta _j^\phi ,\phi _{km})+\widetilde{B}_{\phi [i,j,k]}(r_i^\phi ,\theta _j^\phi ,\phi _{km})\right) +\right. \nonumber \\&\left. \frac{1}{2c_h}\left( \widetilde{\psi }_{[i,j,k-1]}(r_i^\phi ,\theta _j^\phi ,\phi _{km})-\widetilde{\psi }_{[i,j,k]}(r_i^\phi ,\theta _j^\phi ,\phi _{km}) \right) \right\} \end{aligned}$$

(A.17)

Besides the fluxes listed above, other flux components not given here are still computed using those in Eqs. (A.2)–(A.7). For convenience, one can rewrite the fluxes for GLM system here by setting $\mathbf {F}_{r,im,j,k}^{n*}\equiv \left( { F}_{rq,im,j,k}^{n*}\right) ^T$, $\mathbf {F}_{r,ip,j,k}^{n*}\equiv \left( { F}_{rq,ip,j,k}^{n*}\right) ^T$, $\mathbf {F}_{\theta ,i,jm,k}^{n*}\equiv \left( { F}_{\theta q,i,jm,k}^{n*}\right) ^T$, $\mathbf {F}_{\theta ,i,jp,k}^{n*}\equiv \left( {F}_{\theta q,i,jp,k}^{n*}\right) ^T$, $\mathbf {F}_{\phi ,i,j,km}^{n*}\equiv \left( { F}_{\phi q,i,j,km}^{n*}\right) ^T$ and $\mathbf {F}_{\phi ,i,j,kp}^{n*}\equiv \left( \mathrm{F}_{\phi q,i,j,kp}^{n*}\right) ^T$, where $q=1, 2, \ldots , 8,9$:

$$\begin{aligned} \mathbf {F}_{r,im,j,k}^{n*}=~&\frac{1}{2}\left( \mathbf {F}_r\left( \widetilde{\mathbf {U}}_{[i-1,j,k]}\left( r_{im},\theta _j^{r},\phi _k\right) \right) + \mathbf {F}_r\left( \widetilde{\mathbf {U}}_{[i,j,k]}\left( r_{im},\theta _j^{r},\phi _k\right) \right) \right) + \nonumber \\&\frac{1}{2}\left( \sum _\jmath \mathbf {R}_{r\jmath lm}\left| \lambda _{r\jmath lm}\right| \left[ \mathbf {L}_{r\jmath lm} \left( \widetilde{\mathbf {U}}_{[i-1,j,k]}\left( r_{im},\theta _j^{r},\phi _k\right) - \widetilde{\mathbf {U}}_{[i,j,k]}\left( r_{im},\theta _j^{r},\phi _k\right) \right) \right] \right) \nonumber \\ \mathbf {F}_{r,ip,j,k}^{n*}=~&\frac{1}{2}\left( \mathbf {F}_r\left( \widetilde{\mathbf {U}}_{[i,j,k]}\left( r_{ip},\theta _j^{r},\phi _k\right) \right) + \mathbf {F}_r\left( \widetilde{\mathbf {U}}_{[i+1,j,k]}\left( r_{ip},\theta _j^{r},\phi _k\right) \right) \right) + \nonumber \\&\frac{1}{2}\left( \sum _\jmath \mathbf {R}_{r\jmath lp}\left| \lambda _{r\jmath lp}\right| \left[ \mathbf {L}_{r\jmath lp}\left( \widetilde{\mathbf {U}}_{[i,j,k]}\left( r_{ip},\theta _j^{r},\phi _k\right) - \widetilde{\mathbf {U}}_{[i+1,j,k]}\left( r_{ip},\theta _j^{r},\phi _k\right) \right) \right] \right) \nonumber \\ \mathbf {F}_{\theta ,i,jm,k}^{n*}=~&\frac{1}{2}\left( \mathbf {F}_\theta \left( \widetilde{\mathbf {U}}_{[i,j-1,k]}\left( r_{i}^{\theta },\theta _{jm},\phi _k\right) \right) + \mathbf {F}_\theta \left( \widetilde{\mathbf {U}}_{[i,j,k]}\left( r_{i}^{\theta },\theta _{jm},\phi _k\right) \right) \right) + \nonumber \\&\frac{1}{2}\left( \sum _{\jmath }\mathbf {R}_{\theta \jmath lm}\left| \lambda _{\theta \jmath lm}\right| \left[ \mathbf {L}_{\theta \jmath lm}\left( \widetilde{\mathbf {U}}_{[i,j-1,k]}\left( r_{i}^{\theta },\theta _{jm},\phi _k\right) - \widetilde{\mathbf {U}}_{[i,j,k]}\left( r_{i}^{\theta },\theta _{jm},\phi _k\right) \right) \right] \right) \nonumber \\ \mathbf {F}_{\theta ,i,jp,k}^{n*}=~&\frac{1}{2}\left( \mathbf {F}_\theta \left( \widetilde{\mathbf {U}}_{[i,j,k]}\left( r_{i}^{\theta },\theta _{jp},\phi _k\right) \right) + \mathbf {F}_\theta \left( \widetilde{\mathbf {U}}_{[i,j+1,k]}\left( r_{i}^{\theta },\theta _{jp},\phi _k\right) \right) \right) + \nonumber \\&\frac{1}{2}\left( \sum _{\jmath }\mathbf {R}_{\theta \jmath lp}\left| \lambda _{\theta \jmath lp}\right| \left[ \mathbf {L}_{\theta \jmath lp}\left( \widetilde{\mathbf {U}}_{[i,j,k]}\left( r_{i}^{\theta },\theta _{jp},\phi _k\right) - \widetilde{\mathbf {U}}_{[i,j+1,k]}\left( r_{i}^{\theta },\theta _{jp},\phi _k\right) \right) \right] \right) \nonumber \\ \mathbf {F}_{\phi ,i,j,km}^{n*}=~&\frac{1}{2}\left( \mathbf {F}_\phi \left( \widetilde{\mathbf {U}}_{[i,j,k-1]}\left( r_{i}^{\phi },\theta _{j}^{\phi },\phi _{km}\right) \right) + \mathbf {F}_\phi \left( \widetilde{\mathbf {U}}_{[i,j,k]}\left( r_{i}^{\phi },\theta _{j}^{\phi },\phi _{km}\right) \right) \right) + \nonumber \\&\frac{1}{2}\left( \sum _{\jmath }\mathbf {R}_{\phi \jmath lm}\left| \lambda _{\phi \jmath lm}\right| \left[ \mathbf {L}_{\phi \jmath lm} \left( \widetilde{\mathbf {U}}_{[i,j,k-1]}\left( r_{i}^{\phi },\theta _{j}^{\phi },\phi _{km}\right) - \widetilde{\mathbf {U}}_{[i,j,k]}\left( r_{i}^{\phi },\theta _{j}^{\phi },\phi _{km}\right) \right) \right] \right) \nonumber \\ \mathbf {F}_{\phi ,i,j,kp}^{n*}=~&\frac{1}{2}\left( \mathbf {F}_\phi \left( \widetilde{\mathbf {U}}_{[i,j,k]}\left( r_{i}^{\phi },\theta _{j}^{\phi },\phi _{kp}\right) \right) + \mathbf {F}_\phi \left( \widetilde{\mathbf {U}}_{[i,j,k+1]}\left( r_{i}^{\phi },\theta _{j}^{\phi },\phi _{kp}\right) \right) \right) + \nonumber \\&\frac{1}{2}\left( \sum _{\jmath }\mathbf {R}_{\phi \jmath lp}\left| \lambda _{\phi \jmath lp}\right| \left[ \mathbf {L}_{\phi \jmath lp}\left( \widetilde{\mathbf {U}}_{[i,j,k]}\left( r_{i}^{\phi },\theta _{j}^{\phi },\phi _{kp}\right) - \widetilde{\mathbf {U}}_{[i,j,k+1]}\left( r_{i}^{\phi },\theta _{j}^{\phi },\phi _{kp}\right) \right) \right] \right) \end{aligned}$$

(A.18)

where $\mathbf {U}=\left( \rho , \rho v_{r}, \rho v_{\theta }, \rho v_{\phi }r\sin \theta , e, B_r, B_\theta , B_\phi , \psi \right) ^T$, and the summation in $\jmath$ is carried out for all eigenvalues except $\pm c_h$, with the eigenvalues corresponding to r direction being $\lambda _{r1}=v_r-c_{fr}$, $\lambda _{r2}=v_r-|c_{ar}|$, $\lambda _{r3}=v_r-c_{sr}$, $\lambda _{r4}=v_r$, $\lambda _{r5}=v_r+c_{sr}$, $\lambda _{r6}=v_r+|c_{ar}|$, $\lambda _{r7}=v_r+c_{fr}$, and the corresponding eigenvectors $\mathbf {l}_{r\jmath }$ and $\mathbf {r}_{r\jmath }$, with respect to primitive variables $\mathbf {W}=\left( \rho , v_{r}, v_{\theta }, v_{\phi }, p, B_r, B_\theta , B_\phi , \psi \right) ^T$. That is, seven of the nine eigenvalues of the GLM system are the same as the Powell’s eight-wave form Powell et al. [141], with two different eigenvalues $\pm c_h$. The corresponding eigenvectors for these $\lambda _\jmath$’s are almost the same but presently with the ninth component being zero associated with $\psi$, except that the eigenvectors corresponding to the eigenvalues $\pm c_h$ are complex, however, those eigenvectors associated with the eigenvalues $\pm c_h$ are not used actually as seen from Eqs. (A.17). The eigenvalues on the interfaces are expressed in the same way as in Eq. (A.9) by $\lambda _{r\jmath lm}=\lambda _{r\jmath }\left( \mathbf {U}_{lm}\right) =\lambda _{r\jmath }\left( \frac{1}{2}\left( \widetilde{\mathbf {U}}_{[i-1,j,k]}(r_{im},\theta _j^r,\phi _k)+ \widetilde{\mathbf {U}}_{[i,j,k]}(r_{im},\theta _j^r,\phi _k)\right) \right)$, while the corresponding eigenvectors $\mathbf {L}_{r\jmath lm}$ and $\mathbf {R}_{r\jmath lm}$ corresponding to the conservative variables $\mathbf {U}=\left( \rho , \rho v_{r}, \rho v_{\theta }, \rho v_{\phi }r\sin \theta , e, B_r, B_\theta , B_\phi , \psi \right) ^T$ can be calculated by following Eq. (A.10): $\mathbf {L}_{r\jmath lm}=\left( \mathbf {l}_{r\jmath }\frac{\partial \mathbf {W}}{\partial \mathbf {U}}\right) \left( \mathbf {U}_{lm}\right) , \ \mathbf {R}_{r\jmath lm}=\left( \frac{\partial \mathbf {U}}{\partial \mathbf {W}}\mathbf {r}_{r\jmath }\right) \left( \mathbf {U}_{lm}\right)$. Note that $\frac{\partial \mathbf {W}}{\partial \mathbf {U}}$ and $\frac{\partial \mathbf {U}}{\partial \mathbf {W}}$ are $9\times 9$ matrices, whose left-upper blocks are the same as those in Eqs. (A.11) and (A.12), and right-bottom blocks have only one element 1.

Combining the reconstructed fluxes in Eqs. (A.17) and (A.18), one can obtain the final fluxes $\mathbf {F}_{r,im,j,k}^n$, $\mathbf {F}_{r,ip,j,k}^n$, $\mathbf {F}_{\theta ,i,jm,k}^n$, $\mathbf {F}_{\theta ,i,jp,k}^n$, $\mathbf {F}_{\phi ,i,j,km}^n$ and $\mathbf {F}_{\phi ,i,j,kp}^n$ for the GLM system:

$$\begin{aligned} \mathbf {F}_{r,im,j,k}^n&=\left( { F}_{r1,im,j,k}^{n*},{ F}_{r2,im,j,k}^{n*},{F}_{r3,im,j,k}^{n*},{ F}_{r4,im,j,k}^{n*}, { F}_{r5,im,j,k}^{n*},{ F}_{r,B_r,im,j,k}^n,\right. \nonumber \\&\qquad \left. { F}_{r7,im,j,k}^{n*},{F}_{r8,im,j,k}^{n*}, { F}_{r,\psi ,im,j,k}^n\right) ^T \nonumber \\ \mathbf {F}_{r,ip,j,k}^n&=\left( { F}_{r1,ip,j,k}^{n*},\mathrm{F}_{r2,ip,j,k}^{n*},{ F}_{r3,ip,j,k}^{n*}, {F}_{r4,ip,j,k}^{n*}, {F}_{r5,ip,j,k}^{n*}, {F}_{r,B_r,ip,j,k}^n, { F}_{r7,ip,j,k}^{n*},\right. \nonumber \\&\qquad \left. {F}_{r8,ip,j,k}^{n*}, {F}_{r,\psi ,ip,j,k}^n\right) ^T \nonumber \\ \mathbf {F}_{\theta ,i,jm,k}^n&=\left( {F}_{\theta 1,i,jm,k}^{n*}, {F}_{\theta 2,i,jm,k}^{n*}, {F}_{\theta 3,i,jm,k}^{n*}, {F}_{\theta 4,i,jm,k}^{n*}, {F}_{\theta 5,i,jm,k}^{n*}, { F}_{\theta 6,i,jm,k}^{n*},\right. \nonumber \\&\qquad \left. {F}_{\theta ,B_\theta ,i,jm,k}^n, {F}_{r\theta 8,i,jm,k}^{n*}, { F}_{\theta ,\psi ,i,jm,k}^n\right) ^T \nonumber \\ \mathbf {F}_{\theta ,i,jp,k}^n&=\left( { F}_{\theta 1,i,jp,k}^{n*}, {F}_{\theta 2,i,jp,k}^{n*}, {F}_{\theta 3,i,jp,k}^{n*}, {F}_{\theta 4,i,jp,k}^{n*}, {F}_{\theta 5,i,jp,k}^{n*},\right. \nonumber \\&\qquad \left. { F}_{\theta 6,i,jp,k}^{n*}, {F}_{\theta ,B_\theta ,i,jp,k}^n, {F}_{r\theta 8,i,jp,k}^{n*}, {F}_{\theta ,\psi ,i,jp,k}^n\right) ^T \nonumber \\ \mathbf {F}_{\phi ,i,j,km}^n&=\left( {F}_{\phi 1,i,j,km}^{n*}, {F}_{\phi 2,i,j,km}^{n*}, {F}_{\phi 3,i,j,km}^{n*}, {F}_{\phi 4,i,j,km}^{n*}, {F}_{\phi 5,i,j,km}^{n*}, { F}_{\phi 6,i,j,km}^{n*},\right. \nonumber \\&\qquad \left. {F}_{\phi 7,i,j,km}^{n*}, {F}_{\phi ,B_\phi ,i,j,km}^n, {F}_{\phi ,\psi ,i,j,km}^{n}\right) ^T \nonumber \\ \mathbf {F}_{\phi ,i,j,kp}^n&=\left( {F}_{\phi 1,i,j,kp}^{n*}, {F}_{\phi 2,i,j,kp}^{n*},{ F}_{\phi 3,i,j,kp}^{n*}, {F}_{\phi 4,i,j,kp}^{n*}, { F}_{\phi 5,i,j,kp}^{n*}, { F}_{\phi 6,i,j,kp}^{n*}, {F}_{\phi 7,i,j,kp}^{n*},\right. \nonumber \\&\qquad \left. { F}_{\phi ,B_\phi ,i,j,kp}^n, {F}_{\phi ,\psi ,i,j,kp}^{n}\right) ^T \end{aligned}$$

(A.19)

It should be noted that the interface values $\psi _l$ and $B_{rl}$ from Eqs. (A.15) and (A.16) are used as more as possible for flux calculations. According to Harten et al. [70, 141], if a so-called “entropy fix” is not applied to Roe’s scheme, expansion shocks can be permitted. The entropy fix is applied to the magnetosonic waves to bound those eigenvalues away from zero when the flow is expanding. This is done by replacing $|\lambda _{\jmath l}|$ with $|\lambda _{\jmath l}^*|$ in the discretized flux formulations above. Note that the replacement is implemented for the values of $|\lambda _{\jmath l}|$ $(v_r-c_{fr}, v_r-c_{sr}, v_r+c_{sr}, v_r+c_{fr})$ corresponding to the magnetoacoustic waves only, with $|\lambda _{\jmath l}^*|$ given by

$$\begin{aligned} |\lambda _{\jmath l}^*|= \left\{ \begin{array}{cc} |\lambda _{\jmath l}| &{}|\lambda _{\jmath l}|\ge \frac{\delta \lambda _{\jmath }}{2}\\ \frac{\lambda _{\jmath l}^2}{\delta \lambda _{\jmath }}+\frac{\delta \lambda _{\jmath }}{4} &{} |\lambda _{\jmath l}|<\frac{\delta \lambda _{\jmath }}{2} \end{array} \right. \end{aligned}$$

(A.20)

where

$$ \delta \lambda _{\jmath }=\mathrm{max}\left( 4\left( \lambda _{\jmath R}-\lambda _{\jmath L}\right) ,0\right) $$

Another entropy correction formula suggested by Kermani and Plett [95] runs as follows:

$$\begin{aligned} |\lambda _{\jmath l}^*|= \left\{ \begin{array}{cc} \frac{\lambda _{\jmath l}^2+\epsilon ^2}{2\epsilon } &{} |\lambda _{\jmath l}|<\epsilon \\ |\lambda _{\jmath l}| &{} |\lambda _{\jmath l}|\ge \epsilon \end{array} \right. \end{aligned}$$

(A.21)

where $\epsilon$ has two choices with $\epsilon _1=2 \mathrm{max}\left[ 0,\left( \lambda _{\jmath R}-\lambda _{\jmath L}\right) \right]$ and $\epsilon _2=4 \mathrm{max} \left[ 0,\left( \lambda _{\jmath l}-\lambda _{\jmath L}\right) ,\left( \lambda _{\jmath l}-\lambda _{\jmath R}\right) \right]$. All these entropy fixing methods in Eqs. (A.20) and (A.21) are employed in our code.

Up to now, we are ready to solve Eq. (A.1) with the fluxes defined by Eq. (A.19). Finally, supplemented with the initial condition $\bar{\psi }_{i,j,k}^{n*}$ given by the output of the most recent step, one can solve the initial value problem obtained by the last equation of (A.14) without the $\nabla \cdot \mathbf {B}$ term

$$\begin{aligned} \frac{\partial \psi }{\partial t}=-\frac{c_h^2}{c_p^2}\psi \end{aligned}$$

(A.22)

by integrating exactly for a time increment $\varDelta t_n$, yielding

$$\begin{aligned} {\psi }_{i,j,k}^{n+1}={\exp }(-\varDelta t_n\frac{c_h^2}{c_p^2})\bar{\psi }_{i,j,k}^{n*} \end{aligned}$$

(A.23)

The constant $c^2_p$ has the dimension of length squared over time and thus can be regarded as a diffusion coefficient. Dedner et al. [35] chose $c_p=\sqrt{0.18c_h}$, depending on their numerical experiment. However, this definition is incomplete, since $c_p^2/c_h$ has the dimension of length and thus it is not a dimensionless quantity. Thus, another choice defined by the dimensionless parameter $\alpha =\frac{\varDelta hc_h}{c_p^2}\in [0,1]$ is suggested [121, 122, 170], and can be regarded as the ratio of the diffusive and advective time scales, i.e., $\alpha =\frac{\varDelta t_d}{\varDelta t_a}$, where $\varDelta t_d=\varDelta h^2/c_p^2$ and $\varDelta t_a=\varDelta h/c_h$. $\varDelta h$ is the minimum grid spacing $\left( \displaystyle \min _{i,j,k}\left( \varDelta r(i), r(i)\varDelta \theta (j), r(i)\sin \theta (j)\varDelta \phi (k)\right) \right)$. The third choice $c_p=\sqrt{-\frac{\varDelta t}{\ln c_d}c_h^2}$ with $c_d \in (0,1)$ has been used [35, 69, 88, 202], where $c_h=\frac{\text {CFL}}{\varDelta t}\varDelta h$. If one chooses $c_d=\exp (-\alpha \mathrm{CFL})$, then the later two choices becomes the same. In STEP-MHD code $\alpha =0.1$, $\psi$ is set to be zero initially and $\psi$ is fixed at the inner boundary and outer boundary.

3.1.3 HLL Type Flux

$$\begin{aligned} \begin{aligned} \mathbf {F}_{r, im,j,k}^n=&-\frac{a_{im,j^r,k}^+a_{im,j^r,k}^-}{a_{im,j^r,k}^+-a_{im,j^r,k}^-} \left( \widetilde{\mathbf {U}}_{[i-1,j,k]}(r_{im},\theta _j^r,\phi _k)-\widetilde{\mathbf {U}}_{[i,j,k]}(r_{im},\theta _j^r,\phi _k)\right) \\&+\frac{a_{im,j^r,k}^+\mathbf {F}_r\left( \widetilde{\mathbf {U}}_{[i-1,j,k]}(r_{im},\theta _j^r,\phi _k),\widetilde{\mathbf {B}}_{[i-1,j,k]} (r_{im},\theta _j^r,\phi _k)\right) }{a_{im,j^r,k}^+-a_{im,j^r,k}^-} \\&-\frac{a_{im,j^r,k}^-\mathbf {F}_r \left( \widetilde{\mathbf {U}}_{[i,j,k]}(r_{im},\theta _j^r,\phi _k),\widetilde{\mathbf {B}}_{[i,j,k]} (r_{im},\theta _j^r,\phi _k)\right) }{a_{im,j^r,k}^+-a_{im,j^r,k}^-} \end{aligned} \end{aligned}$$

$$\begin{aligned} \mathbf {F}_{\theta , i,jm,k}^n=&-\frac{b_{i^\theta ,jm,k}^+b_{i^\theta ,jm,k}^-}{b_{i^\theta ,jm,k}^+-b_{i^\theta ,jm,k}^-} \left( \widetilde{\mathbf {U}}_{[i,j-1,k]}(r_i^\theta ,\theta _{jm},\phi _k)-\widetilde{\mathbf {U}}_{[i,j,k]}(r_i^\theta ,\theta _{jm},\phi _k)\right) \\&+\frac{b_{i^\theta ,jm,k}^+\mathbf {F}_\theta \left( \widetilde{\mathbf {U}}_{[i,j-1,k]}(r_i^\theta ,\theta _{jm},\phi _k),\widetilde{\mathbf {B}}_{[i,j-1,k]} (r_i^\theta ,\theta _{jm},\phi _k)\right) }{b_{i^\theta ,jm,k}^+-b_{i^\theta ,jm,k}^-} \\&-\frac{b_{i^\theta ,jm,k}^-\mathbf {F}_\theta \left( \widetilde{\mathbf {U}}_{[i,j,k]}(r_i^\theta ,\theta _{jm},\phi _k),\widetilde{\mathbf {B}}_{[i,j,k]} (r_i^\theta ,\theta _{jm},\phi _k)\right) }{b_{i^\theta ,jm,k}^+-b_{i^\theta ,jm,k}^-} \end{aligned}$$

$$\begin{aligned} \begin{aligned} \mathbf {F}_{\phi , i,j,km}^n =&-\frac{c_{i^\phi ,j^\phi ,km}^+c_{i^\phi ,j^\phi ,km}^-}{c_{i^\phi ,j^\phi ,km}^+-c_{i^\phi ,j^\phi ,km}^-} \left( \widetilde{\mathbf {U}}_{[i,j,k-1]}(r_i^\phi ,\theta _j^\phi ,\phi _{km})-\widetilde{\mathbf {U}}_{[i,j,k]}(r_i^\phi ,\theta _j^\phi ,\phi _{km})\right) \\&+\frac{c_{i^\phi ,j^\phi ,km}^+\mathbf {F}_\phi \left( \widetilde{\mathbf {U}}_{[i,j,k-1]}(r_i^\phi ,\theta _j^\phi ,\phi _{km}),\widetilde{\mathbf {B}}_{[i,j,k-1]} (r_i^\phi ,\theta _j^\phi ,\phi _{km})\right) }{c_{i^\phi ,j^\phi ,km}^+-c_{i^\phi ,j^\phi ,km}^-} \\&-\frac{c_{i^\phi ,j^\phi ,km}^-\mathbf {F}_\phi \left( \widetilde{\mathbf {U}}_{[i,j,k]}(r_i^\phi ,\theta _j^\phi ,\phi _{km}),\widetilde{\mathbf {B}}_{[i,j,k]} (r_i^\phi ,\theta _j^\phi ,\phi _{km})\right) }{c_{i^\phi ,j^\phi ,km}^+-c_{i^\phi ,j^\phi ,km}^-} \end{aligned} \end{aligned}$$

Parameters $a_{im,j^r,k}^+, a_{im,j^r,k}^-, b_{i^\theta ,jm,k}^+, b_{i^\theta ,jm,k}^-, c_{i^\phi ,j^\phi ,km}^+, c_{i^\phi ,j^\phi ,km}^-$ involved in the flux functions are defined by the maximum and minimum speed on the centers of six faces

$$\begin{aligned} \begin{aligned}&a_{im,j^r,k}^+=\mathrm{max}\left\{ (\widetilde{v}_r+\widetilde{c}_{fr})_{[i-1,j,k]}(r_{im},\theta _j^r,\phi _k), (\widetilde{v}_r+\widetilde{c}_{fr})_{[i,j,k]}(r_{im},\theta _j^r,\phi _k),0\right\} \\&a_{im,j^r,k}^-=\mathrm{min}\left\{ (\widetilde{v}_r-\widetilde{c}_{fr})_{[i-1,j,k]}(r_{im},\theta _j^r,\phi _k), (\widetilde{v}_r-\widetilde{c}_{fr})_{[i,j,k]}(r_{im},\theta _j^r,\phi _k),0\right\} \\&b_{i^\theta ,jm,k}^+=\mathrm{max}\left\{ (\widetilde{v}_\theta +\widetilde{c}_{f\theta })_{[i,j-1,k]}(r_i^\theta ,\theta _{jm},\phi _k), (\widetilde{v}_\theta +\widetilde{c}_{f\theta })_{[i,j,k]}(r_i^\theta ,\theta _{jm},\phi _k),0\right\} \\&b_{i^\theta ,jm,k}^-=\mathrm{min}\left\{ (\widetilde{v}_\theta -\widetilde{c}_{f\theta })_{[i,j-1,k]}(r_i^\theta ,\theta _{jm},\phi _k), (\widetilde{v}_\theta -\widetilde{c}_{f\theta })_{[i,j,k]}(r_i^\theta ,\theta _{jm},\phi _k),0\right\} \\&c_{i^\phi ,j^\phi ,km}^+=\mathrm{max}\left\{ (\widetilde{v}_\phi +\widetilde{c}_{f\phi })_{[i,j,k-1]}(r_i^\phi ,\theta _j^\phi ,\phi _{km}), (\widetilde{v}_\phi +\widetilde{c}_{f\phi })_{[i,j,k]}(r_i^\phi ,\theta _j^\phi ,\phi _{km}),0\right\} \\&c_{i^\phi ,j^\phi ,km}^-=\mathrm{min}\left\{ (\widetilde{v}_\phi -\widetilde{c}_{f\phi })_{[i,j,k-1]}(r_i^\phi ,\theta _j^\phi ,\phi _{km}), (\widetilde{v}_\phi -\widetilde{c}_{f\phi })_{[i,j,k]}(r_i^\phi ,\theta _j^\phi ,\phi _{km}),0\right\} \\ \end{aligned} \end{aligned}$$

(A.24)

3.1.4 CT Formulation

The CT method advances the variables $\mathbf {V}=\left( \rho , \rho v_{r}, \rho v_{\theta }, \rho v_{\phi }r\sin \theta , e\right) ^T$ at the centroid point $(\overline{r}_i, \overline{\theta }_j, \phi _k)$ and $\mathbf {B}=\left( B_r, B_\theta , B_\phi \right)$ at the face center. In the derivation of numerical scheme with CT, the governing MHD equations are divided into two coupled subsystems of equations: a fluid part Eq. (3.14) and a magnetic induction part Eq. (3.15). The numerical formulation of fluid part Eq. (3.14) for the vector $\mathbf {V}=\left( \rho , \rho v_{r}, \rho v_{\theta }, \rho v_{\phi }r\sin \theta , e\right) ^T$ can be derived from the first five arguments’s equations for $\mathbf {U}$ in Eq. (3.2), where $\mathbf {G}_r, \mathbf {G}_\theta , \mathbf {G}_\phi , \mathbf {Q}$ are the first five arguments in $\mathbf {F}_r, \mathbf {F}_\theta , \mathbf {F}_\phi$, and $\mathbf {S}_1+\mathbf {S}_2$. Hence, the discretized formulation (3.20) for $\mathbf {V}$ follows from those of the first five arguments associated with Eq. (A.1).

Following Eq. (3.24) in Ziegler [219], by averaging Eqs. (3.15) over facial areas of a cell with geometrical center (i, j, k), one can obtain the semi-discrete form Eq. (3.21) separately as follows

$$\begin{aligned} \begin{aligned}&\frac{dB_{r,im,j^r,k}}{dt}=-\frac{\sin \theta _{jp}E_{\phi , im,jp,k}^*- \sin \theta _{jm}E_{\phi , im,jm,k}^*}{r_{im}(\cos \theta _{jm}-\cos \theta _{jp})}\\&\qquad \qquad \qquad \quad + \frac{\varDelta \theta (j)}{r_{im}(\cos \theta _{jm}-\cos \theta _{jp})} \frac{E_{\theta , im,j,kp}^*-E_{\theta , im,j,km}^*}{\varDelta \phi (k)}\\&\frac{dB_{\theta ,i^\theta ,jm,k}}{dt}=-\frac{2}{ (r_{im}+r_{ip})\sin \theta _{jm}}\frac{E_{r, i,jm,kp}^*-E_{r, i,jm,km}^*}{\varDelta \phi (k)}\\&\qquad \qquad \qquad \quad + \frac{2\left( r_{ip}E_{\phi , ip,jm,k}^*-r_{im}E_{\phi , im,jm,k}^*\right) }{r_{ip}^2-r_{im}^2}\\&\frac{dB_{\phi ,i^\phi ,j^\phi ,km}}{dt}=-\frac{2\left( r_{ip}E_{\theta , ip,j,km}^*-r_{im}E_{\theta , im,j,km}^*\right) }{r_{ip}^2-r_{im}^2}\\&\qquad \qquad \qquad \qquad + \frac{2}{r_{im}+r_{ip}}\frac{E_{r, i,jp,km}^*-E_{r, i, jm, km}^*}{\varDelta \theta (j)} \end{aligned} \end{aligned}$$

Here, $E_{r, i,jm,km}^*, E_{\theta , im, j, km}^*, E_{\phi , im, jm, k}^*$ are the reconstructed values of the electric fields at cell edge $(r_i, \theta _{jm}, \phi _{km})$, $(r_{im}, \theta _j, \phi _{km})$, $(r_{im},\theta _{jm},\phi _k)$, defined respectively by

$$\begin{aligned} \begin{aligned} E_{r, i,jm,km}^*&=-\frac{b_{i,jm,km}^-b_{i,jm,km}^+}{b_{i,jm,km}^+-b_{i,jm,km}^-}\left( \widetilde{B}_{\phi , i,jm+,km}-\widetilde{B}_{\phi , i,jm-,km}\right) +\frac{c_{i,jm,km}^-c_{i,jm,km}^+}{c_{i,jm,km}^+-c_{i,jm,km}^-}\\&\times \left( \widetilde{B}_{\theta , i,jm,km+}-\widetilde{B}_{\theta , i,jm,km-}\right) -\frac{1}{(b_{i,jm,km}^+-b_{i,jm,km}^-)(c_{i,jm,km}^+-c_{i,jm,km}^-)}\\&\times \left( b_{i,jm,km}^+c_{i,jm,km}^-\widetilde{E}_{r,i,jm-,km+}-b_{i,jm,km}^-c_{i,jm,km}^-\widetilde{E}_{r,i,jm+,km+}\right. \\&\left. -\ b_{i,jm,km}^+c_{i,jm,km}^+\widetilde{E}_{r,i,jm-,km-} +b_{i,jm,km}^-c_{i,jm,km}^+\widetilde{E}_{r,i,jm+,km-}\right) \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} E_{\theta , im,j,km}^*&=\frac{a_{im,j,km}^-a_{im,j,km}^+}{a_{im,j,km}^+-a_{im,j,km}^-}\left( \widetilde{B}_{\phi , im+,j,km}-\widetilde{B}_{\phi , im-,j,km}\right) -\frac{c_{im,j,km}^-c_{im,j,km}^+}{c_{im,j,km}^+-c_{im,j,km}^-}\\&\times \left( \widetilde{B}_{r, im,j,km+}-\widetilde{B}_{r, im,j,km-}\right) -\frac{1}{(a_{im,j,km}^+-a_{im,j,km}^-)(c_{im,j,km}^+-c_{im,j,km}^-)}\\&\times \left( a_{im,j,km}^+c_{im,j,km}^-\widetilde{E}_{\theta , im-,j,km+}-a_{im,j,km}^-c_{im,j,km}^-\widetilde{E}_{\theta , im+,j,km+}\right. \\&\left. -\ a_{im,j,km}^+c_{im,j,km}^+\widetilde{E}_{\theta , im-,j,km-} +a_{im,j,km}^-c_{im,j,km}^+\widetilde{E}_{\theta , im+,j,km-}\right) \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} E_{\phi , im,jm,k}^*&=-\frac{a_{im,jm,k}^-a_{im,jm,k}^+}{a_{im,jm,k}^+-a_{im,jm,k}^-}\left( \widetilde{B}_{\theta , im+,jm,k}-\widetilde{B}_{\theta , im-,jm,k}\right) +\frac{b_{im,jm,k}^-b_{im,jm,k}^+}{b_{im,jm,k}^+-b_{im,jm,k}^-}\\&\times \left( \widetilde{B}_{r, im,jm+,k}-\widetilde{B}_{r, im,jm-,k}\right) -\frac{1}{(b_{im,jm,k}^+-b_{im,jm,k}^-)(a_{im,jm,k}^+-a_{im,jm,k}^-)}\\&\times \left( a_{im,jm,k}^+b_{im,jm,k}^-\widetilde{E}_{\phi , im-,jm+,k}-a_{im,jm,k}^-b_{im,jm,k}^-\widetilde{E}_{\phi , im+,jm+,k}\right. \\&\left. -\ a_{im,jm,k}^+b_{im,jm,k}^+\widetilde{E}_{\phi , im-,jm-,k} +a_{im,jm,k}^-b_{im,jm,k}^+\widetilde{E}_{\phi , im+,jm-,k}\right) \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned}&\widetilde{E}_{r,i,jm+,km+}=-\left( \widetilde{v}_\theta \widetilde{B}_\phi -\widetilde{v}_\phi \widetilde{B}_\theta \right) _{[i,j,k]}(r_i,\theta _{jm},\phi _{km}) \\&\widetilde{E}_{r,i,jm-,km+}=-\left( \widetilde{v}_\theta \widetilde{B}_\phi -\widetilde{v}_\phi \widetilde{B}_\theta \right) _{[i,j-1,k]}(r_i,\theta _{jm},\phi _{km}) \\&\widetilde{E}_{r, i, jm+, km-}=-\left( \widetilde{v}_\theta \widetilde{B}_\phi -\widetilde{v}_\phi \widetilde{B}_\theta \right) _{[i,j,k-1]}(r_i,\theta _{jm},\phi _{km}) \\&\widetilde{E}_{r, i, jm-, km-}=-\left( \widetilde{v}_\theta \widetilde{B}_\phi -\widetilde{v}_\phi \widetilde{B}_\theta \right) _{[i,j-1,k-1]}(r_i,\theta _{jm},\phi _{km}) \\&\widetilde{B}_{r[i,j,k]}(r_{im},\theta _{jm},\phi _k)=\widetilde{B}_{1r[i,j,k]}(r_{im},\theta _{jm},\phi _k)+\widetilde{B}_{0r[i,j,k]}(r_{im},\theta _{jm},\phi _k) \\&\widetilde{B}_{r, im, j, km+}=\widetilde{B}_{r[i,j,k]}(r_{im},\theta _j,\phi _{km}),\ \widetilde{B}_{r, im, j, km-}=\widetilde{B}_{r[i,j,k-1]}(r_{im},\theta _j,\phi _{km}) \\&\widetilde{B}_{r, im, jm+, k}=\widetilde{B}_{r[i,j,k]}(r_{im},\theta _{jm},\phi _k),\ \widetilde{B}_{r, im, jm-, k}=\widetilde{B}_{r[i,j-1,k]}(r_{im},\theta _{jm},\phi _k) \end{aligned} \end{aligned}$$

The formulation in $\theta$ and $\phi$ directions is similar to that in the r direction, which is omitted here. It is noted that, $(\widetilde{v}_r, \widetilde{v}_\theta , \widetilde{v}_\phi )$ is defined by Eq. (A.8) as above, while $(\widetilde{B}_r, \widetilde{B}_\theta , \widetilde{B}_\phi )$ is decided by the second-order magnetic field construction proposed in Sect. 3.2.4 by Ziegler [219]

$$\begin{aligned} \begin{aligned} \widetilde{B}_{r[i,j,k]}(\mathbf {r})&=\frac{1}{2}\left( B_{r,ip,j^r,k}+B_{r,im,j^r,k}\right) +\left( B_{r,ip,j^r,k}-B_{r,im,j^r,k}\right) \frac{r-r_i}{\varDelta r(i)} \\&+\left[ \widetilde{(\partial _\theta B_r)}_{ip,j,k}\frac{r-r_{im}}{\varDelta r(i)}+\widetilde{(\partial _\theta B_r)}_{im,j,k}\frac{r_{ip}-r}{\varDelta r(i)}\right] (\theta -\theta _j^r) \\&+\left[ \widetilde{(\partial _\phi B_r)}_{ip,j,k}\frac{r-r_{im}}{\varDelta r(i)}+\widetilde{(\partial _\phi B_r)}_{im,j,k}\frac{r_{ip}-r}{\varDelta r(i)}\right] (\phi -\phi _k) \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} \widetilde{B}_{\theta [i,j,k]}(\mathbf {r})&=\frac{1}{2}\left( B_{\theta ,i^\theta ,jp,k}+B_{\theta ,i^\theta ,jm,k}\right) +\left( B_{\theta ,i^\theta ,jp,k}-B_{\theta ,i^\theta ,jm,k}\right) \frac{\theta -\theta _j}{\varDelta \theta (j)} \\&+\left[ \widetilde{(\partial _r B_\theta )}_{i,jp,k}\frac{\theta -\theta _{jm}}{\varDelta \theta (j)}+\widetilde{(\partial _r B_\theta )}_{i,jm,k}\frac{\theta _{jp}-\theta }{\varDelta \theta (j)}\right] (r-r_i^\theta ) \\&+\left[ \widetilde{(\partial _\phi B_\theta )}_{i,jp,k}\frac{\theta -\theta _{jm}}{\varDelta \theta (j)}+\widetilde{(\partial _\phi B_\theta )}_{i,jm,k}\frac{\theta _{jp}-\theta }{\varDelta \theta (j)}\right] (\phi -\phi _k) \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} \widetilde{B}_{\phi [i,j,k]}(\mathbf {r})&=\frac{1}{2}\left( B_{\phi ,i^\phi ,j^\phi ,kp}+B_{\phi ,i^\phi ,j^\phi ,km}\right) +\left( B_{\phi ,i^\phi ,j^\phi ,kp}-B_{\phi ,i^\phi ,j^\phi ,km}\right) \frac{\phi -\phi _k}{\varDelta \phi (k)} \\&+\left[ \widetilde{(\partial _r B_\phi )}_{i,j,kp}\frac{\phi -\phi _{km}}{\varDelta \phi (k)} +\widetilde{(\partial _r B_\phi )}_{i,j,km}\frac{\phi _{kp}-\phi }{\varDelta \phi (k)}\right] (r-r_i^\phi ) \\&+\left[ \widetilde{(\partial _\theta B_\phi )}_{i,j,kp}\frac{\phi -\phi _{km}}{\varDelta \phi (k)}+\widetilde{(\partial _\theta B_\phi )}_{i,j,km}\frac{\phi _{kp}-\phi }{\varDelta \phi (k)}\right] (\theta -\theta _j^\phi ) \end{aligned} \end{aligned}$$

where $\widetilde{(\partial _\theta B_r)}_{i\pm \frac{1}{2},j,k}$ and other cross derivative terms are slope-limited approximations to the exact derivatives at cell faces. For example, using the minmod limiter,

$$\begin{aligned} \begin{aligned}&\widetilde{(\partial _\theta B_r)}_{i\pm \frac{1}{2},j,k}=\mathrm{minmod}\left\{ \frac{B_{r,i\pm \frac{1}{2},(j+1)^r,k}-B_{r,i\pm \frac{1}{2},j^r,k}}{\theta _{j+1}^r-\theta _j^r}, \frac{B_{r,i\pm \frac{1}{2},j^r,k}-B_{r,i\pm \frac{1}{2},(j-1)^r,k}}{\theta _j^r-\theta _{j-1}^r}\right\} \\&\widetilde{(\partial _\phi B_r)}_{i\pm \frac{1}{2},j,k}=\mathrm{minmod}\left\{ \frac{B_{r,i\pm \frac{1}{2},j^r,k+1}-B_{r,i\pm \frac{1}{2},j^r,k}}{\phi _{k+1}-\phi _k}, \frac{B_{r,i\pm \frac{1}{2},j^r,k}-B_{r,i\pm \frac{1}{2},j^r,k-1}}{\phi _{k}-\phi _{k-1}} \right\} \\&\widetilde{(\partial _r B_\theta )}_{i,j\pm \frac{1}{2},k}=\mathrm{minmod}\left\{ \frac{B_{\theta ,(i+1)^\theta ,j\pm \frac{1}{2},k}-B_{\theta ,i^\theta ,j\pm \frac{1}{2},k}}{r_{i+1}^\theta -r_i^\theta }, \frac{B_{\theta ,i^\theta ,j\pm \frac{1}{2},k}-B_{\theta ,(i-1)^\theta ,j\pm \frac{1}{2},k}}{r_i^\theta -r_{i-1}^\theta } \right\} \\&\widetilde{(\partial _\phi B_\theta )}_{i,j\pm \frac{1}{2},k}=\mathrm{minmod}\left\{ \frac{B_{\theta ,i^\theta ,j\pm \frac{1}{2},k+1}-B_{\theta ,i^\theta ,j\pm \frac{1}{2},k}}{\phi _{k+1}-\phi _k}, \frac{B_{\theta ,i^\theta ,j\pm \frac{1}{2},k}-B_{\theta ,i^\theta ,j\pm \frac{1}{2},k-1}}{\phi _{k}-\phi _{k-1}} \right\} \\&\widetilde{(\partial _r B_\phi )}_{i,j,k\pm \frac{1}{2}}=\mathrm{minmod}\left\{ \frac{B_{\phi ,(i+1)^\phi ,j^\phi ,k\pm \frac{1}{2}}-B_{\phi ,i^\phi ,j^\phi ,k\pm \frac{1}{2}}}{r_{i+1}^\phi -r_i^\phi }, \frac{B_{\phi ,i^\phi ,j^\phi ,k\pm \frac{1}{2}}-B_{\phi ,(i-1)^\phi ,j^\phi ,k\pm \frac{1}{2}}}{r_i^\phi -r_{i-1}^\phi } \right\} \\&\widetilde{(\partial _\theta B_\phi )}_{i,j,k\pm \frac{1}{2}}=\mathrm{minmod}\left\{ \frac{B_{\phi ,i^\phi ,(j+1)^\phi ,k\pm \frac{1}{2}}-B_{\phi ,i^\phi ,j^\phi ,k\pm \frac{1}{2}}}{\theta _{j+1}^\phi -\theta _j^\phi }, \frac{B_{\phi ,i^\phi ,j^\phi ,k\pm \frac{1}{2}}-B_{\phi ,i^\phi ,(j-1)^\phi ,k\pm \frac{1}{2}}}{\theta _j^\phi -\theta _{j-1}^\phi } \right\} \end{aligned} \end{aligned}$$

As claimed by Ziegler [219], the above magnetic field reconstruction meets the following three features: (1) magnetic flux conservation

$$\begin{aligned} \begin{aligned}&\langle B_r \rangle _{i\pm \frac{1}{2},j,k}^r=\frac{1}{S_{r,i\pm \frac{1}{2},j,k}}\int _{ S_{r,i\pm \frac{1}{2},j,k}}\widetilde{B}_{r[i,j,k]}(i\pm \frac{1}{2},\theta ', \phi ') dS' \approx B_r(r_{i\pm \frac{1}{2}},\theta _j^r,\phi _k)\\ {}&\qquad \quad \quad \quad \,\,=B_{r,i\pm \frac{1}{2},j^r,k} \\&\langle B_\theta \rangle _{i,j\pm \frac{1}{2},k}^\theta =\frac{1}{ S_{\theta ,i,j\pm \frac{1}{2},k}}\int _{ S_{\theta ,i,j\pm \frac{1}{2},k}} \widetilde{B}_{\theta [i,j,k]} (r',j\pm \frac{1}{2}, \phi ')dS' \approx B_\theta (r_i^\theta ,\theta _{j\pm \frac{1}{2}},\phi _k)\\ {}&\qquad \quad \quad \quad \,\,=B_{\theta ,i^\theta ,j\pm \frac{1}{2},k} \\&\langle B_\phi \rangle _{i,j,k\pm \frac{1}{2}}^\phi =\frac{1}{ S_{\phi ,i,j,k\pm \frac{1}{2}}}\int _{ S_{\phi ,i,j,k\pm \frac{1}{2}}} \widetilde{B}_{\phi [i,j,k]}(r', \theta ', k\pm \frac{1}{2})dS' \approx B_\phi (r_i^\phi ,\theta _j^\phi ,\phi _{k\pm \frac{1}{2}})\\ {}&\qquad \quad \quad \quad \,\,=B_{\phi ,i^\phi ,j^\phi ,k\pm \frac{1}{2}} \end{aligned} \end{aligned}$$

(2) continuity of $\widetilde{B}_r\ (\widetilde{B}_\theta ,\widetilde{B}_\phi )$ in the $r\ (\theta , \phi )$-coordinate, i.e., $\widetilde{B}_{r[i, j, k]}(r_{im},\theta , \phi )=\widetilde{B}_{r[i-1,j,k]}(r_{im}, \theta , \phi ), \widetilde{B}_{\theta [i,j, k]}(r, \theta _{jm}, \phi )=\widetilde{B}_{\theta [i,j-1,k]}(r, \theta _{jm}, \phi )$, and $\widetilde{B}_{\phi [i, j, k]}(r, \theta , \phi _{km})=\widetilde{B}_{\phi [i,j, k-1]}(r, \theta , \phi _{km})$. Due to the continuity property of magnetic field components the longitudinal derivatives $\partial _r B_r, \partial _\theta B_\theta , \partial _\phi B_\phi$ can be approximated in a straightforward manner by using the facial magnetic field values without any slope limitation needed.

(3) Solenoidality of $\widetilde{ \mathbf {B} }_{[i,j k]}$ on the discretization scale, i.e., $\left( \overline{ \nabla \cdot \widetilde{ \mathbf {B} }_{ [i,j k]} } \right) _{i,j,k}=0$. It is noted that the application of Gauss’ law to $\left( \overline{\nabla \cdot \widetilde{\mathbf {B}}_{[i,j k]}}\right) _{i,j,k}=0$ yields $\left[ \varDelta S_r \langle \widetilde{B}_{r[i,j,k]}\rangle ^r\right] _{im}^{ip} +\left[ \varDelta S_\theta \langle \widetilde{B}_{\theta [i,j,k]}\rangle ^\theta \right] _{jm}^{jp}+\left[ \varDelta S_\phi \langle \widetilde{B}_{\phi [i,j,k]}\rangle ^\phi \right] _{km}^{kp}=0$, which follows from the magnetic flux conservation and continuity stated here. Thus, one can infer that $\left( \overline{ \nabla \cdot \widetilde{ \mathbf {B} }_{ [i,j k]} } \right) _{i,j,k}=\left( \overline{ \nabla \cdot { \mathbf {B} } } \right) _{i,j,k}$ keeps its initial zero value on the discretization scale in the computation, since the initial magnetic field is cellwisely divergence-free and $\frac{d}{dt}\left( \overline{ \nabla \cdot { \mathbf {B} } } \right) _{i,j,k}=0$ by the present construction of spatial discretization. However, this method can not claim the more strict condition of pointwise solenoidality.

Similar to the above formulation given by Eq. (A.24), $a_{im,j,k}^+$, $a_{im,j,k}^-$, $b_{i,jm,k}^+$, $b_{i,jm,k}^-$, $c_{i,j,km}^+$, and $c_{i,j,km}^-$ are defined by the maximum and minimum speeds on the geometrical center of six faces

$$\begin{aligned} \begin{aligned}&a_{im,j,k}^+=\mathrm{max}\left\{ (\widetilde{v}_r+\widetilde{c}_{fr})_{[i-1,j,k]}(r_{im},\theta _j,\phi _k), \ (\widetilde{v}_r+\widetilde{c}_{fr})_{[i,j,k]}(r_{im},\theta _j,\phi _k),0 \right\} \\&a_{im,j,k}^-=\mathrm{min}\left\{ (\widetilde{v}_r-\widetilde{c}_{fr})_{[i-1,j,k]}(r_{im},\theta _j,\phi _k), \ (\widetilde{v}_r-\widetilde{c}_{fr})_{[i,j,k]}(r_{im},\theta _j,\phi _k),0 \right\} \\&b_{i,jm,k}^+=\mathrm{max}\left\{ (\widetilde{v}_\theta +\widetilde{c}_{f\theta })_{[i,j-1,k]}(r_i,\theta _{jm},\phi _k),\ (\widetilde{v}_\theta +\widetilde{c}_{f\theta })_{[i,j,k]}(r_i,\theta _{jm},\phi _k),0 \right\} \\&b_{i,jm,k}^-=\mathrm{min}\left\{ (\widetilde{v}_\theta -\widetilde{c}_{f\theta })_{[i,j-1,k]}(r_i,\theta _{jm},\phi _k), \ (\widetilde{v}_\theta -\widetilde{c}_{f\theta })_{[i,j,k]}(r_i,\theta _{jm},\phi _k),0 \right\} \\&c_{i,j,km}^+=\mathrm{max}\left\{ (\widetilde{v}_\phi +\widetilde{c}_{f\phi })_{[i,j,k-1]}(r_i,\theta _j,\phi _{km}), \ (\widetilde{v}_\phi +\widetilde{c}_{f\phi })_{[i,j,k]}(r_i,\theta _j,\phi _{km}),0 \right\} \\&c_{i,j,km}^-=\mathrm{min}\left\{ (\widetilde{v}_\phi -\widetilde{c}_{f\phi })_{[i,j,k-1]}(r_i,\theta _j,\phi _{km}), \ (\widetilde{v}_\phi -\widetilde{c}_{f\phi })_{[i,j,k]}(r_i,\theta _j,\phi _{km}),0 \right\} \end{aligned} \end{aligned}$$

In the same way, $a_{im,j-1,k}^+, a_{im,j-1,k}^-, a_{im,j,k-1}^+, a_{im,j,k-1}^-, b_{i-1,jm,k}^+, b_{i-1,jm,k}^-, b_{i,jm,k-1}^+, b_{i,jm,k-1}^-$, $c_{i-1,j,km}^+$, $c_{i-1,j,km}^-, c_{i,j-1,km}^+, c_{i,j-1,km}^-$ can be given. By using these maximum and minimum speeds on the geometrical center of six faces, the maximum and minimum speeds in the edges involved in $E_{r, i,jm,km}^*, E_{\theta , im,j,km}^*, E_{\phi , im,jm,k}^*$ can be defined as follows

$$\begin{aligned} \begin{aligned}&a_{im,jm,k}^+=\mathrm{max}\left\{ a_{im,j,k}^+, a_{im,j-1,k}^+\right\} , \ a_{im,jm,k}^-=\mathrm{min}\left\{ a_{im,j,k}^-, a_{im,j-1,k}^- \right\} \\&a_{im,j,km}^+=\mathrm{max}\left\{ a_{im,j,k}^+, a_{im,j,k-1}^+ \right\} , \ a_{im,j,km}^-=\mathrm{min}\left\{ a_{im,j,k}^-, a_{im,j,k-1}^- \right\} \\&b_{im,jm,k}^+=\mathrm{max}\left\{ b_{i,jm,k}^+, b_{i-1,jm,k}^+\right\} , \ b_{im,jm,k}^-=\mathrm{min}\left\{ b_{i,jm,k}^-, b_{i-1,jm,k}^-\right\} \\&b_{i,jm,km}^+=\mathrm{max}\left\{ b_{i,jm,k}^+, b_{i,jm,k-1}^+\right\} , \ b_{i,jm,km}^-=\mathrm{min}\left\{ b_{i,jm,k}^-, b_{i,jm,k-1}^- \right\} \\&c_{im,j,km}^+=\mathrm{max}\left\{ c_{i,j,km}^+, c_{i-1,j,km}^+\right\} , \ c_{im,j,km}^-=\mathrm{min}\left\{ c_{i,j,km}^-, c_{i-1,j,km}^- \right\} \\&c_{i,jm,km}^+=\mathrm{max}\left\{ c_{i,j,km}^+, c_{i,j-1,km}^+\right\} , \ c_{i,jm,km}^-=\mathrm{min}\left\{ c_{i,j,km}^-, c_{i,j-1,km}^-\right\} \end{aligned} \end{aligned}$$

3.1.5 MPP Flux-corrected Method

Positivity-preserving stands for the capability to preserve the positivity of the determinative physical constraints for the solution to be positive within physically-permitted bounds. In ideal MHD, density $\rho$ and pressure p should be positive, but one can encounter the negativeness of density and pressure when simulating problems with low plasma $\beta$, high Mach number, or much large magnetic energy compared with internal energy. Even with divergence-free methods, such positivity property is not always satisfied by approximated solutions. When negative density or/and pressure occur, numerical instability may develop so as to cause the breakdown of numerical simulations. Therefore, guaranteeing such determinative properties is a must under consideration in a numerical design.

The positivity-preserving approach introduced in Sect. 3.5.4 is usually the so-called “dual energy” formulation. According to some switching conditions based on the ratio $\epsilon /e$ between the internal energy $\epsilon$ and the total energy e or the magnitude of plasma $\beta$ (usually in small internal energy and/or low plasma $\beta$ regions), the entropy equation or pressure equation or the internal energy density equation is used to replace the conservative energy equation [10, 17, 20, 50, 57, 58, 106, 153, 176, 184]. Evidently, this kind of dual energy method is sensitive to the “switching conditions” besides its computational overhead. For some comments about the use of conservative versus nonconservative forms of the energy equations, the reader may refer to Sect. 2.7 of Usmanov et al. [180].

Recent years have witnessed some progresses in constructing conservative positivity-preserving schemes for hyperbolic conservation laws. Specifically, some efforts work on the positivity-preserving in Riemann solvers or the evolved solution polynomials in a finite volume or discontinuous Galerkin (DG) method for both hydrodynamic and MHD flows [19, 22, 42, 60, 86, 105, 116, 128, 139, 198, 199, 204, 205, 209, 209, 212,213,214,215, 217]. As demonstrated formerly (e.g., [13, 14, 139, 188]), the positivity property may fail to hold even with positivity-preserving numerical fluxes considered, positivity can be retained by further restricting the reconstructed states (e.g., [8, 59, 61, 62, 188]).

Also, parametrized MPP flux-corrected limiting procedure is an effective measure to preserve the positivity of density and the pressure for finite difference, finite volume and DG methods. Hu et al. [79] suggested a flux cut-off limiter for finite difference WENO schemes to maintain the positivity of density and pressure for the compressible Euler system. It should be mentioned that the MPP flux limiter [108, 203] or later improvement through the parametrized positivity-preserving flux limiter [201] has been developed for the compressible Euler system or hyperbolic conservation laws. Parent [136] suggested a class of flux-limited schemes for conservation laws of both high-resolution and positivity-preserving property through the use of the Steger-Warming method and TVD flux limiters. Christlieb et al. [29] developed a class of high order positivity-preserving finite difference WENO methods for the ideal MHD equations under the CT framework. Moreover, Wu [194] designed a high-order, density and pressure positivity-preserving methods for the general relativistic hydrodynamic (GRHD) equations with a general equation of state and carried it out at every step of the high-order strong stability preserving (SSP) time integration methods. For more details about the positivity-preserving limiters or MPP flux-corrected methods, see the existing literature (e.g., [26, 28, 29, 89, 109, 194, 200, 201, 203, 204, 209]) and references therein.

The positivity-preserving flux limiting approach is to seek a convex combination of the first-order monotone flux with the high-order flux, in the hope of that such combination can achieve both positivity-preserving property and high-order accuracy under certain conditions, e.g. some mild time step constraint. Popularly, positivity-preserving of the first-order Lax-Friedrichs scheme plays a pivotal role in the parametrized positivity-preserving flux-corrected limiting procedure. As pointed out by Zhang and Shu [212,213,214] (also Remark 2.4 of Zhang and Shu [212]), the local Lax-Friedrichs flux preserves positivity of both density and pressure. Xiong et al. [200, 201] showed that the global maximum principle can be preserved while the high order accuracy of the base scheme is maintained, when applied in incompressible flows and the compressible Euler equations, respectively. Through error analysis on one-dimensional non-uniform meshes, Christlieb et al. [28] stated that the proposed MPP flux limiting schemes can maintain high order of accuracy. Cheng et al. [22, 22] proved that the simple first-order local Lax-Friedrichs scheme coupled with forward Euler time discretization is positivity-preserving for one-dimensional MHD under the restriction CFL $\le$ 0.5. Christlieb et al. [26, 29] also demonstrated that the corrected flux maintains the designed order of accuracy of the high-order flux for ideal MHD equations. Moreover, the overall scheme can also take the same CFL constraint of 0.5 as that of the local Lax-Friedrichs scheme [29].

The following is devoted to the MPP flux limiting technique for a second order positivity-preserving Godunov finite volume HLL method implemented for the solar wind plasma MHD equations. Based on the underlying first order building block of positivity-preserving Lax-Friedrichs, and the CT or GLM framework, the MPP solver described here is built with slightly extra CFL constraints due to spherical geometrical factor. In practice, the construction of the following positivity-preserving formulation abides by the framework of Christlieb et al. [29].

In order to establish positivity-preserving HLL scheme, Eqs. (3.17) or Eq. (A.1) with HLL type flux defined above can be used as the base scheme, where the second-order accurate linear reconstruction Eq. (A.8) is employed. Then, the variables at the centroid point $(\overline{r}_i, \overline{\theta }_j, \phi _k)$ can be advanced with a second-order Runge–Kutta scheme in Eq. (3.18). For clarity, rewrite these two stages explicitly

$$\begin{aligned} \overline{{\mathbf U}}^*=\overline{\mathbf U}^n+\varDelta t\mathcal {R}_{\mathbf U}[\overline{\mathbf U}^n] \end{aligned}$$

(A.25)

$$\begin{aligned} \overline{{\mathbf U}}^{**}=\overline{{\mathbf U}}^*+\varDelta t\mathcal {R}_{\mathbf U}[\overline{{\mathbf U}}^*] \end{aligned}$$

(A.26)

$$\begin{aligned} \overline{\mathbf U}^{n+1}=\frac{1}{2}\overline{\mathbf U}^n+\frac{1}{2}\overline{{\mathbf U}}^{**} =\overline{\mathbf U}^n +\frac{\varDelta t}{2} \left( \mathcal {R}_{\mathbf U}[\overline{\mathbf U}^n] +\mathcal {R}_{\mathbf U}[\overline{{\mathbf U}}^*]\right) \end{aligned}$$

(A.27)

By writing out Eq. (A.26) one has

$$\begin{aligned} \begin{aligned} \overline{{\mathbf U}}^{**}=&\ \overline{{\mathbf U}}^*-\varDelta t \left. \left[ \frac{3\left( r_{ip}^2\mathbf {F}_{r,ip,j,k}^*-r_{im}^2\mathbf {F}_{r,im,j,k}^*\right) }{\left( r_{ip}^3-r_{im}^3\right) } + \frac{\sin \theta _{jp}\mathbf {F}_{\theta , i,jp,k}^*-\sin \theta _{jm}\mathbf {F}_{\theta , i,jm,k}^*}{r_i^\theta \left( \cos \theta _{jm}-\cos \theta _{jp}\right) } \right. \right. \\&\left. \left. +\ \frac{\varDelta \theta (j)}{r_i^\phi \left( \cos \theta _{jm}-\cos \theta _{jp} \right) }\frac{\mathbf {F}_{\phi , i,j,kp}^*-\mathbf {F}_{\phi , i,j,km}^*}{\varDelta \phi (k)}\right] \right| _{\overline{{\mathbf U}}^*} +\left. \varDelta t\overline{\mathbf {S}}^{*}_{i,j,k}\right| _{\overline{{\mathbf U}}^*} \end{aligned} \end{aligned}$$

where $\mathbf {F}^*$ is the flux evaluated at $\overline{{\mathbf U}}^*$ and $\mathbf {S}^*$ takes its value at $\overline{{\mathbf U}}^*$. Then substitute $\overline{{\mathbf U}}^{**}$ into Eq. (A.27) to obtain

$$\begin{aligned} \begin{aligned} \overline{\mathbf U}^{n+1}=&\ \overline{\mathbf U}^n- {\varDelta t} \left[ \frac{3\left( r_{ip}^2\mathbf {F}_{r,ip,j,k}^{rk}-r_{im}^2\mathbf {F}_{r,im,j,k}^{rk}\right) }{\left( r_{ip}^3-r_{im}^3\right) } + \frac{\sin \theta _{jp}\mathbf {F}_{\theta , i,jp,k}^{rk}-\sin \theta _{jm}\mathbf {F}_{\theta , i,jm,k}^{rk}}{r_i^\theta \left( \cos \theta _{jm}-\cos \theta _{jp}\right) } \right. \\&\left. +\ \frac{\varDelta \theta (j)}{r_i^\phi \left( \cos \theta _{jm}-\cos \theta _{jp} \right) }\frac{\mathbf {F}_{\phi , i,j,kp}^{rk}-\mathbf {F}_{\phi , i,j,km}^{rk}}{\varDelta \phi (k)}\right] +\varDelta t\mathbf {S}^{rk} _{i,j,k} \end{aligned} \end{aligned}$$

(A.28)

where $\mathbf {F}^{rk}_{r, ip,j,k}{=}\frac{1}{2}(\mathbf {F}^{n}_{r, ip,j,k}{+}\mathbf {F}^{*}_{r, ip,j,k}), \mathbf {F}^{rk}_{\theta , i,jp,k}=\frac{1}{2}(\mathbf {F}^{n}_{\theta , i,jp,k}+\mathbf {F}^{*}_{\theta , i,jp,k}), \mathbf {F}^{rk}_{\phi , i,j,kp}=\frac{1}{2}(\mathbf {F}^{n}_{\phi , i,j,kp}+\mathbf {F}^{*}_{\phi , i,j,kp})$, $\mathbf {S}^{rk} _{i,j,k}= \frac{1}{2}\left[ \overline{\mathbf {S}}_{i,j,k}+\overline{\mathbf {S}}^{*}_{i,j,k}\right]$. Here, rk stands for Runge–Kutta integration method with two stages.

Before proceeding to state 3D case of positivity-preserving formulation, it is convenient to denote fluxes $\mathbf {F}_r, \mathbf {F}_\theta , \mathbf {F}_\phi$ as $\mathbf {f}, \mathbf {g}, \mathbf {h}$ when used in the first-order Lax-Friedrichs scheme. By combining the fluxes $\mathbf {f}, \mathbf {g}, \mathbf {h}$ in the first-order Lax-Friedrichs scheme and those in our HLL type scheme, one can configure the following fluxes

$$\begin{aligned} {\left\{ \begin{array}{ll} \hat{\mathbf {F}}_{ip}=\omega _{ip,j,k}\left( {\mathbf {F}}^{rk}_{r, ip,j,k} -{\mathbf {f}}^n_{ip, j,k}\right) +{\mathbf {f}}^n_{ip, j,k}\\ \hat{\mathbf {F}}_{im}=\omega _{im,j,k}\left( {\mathbf {F}}^{rk}_{r, im,j,k} -{\mathbf {f}}^n_{im, j,k}\right) +{\mathbf {f}}^n_{im, j,k}\\ \hat{\mathbf {F}}_{jp}=\omega _{i,jp,k}\left( {\mathbf {F}}^{rk}_{\theta , i,jp,k}-{\mathbf {g}}^n_{i,jp,k}\right) +{\mathbf {g}}^n_{i,jp,k}\\ \hat{\mathbf {F}}_{jm}=\omega _{i,jm,k}\left( {\mathbf {F}}^{rk}_{\theta , i,jm,k}-{\mathbf {g}}^n_{i,jm,k}\right) +{\mathbf {g}}^n_{i,jm,k}\\ \hat{\mathbf {F}}_{kp}=\omega _{i,j,kp}\left( {\mathbf {F}}^{rk}_{\phi , i,j,kp}-{\mathbf {h}}^n_{i,j,kp}\right) +{\mathbf {h}}^n_{i,j,kp}\\ \hat{\mathbf {F}}_{km}=\omega _{i,j,km}\left( {\mathbf {F}}^{rk}_{\phi , i,j,km}-{\mathbf {h}}^n_{i,j,km}\right) +{\mathbf {h}}^n_{i,j,km} \end{array}\right. } \end{aligned}$$

(A.29)

By replacing $\mathbf {F}^{rk}$ in Eq. (A.28) with $\hat{\mathbf {F}}$ in Eqs. (A.29), (A.28) becomes

$$\begin{aligned} \begin{aligned} \overline{\mathbf U}^{n+1}({\omega })=&\ \overline{\mathbf U}^{n}-\varDelta t \left[ \frac{3\left( r_{ip}^2\hat{\mathbf {F}}_{ip}-r_{im}^2\hat{\mathbf {F}}_{im}\right) }{\left( r_{ip}^3-r_{im}^3\right) }+ \frac{\sin \theta _{jp}\hat{\mathbf {F}}_{jp}-\sin \theta _{jm}\hat{\mathbf {F}}_{jm}}{r_i^\theta \left( \cos \theta _{jm}-\cos \theta _{jp}\right) }+\right. \\&\left. \frac{\varDelta \theta (j)}{r_i^\phi \left( \cos \theta _{jm}-\cos \theta _{jp} \right) }\frac{\hat{\mathbf {F}}_{kp}-\hat{\mathbf {F}}_{km}}{\varDelta \phi (k)}\right] +\varDelta t\mathbf {S}^{rk} _{i,j,k} \end{aligned} \end{aligned}$$

(A.30)

where $\overline{\mathbf U}^{n+1}({\omega })$ stands for the solution obtained from Eq. (A.30) with the modified fluxes in Eq. (A.29) and ${\omega }=(\omega _{im,j,k}, \omega _{ip,j,k}, \omega _{i,jm,k}, \omega _{i, jp,k}, \omega _{i,j,km}, \omega _{i,j,kp} )$. At this time, $\hat{\mathbf {F}}$ in Eq. (A.29) completes the flux modification of $\mathbf {F}^{rk}$ in Eq. (A.28). This procedure modifies the fluxes only at the final state (A.27), and is in practice favorable due to the time cost consideration. It should be mentioned that this kind of flux modification can be carried out in both time integration stages (A.25) and (A.26). Then to guarantee the positivity, one needs to find suitable $\omega _{ip,j,k}, \omega _{im,j,k}, \omega _{i, jp,k}, \omega _{i,jm,k}, \omega _{i,j,kp}, \omega _{i,j,km}$ such that the solutions satisfy

$$\begin{aligned}\begin{gathered} {\left\{ \begin{array}{ll} \rho ^{n+1}_{i,j,k}({\omega })\ge \epsilon _{\rho }\\ p ^{n+1}_{i,j,k}({\omega })\ge \epsilon _{p} \end{array}\right. } \end{gathered}\end{aligned}$$

where

$$\begin{aligned}\begin{gathered} {\left\{ \begin{array}{ll} \epsilon _{\rho }=\displaystyle \min _{i,j,k}\left( {}^\mathrm{LLF}\rho ^{n+1}_{i,j,k},10^{-13}\right) \\ \epsilon _{p}=\displaystyle \min _{i,j,k}\left( {}^\mathrm{LLF}p^{n+1}_{i,j,k},10^{-13}\right) \end{array}\right. } \end{gathered}\end{aligned}$$

with ${}^\mathrm{LLF}\rho ^{n+1}_{i,j,k}, {}^\mathrm{LLF}p^{n+1}_{i,j,k}$ obtained from the 1st order Lax-Fredrichs scheme.

To facilitate the description, the flux component associated with density equation is labeled with upper left superscript $\rho$. Then

$$\begin{aligned}\begin{aligned} {}^\mathrm{LLF}\rho ^{n+1} =&\ \overline{\rho }^{n}-\varDelta t \left[ \frac{3\left( r_{ip}^2{}^{\rho }{f}^n_{ip,j,k}-r_{im}^2 {}^{\rho }{f}^n_{im,j,k}\right) }{\left( r_{ip}^3-r_{im}^3\right) }+ \frac{\sin \theta _{jp}{}^{\rho }{g}^n_{i,jp,k}-\sin \theta _{jm}{}^{\rho }{g}^n_{i,jm,k}}{r_i^\theta \left( \cos \theta _{jm}-\cos \theta _{jp}\right) } +\right. \\&\left. \frac{\varDelta \theta (j)}{r_i^\phi \left( \cos \theta _{jm}-\cos \theta _{jp} \right) }\frac{{}^{\rho }{h}_{i,j,kp}^n-{}^{\rho }{h}_{i,j,km}^n}{\varDelta \phi (k)} \right] \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned} \overline{\rho }^{n+1}({\omega })=&\ \overline{\rho }^{n}-\varDelta t \left[ \frac{3\left( r_{ip}^2 {}^{\rho }\hat{{F}}_{ip}-r_{im}^2{}^{\rho }\hat{{F}}_{im}\right) }{\left( r_{ip}^3-r_{im}^3\right) }+ \frac{\sin \theta _{jp}{}^{\rho }\hat{{F}}_{jp}-\sin \theta _{jm}{}^{\rho }\hat{{F}}_{jm}}{r_i^\theta \left( \cos \theta _{jm}-\cos \theta _{jp}\right) }+\right. \\&\left. \frac{\varDelta \theta (j)}{r_i^\phi \left( \cos \theta _{jm}-\cos \theta _{jp} \right) }\frac{{}^{\rho }\hat{{F}}_{kp}-{}^{\rho }\hat{{F}}_{km}}{\varDelta \phi (k)}\right] \end{aligned} \end{aligned}$$

(A.31)

To preserve positive density, it is necessary to find the upper bounds $\varOmega ^{\rho }_{im,j,k}$, $\varOmega ^{\rho }_{ip,j,k}$, $\varOmega ^{\rho }_{i,jm,k}$, $\varOmega ^{\rho }_{i,jp,k}$, $\varOmega ^{\rho }_{i,j,km}$, $\varOmega ^{\rho }_{i,j,kp}$ of the limiting parameters $\omega _{im,j,k}, \omega _{ip,j,k}$, $\omega _{i,jm,k}, \omega _{i,jp,k}$, $\omega _{i,j,km}, \omega _{i,j,kp}$ such that any combination $\left( \omega _{im,j,k}, \omega _{ip,j,k},\omega _{i,jm,k},\right.$$\left. \omega _{i,jp,k},\omega _{i,j,km},\omega _{i,j,kp}\right)$ belongs to $S_{\rho }$, where

$$\begin{aligned} S_{\rho }&= \left[ 0,\varOmega ^{\rho }_{im,j,k}\right] \times \left[ 0,\varOmega ^{\rho }_{ip,j,k}\right] \times \left[ 0,\varOmega ^{\rho }_{i,jm,k}\right] \times \left[ 0,\varOmega ^{\rho }_{i,jp,k}\right] \times \left[ 0,\varOmega ^{\rho }_{i,j,km}\right] \\&\quad \times \left[ 0,\varOmega ^{\rho }_{i,j,kp}\right] \end{aligned}$$

The requirement of positivity preserving of density implies the following inequality

$$\begin{aligned} \begin{aligned} \overline{\rho }^{n+1}({\omega })=&\ \overline{\rho }^{n}-\varDelta t \left[ \frac{3\left( r_{ip}^2 {}^{\rho }\hat{{F}}_{ip}-r_{im}^2{}^{\rho }\hat{{F}}_{im}\right) }{\left( r_{ip}^3-r_{im}^3\right) }+ \frac{\sin \theta _{jp}{}^{\rho }\hat{{F}}_{jp}-\sin \theta _{jm}{}^{\rho }\hat{{F}}_{jm}}{r_i^\theta \left( \cos \theta _{jm}-\cos \theta _{jp}\right) }+\right. \\&\left. \frac{\varDelta \theta (j)}{r_i^\phi \left( \cos \theta _{jm}-\cos \theta _{jp} \right) }\frac{{}^{\rho }\hat{{F}}_{kp}-{}^{\rho }\hat{{F}}_{km}}{\varDelta \phi (k)}\right] \ge \epsilon _{\rho } \end{aligned} \end{aligned}$$

(A.32)

with fluxes given by Eq. (A.29), i.e.,

$$\begin{aligned}\begin{gathered} {\left\{ \begin{array}{ll} {}^{\rho }\hat{{F}}_{ip}=\omega _{ip,j,k}\left( {}^{\rho }{F}^{rk}_{r, ip,j,k}-{}^{\rho }{f}^{n}_{ip,j,k} \right) +{}^{\rho }{f}^{n}_{ip,j,k}\\ {}^{\rho }\hat{{F}}_{im}=\omega _{im,j,k}\left( {}^{\rho }{F}^{rk}_{r, im,j,k} -{}^{\rho }{f}^{n}_{im,j,k}\right) +{}^{\rho }{f}^{n}_{im,j,k}\\ {}^{\rho }\hat{{F}}_{jp}=\omega _{i,jp,k}\left( {}^{\rho }{F}^{rk}_{\theta , i,jp,k}-{}^{\rho }{g}^{n}_{i,jp,k}\right) +{}^{\rho }{g}^{n}_{i,jp,k}\\ {}^{\rho }\hat{{F}}_{jm}=\omega _{i,jm,k}\left( {}^{\rho }{F}^{rk}_{\theta , i,jm,k}-{}^{\rho }{g}^{n}_{i,jm,k}\right) +{}^{\rho }{g}^{n}_{i,jm,k}\\ {}^{\rho }\hat{{F}}_{kp}=\omega _{i,j,kp}\left( {}^{\rho }{F}^{rk}_{\phi , i,j,kp}-{}^{\rho }{h}^{n}_{i,j,kp}\right) +{}^{\rho }{h}^{n}_{i,j,kp}\\ {}^{\rho }\hat{{F}}_{km}=\omega _{i,j,km}\left( {}^{\rho }{F}^{rk}_{\phi , i,j,km}-{}^{\rho }{h}^{n}_{i,j,km}\right) +{}^{\rho }{h}^{n}_{i,j,km} \end{array}\right. } \end{gathered}\end{aligned}$$

Then Eq. (A.32) becomes

$$\begin{aligned} \begin{aligned}&\left( \omega _{im,j,k}{\delta }{f}_{im,j,k}+\omega _{ip,j,k}{\delta }{f}_{ip,j,k}\right) + \left( \omega _{i,jm,k}{\delta }{f}_{i,jm,k}+\omega _{i,jp,k}{\delta }{f}_{i,jp,k}\right) \\&+\left( \omega _{i,j,km}{\delta }{f}_{i,j,km}+\omega _{i,j,kp}{\delta }{f}_{i,j,kp}\right) \ge \epsilon _{\rho }-{}^\mathrm{LLF}\rho ^{n+1}_{i,j,k} \end{aligned} \end{aligned}$$

(A.33)

where

$$\begin{aligned}\begin{gathered} {\left\{ \begin{array}{ll} \delta {f}_{im,j,k}=\dfrac{3\varDelta t r_{im}^2}{r_{ip}^3-r_{im}^3}\left( {}^{\rho }{F}^{rk}_{r, im,j,k}-{}^{\rho }{f}^{n}_{im,j,k}\right) \\ {\delta }{f}_{i,jm,k}=\dfrac{\varDelta t\sin \theta _{jm}}{r_i^\theta \left( \cos \theta _{jm}-\cos \theta _{jp}\right) } \left( {}^{\rho }{F}^{rk}_{\theta , i,jm,k}-{}^{\rho }{g}^{n}_{i,jm,k}\right) \\ {\delta }{f}_{i,j,km}=\dfrac{\varDelta t \varDelta \theta (j)}{r_i^\phi \left( \cos \theta _{jm}-\cos \theta _{jp} \right) \varDelta \phi (k) } \left( {}^{\rho }{F}^{rk}_{\phi , i,j,km}-{}^{\rho }{h}^{n}_{i,j,km} \right) \\ \end{array}\right. } \end{gathered}\end{aligned}$$

$$\begin{aligned}\begin{gathered} {\left\{ \begin{array}{ll} {\delta }{f}_{ip,j,k}=- \dfrac{3 \varDelta t r_{ip}^2}{r_{ip}^3-r_{im}^3}\left( {}^{\rho }{F}^{rk}_{r, ip,j,k}-{}^{\rho }{f}^{n}_{ip,j,k}\right) \\ {\delta }{f}_{i,jp,k}=-\dfrac{\varDelta t\sin \theta _{jp}}{r_i^\theta \left( \cos \theta _{jm}-\cos \theta _{jp}\right) } \left( {}^{\rho }{F}^{rk}_{\theta , i,jp,k}-{}^{\rho }{g}^{n}_{i,jp,k}\right) \\ {\delta }{f}_{i,j,kp}=-\dfrac{\varDelta t \varDelta \theta (j)}{r_i^\phi \left( \cos \theta _{jm}-\cos \theta _{jp} \right) \varDelta \phi (k)}\left( {}^{\rho }{F}^{rk}_{\phi , i,j,kp}-{}^{\rho }{h}^{n}_{i,j,kp}\right) \end{array}\right. } \end{gathered}\end{aligned}$$

Due to the positivity-preserving property of the 1th-order LLF scheme (proved below), ${}^\mathrm{LLF}\rho ^{n+1}_{i,j,k}\ge \epsilon _{\rho }$, the right hand side $\epsilon _{\rho }-{}^\mathrm{LLF}\rho ^{n+1}_{i,j,k}\le 0$. Then we decouple the inequalities (A.33) based on the sign of the six defined items: ${\delta }{f}_{im,j,k}$, ${\delta }{f}_{ip,j,k}$, ${\delta }{f}_{i,jm,k}$, ${\delta }{f}_{i,jm,k}$, ${\delta }{f}_{i,j, km}$, ${\delta }{f}_{i, j, km}$, and define the upper bounds by limiting parameters as follows:

$$\begin{aligned}\begin{gathered} \varOmega ^{\rho }_\mathrm{index}= {\left\{ \begin{array}{ll} 1 \quad &{} \text {If} \ \ {\delta }{f}_\mathrm{index}\ge 0 \\ \min \left( 1,\dfrac{\epsilon _{\rho }-{}^{\text {LLF}}\rho ^{n+1}_{i,j,k}}{\displaystyle \sum _{{\delta }{f}_ \mathrm{index}<0} {\delta }{f}_\mathrm{index}}\right) \quad &{} \text {If} \ \ {\delta }{f}_ \mathrm{index}<0 \end{array}\right. } \end{gathered}\end{aligned}$$

where the subscript “index” belongs to the index set $\left\{ \{im,j,k\}, \{ip,j,k\}, \{i,jm,k\},\right.$$\left. \{i,jp,k\}, \{i,j,km\}, \{i,j,kp\} \right\}$, and ${\delta }{f}_\mathrm{index}\in \left\{ {\delta }{f}_{im,j,k},{\delta }{f}_{ip,j,k}, {\delta }{f}_{i,jm,k}, {\delta }{f}_{i,jp,k}, \right.$$\left. {\delta }{f}_{i,j,km}, {\delta }{f}_{i,j,kp}\right\}$. Now we denote the sixty four vertices of

$$\begin{aligned} S_{\rho }&= \left[ 0,\varOmega ^{\rho }_{im,j,k}\right] \times \left[ 0,\varOmega ^{\rho }_{ip,j,k}\right] \times \left[ 0,\varOmega ^{\rho }_{i,jm,k}\right] \times \left[ 0,\varOmega ^{\rho }_{i,jp,k}\right] \\&\quad \times \left[ 0,\varOmega ^{\rho }_{i,j,km}\right] \times \left[ 0,\varOmega ^{\rho }_{i,j,kp}\right] \end{aligned}$$

to be $\mathcal{A}^{k_1,k_2,k_3,k_4,k_5,k_6}$, that is,

$$\begin{aligned}\begin{gathered} \mathcal{A}^{k_1,k_2,k_3,k_4,k_5,k_6}= \left( k_1\varOmega ^{\rho }_{im,j,k},k_2\varOmega ^{\rho }_{ip,j,k}, k_3\varOmega ^{\rho }_{i,jm,k},k_4\varOmega ^{\rho }_{i,jp,k}, k_5\varOmega ^{\rho }_{i,j,km},k_6\varOmega ^{\rho }_{i,j,kp} \right) \end{gathered}\end{aligned}$$

with $k_1, k_2, k_3, k_4, k_5, k_6$ being 0 or 1. At present, for any set ${\omega }\in S_\rho$, $\overline{\rho }^{n+1}({\omega })$ obtained from Eq. (A.31) is obviously positive.

Under the condition of preserving positive density, it is ready to describe how to preserve the positivity of the pressure/internal energy. It is known that the pressure can be derived from the energy equation as follows:

$$\begin{aligned}\begin{gathered} p=(\gamma -1)\left( e-\dfrac{1}{2}\frac{(\rho \mathbf {v})^{2}}{\rho }-\dfrac{1}{2\mu }\mathbf {B}^2\right) \end{gathered}\end{aligned}$$

Next a strategy is proposed to obtain positive pressure. More specifically, a subset $S_p$ of $S_\rho$ is identified such that $p^{n+1}_{i,j,k}$ is positive, i.e.,

$$ S_p = \left\{ {\omega }=\left( \omega _{im,j,k}, \omega _{ip,j,k}, \omega _{i,jm,k}, \omega _{i,jp,k}, \omega _{i,j,km}, \omega _{i,j,kp} \right) \in S_\rho : p^{n+1}_{i,j,k}({\omega }) \ge \epsilon _p \right\} $$

In this case, the vertices of $S_{p}$ consist of

$$\begin{aligned} \mathcal{B}^{k_1,k_2,k_3,k_4,k_5,k_6}= r^{k_1,k_2,k_3,k_4,k_5,k_6} \mathcal{A}^{k_1,k_2,k_3,k_4,k_5,k_6} \end{aligned}$$

(A.34)

and satisfy the following inequality:

$$\begin{aligned} p^{n+1}_{i,j,k}(r^{k_1,k_2,k_3,k_4,k_5,k_6}\mathcal{A}^{k_1,k_2,k_3,k_4,k_5,k_6})\ge \epsilon _{p} \end{aligned}$$

(A.35)

From (A.34) one can obtain $\mathcal{B}^{k_1,k_2,k_3,k_4,k_5,k_6}$ if ${r}^{k_1,k_2,k_3,k_4,k_5,k_6}$ is defined as below:

• 1. If $p^{n+1}_{i,j,k}(\mathcal{A}^{k_1,k_2,k_3,k_4,k_5,k_6})\ge \epsilon _{p}$, then

  $$\begin{aligned}\begin{gathered} r^{k_1,k_2,k_3,k_4,k_5,k_6}=1 \end{gathered}\end{aligned}$$

• 2. If $p^{n+1}_{i,j,k}(\mathcal{A}^{k_1,k_2,k_3,k_4,k_5,k_6})< \epsilon _{p}$, one can obtain $r^{k_1,k_2,k_3,k_4,k_5,k_6}$ by solving the cubic equation (A.35) for the unknown variable $r\in [0,1)$. Thanks to the concave property of pressure as a function of $\mathbf {U}$, we can get r in a simple way:

  $$ p(\alpha \mathbf {U}_1+(1-\alpha ) \mathbf {U}_2)\ge \alpha p(\mathbf {U}_1)+ (1-\alpha ) p(\mathbf {U}_2) \ \ \mathrm{if} \ \ \rho (\mathbf {U}_1)>0, \rho (\mathbf {U}_2)>0 $$

  for $\alpha \in [0,1]$. Evidently it follows that

  $$\begin{aligned} p(\mathbf {U}\left( \alpha {\omega }_1+(1-\alpha ){\omega }_2\right) )&=p(\alpha \mathbf {U}({\omega }_1)+(1-\alpha )\mathbf {U}({\omega }_2))\ge \alpha p(\mathbf {U}({\omega }_1))\\&\quad + (1-\alpha )p(\mathbf {U}({\omega }_2)) \end{aligned}$$

  That is,

  $$\begin{aligned} p(\alpha {\omega }_1+(1-\alpha ) {\omega }_2)\ge \alpha p({\omega }_1)+(1-\alpha ) p({\omega }_2) \ \ \mathrm{if} \ \ \rho ({\omega }_1)>0, \rho ({\omega }_2)>0 \end{aligned}$$

  (A.36)

  where $\mathbf {U}({\omega }), \rho ({\omega })$ and $p({\omega })$ are used to denote the solutions obtained from Eq. (A.30) with

  $${\omega }= \left( \omega _{im,j,k}, \omega _{ip,j,k}, \omega _{i,jm,k}, \omega _{i,jp,k}, \omega _{i,j,km}, \omega _{i,j,kp} \right) $$

  If taking ${\omega }_1=\mathcal{A}^{k_1,k_2, k_3, k_4, k_5, k_6}((k_1,k_2, k_3, k_4, k_5, k_6)\ne \mathbf {0} ), {\omega }_2=\mathcal{A}^{\mathbf {0}}=(0,0,0,0,0,0)$, and $\alpha =r^{k_1,k_2, k_3, k_4, k_5, k_6}$, then $p\left[ \alpha {\omega }_1+(1-\alpha ) {\omega }_2\right] =p\left[ r^{k_1,k_2, k_3, k_4, k_5, k_6}\right.$$\left. \mathcal{A}^{k_1,k_2, k_3, k_4, k_5, k_6}+(1-r^{k_1,k_2, k_3, k_4, k_5, k_6})\mathcal{A}^{\mathbf {0}}\right] =p\left( r^{k_1,k_2, k_3, k_4, k_5, k_6}\mathcal{A}^{k_1,k_2, k_3, k_4, k_5, k_6}\right)$. Thus, Eq. (A.36) becomes

  $$\begin{aligned}&p\left( r^{k_1,k_2, k_3, k_4, k_5, k_6}\mathcal{A}^{k_1,k_2, k_3, k_4, k_5, k_6}\right) \\&\ge r^{k_1,k_2, k_3, k_4, k_5, k_6} p(\mathcal{A}^{k_1,k_2, k_3, k_4, k_5, k_6})+(1-r^{k_1,k_2, k_3, k_4, k_5, k_6}) p(\mathcal{A}^{\mathbf {0}}) \end{aligned}$$

  When the right-hand side of the above inequality $\ge \epsilon _{p}$ is assumed, it follows that

  $$\begin{aligned} \begin{aligned}&p\left( r^{k_1,k_2, k_3, k_4, k_5, k_6}\mathcal{A}^{k_1,k_2, k_3, k_4, k_5, k_6}\right) \\&\ge r^{k_1,k_2, k_3, k_4, k_5, k_6} p(\mathcal{A}^{k_1,k_2, k_3, k_4, k_5, k_6})+(1-r^{k_1,k_2, k_3, k_4, k_5, k_6}) p(\mathcal{A}^{\mathbf {0}})\ge \epsilon _{p} \end{aligned} \end{aligned}$$

  (A.37)

  Since

  $$\begin{aligned} {\left\{ \begin{array}{ll} p(\mathcal{A}^{\mathbf {0}})\ge \epsilon _{p}\\ p(\mathcal{A}^{k_1,k_2, k_3, k_4, k_5, k_6})<\epsilon _{p} \end{array}\right. } \end{aligned}$$

  from (A.37) we get

  $$\begin{aligned} r^{k_1,k_2, k_3, k_4, k_5, k_6}\le \dfrac{p(\mathcal{A}^{\mathbf {0}})-\epsilon _p}{p(\mathcal{A}^{\mathbf {0}})-p(\mathcal{A}^{k_1,k_2, k_3, k_4, k_5, k_6})}\in [0,1) \end{aligned}$$

  or equivalently

  $$\begin{aligned} r^{k_1,k_2,k_3,k_4,k_5,k_6}\le \dfrac{\epsilon _p-p^{n+1}_{i,j,k}(\mathcal{A}^{0,0,0,0,0,0})}{p^{n+1}_{i,j,k}(\mathcal{A}^{k_1,k_2,k_3,k_4,k_5,k_6})-p^{n+1}_{i,j,k}(\mathcal{A}^{0,0,0,0,0,0})} \end{aligned}$$

We can embark a rectangle subset $S_{\rho ,p}$ inside $S_{p}$

$$\begin{aligned} S_{\rho ,p}&= \left[ 0,\varOmega ^{\rho ,p}_{im,j,k}\right] \times \left[ 0,\varOmega ^{\rho ,p}_{ip,j,k}\right] \times \left[ 0,\varOmega ^{\rho ,p}_{i,jm,k}\right] \times \left[ 0,\varOmega ^{\rho ,p}_{i,jp,k}\right] \\&\quad \times \left[ 0,\varOmega ^{\rho ,p}_{i,j,km}\right] \times \left[ 0,\varOmega ^{\rho ,p}_{i,j,kp}\right] \subset S_{p}, \end{aligned}$$

where

$$\begin{aligned}\begin{gathered} {\left\{ \begin{array}{ll} \varOmega ^{\rho ,p}_{im,j,k}=\displaystyle \min _{k_{2,3,4,5,6}=0,1}\left( \mathcal{B}^{1,k_2,k_3,k_4,k_5,k_6}\right) \\ \varOmega ^{\rho ,p}_{ip,j,k}=\displaystyle \min _{k_{1,3,4,5,6}=0,1}\left( \mathcal{B}^{k_1,1,k_3,k_4,k_5,k_6}\right) \\ \varOmega ^{\rho ,p}_{i,jm,k}=\displaystyle \min _{k_{1,2,4,5,6}=0,1}\left( \mathcal{B}^{k_1,k_2,1,k_3,k_5,k_6}\right) \\ \varOmega ^{\rho ,p}_{i,jp,k}=\displaystyle \min _{k_{1,2,3,5,6}=0,1}\left( \mathcal{B}^{k_1,k_2,k_3,1,k_5,k_6}\right) \\ \varOmega ^{\rho ,p}_{i,j,km}=\displaystyle \min _{k_{1,2,3,4,6}=0,1}\left( \mathcal{B}^{k_1,k_2,k_3,k_4,1,k_6}\right) \\ \varOmega ^{\rho ,p}_{i,j,kp}=\displaystyle \min _{k_{1,2,3,4,5}=0,1}\left( \mathcal{B}^{k_1,k_2,k_3,k_4,k_5,1}\right) \end{array}\right. } \end{gathered}\end{aligned}$$

Finally, one can set

$$\begin{aligned} {\left\{ \begin{array}{ll} \omega _{ip,j,k}=\min \left( \varOmega ^{\rho ,p}_{ip,j,k},\varOmega ^{\rho ,p}_{(i+1)m,j,k}\right) \\ \omega _{im,j,k}=\min \left( \varOmega ^{\rho ,p}_{im,j,k},\varOmega ^{\rho ,p}_{(i-1)p,j,k}\right) \\ \omega _{i,jp,k}=\min \left( \varOmega ^{\rho ,p}_{i,jp,k},\varOmega ^{\rho ,p}_{i,(j+1)m,k}\right) \\ \omega _{i,jm,k}=\min \left( \varOmega ^{\rho ,p}_{i,jm,k},\varOmega ^{\rho ,p}_{i,(j-1)p,k}\right) \\ \omega _{i,j,kp}=\min \left( \varOmega ^{\rho ,p}_{i,j,kp},\varOmega ^{\rho ,p}_{i,j,(k+1)m}\right) \\ \omega _{i,j,km}=\min \left( \varOmega ^{\rho ,p}_{i,j,km},\varOmega ^{\rho ,p}_{i,j,(k-1)p}\right) \end{array}\right. } \end{aligned}$$

(A.38)

With the above limiting bounds $\omega _{ip,j,k}, \omega _{im,j,k}, \omega _{i, jp,k}, \omega _{i,jm,k}, \omega _{i,j,kp}, \omega _{i,j,km}$ given by Eq. (A.38), one can restart to update $\overline{\mathbf U}^{n+1}$ from Eq. (A.30). This finishes the positivity-preserving procedure.

Since the first-order local Lax-Friedrichs scheme serves as the building block for the above MPP flux-corrected limiting formulation, we are left to prove positivity-preserving for the local Lax-Friedrichs scheme. In the above context, Lax-Friedrichs scheme can be written as follows:

$$\begin{aligned} \begin{aligned} \frac{d\overline{\mathbf {U}}}{dt}=&-\frac{3\left( r_{ip}^2\mathbf {f}_{ip,j,k}^n-r_{im}^2\mathbf {f}_{im,j,k}^n\right) }{\left( r_{ip}^3-r_{im}^3\right) }- \frac{\sin \theta _{jp}\mathbf {g}_{i,jp,k}^n-\sin \theta _{jm}\mathbf {g}_{i,jm,k}^n}{r_i^\theta \left( \cos \theta _{jm}-\cos \theta _{jp}\right) }\\&-\frac{\varDelta \theta (j)}{r_i^\phi \left( \cos \theta _{jm}-\cos \theta _{jp} \right) }\frac{\mathbf {h}_{i,j,kp}^n-\mathbf {h}_{i,j,km}^n}{\varDelta \phi (k)} +\overline{\mathbf {S}}_{i,j,k} \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned}&\mathbf {f}_{im,j,k}^n=\frac{1}{2} \lambda _{im} \left( \overline{\mathbf {U}}_{i-1,j,k}-\overline{\mathbf {U}}_{i,j,k}\right) + \frac{1}{2} \left[ \mathbf {f}\left( \overline{\mathbf {U}}_{i-1,j,k}\right) +\mathbf {f} \left( \overline{\mathbf {U}}_{i,j,k}\right) \right] \\&\mathbf {g}_{i,jm,k}^n=\frac{1}{2} \lambda _{jm} \left( \overline{\mathbf {U}}_{i,j-1,k}-\overline{\mathbf {U}}_{i,j,k}\right) + \frac{1}{2} \left[ \mathbf {g} \left( \overline{\mathbf {U}}_{i,j-1,k}\right) +\mathbf {g}\left( \overline{\mathbf {U}}_{i,j,k}\right) \right] \\&\mathbf {h}_{i,j,km}^n=\frac{1}{2} \lambda _{km} \left( \overline{\mathbf {U}}_{i,j,k-1}-\overline{\mathbf {U}}_{i,j,k}\right) + \frac{1}{2}\left[ \mathbf {h}\left( \overline{\mathbf {U}}_{i,j,k-1}\right) +\mathbf {h}\left( \overline{\mathbf {U}}_{i,j,k}\right) \right] \end{aligned} \end{aligned}$$

with

$$\begin{aligned} \begin{aligned}&\lambda _{im}=\mathrm{max}\left\{ (\overline{v}_r+\overline{c}_{fr})_{i-1,j,k}, (\overline{v}_r+\overline{c}_{fr})_{i,j,k}\right\} , \\&\lambda _{ip}=\mathrm{max}\left\{ (\overline{v}_r+\overline{c}_{fr})_{i,j,k}, (\overline{v}_r+\overline{c}_{fr})_{i+1,j,k}\right\} \\&\lambda _{jm}=\mathrm{max}\left\{ (\overline{v}_\theta +\overline{c}_{f\theta })_{i,j-1,k}, (\overline{v}_\theta +\overline{c}_{f\theta })_{i,j,k}\right\} , \\&\lambda _{jp}=\mathrm{max}\left\{ (\overline{v}_\theta +\overline{c}_{f\theta })_{i,j,k}, (\overline{v}_\theta +\overline{c}_{f\theta })_{i,j+1,k}\right\} \\&\lambda _{km}=\mathrm{max}\left\{ (\overline{v}_\phi +\overline{c}_{f\phi })_{i,j,k-1}, (\overline{v}_\phi +\overline{c}_{f\phi })_{i,j,k}\right\} , \\&\lambda _{kp}=\mathrm{max}\left\{ (\overline{v}_\phi +\overline{c}_{f\phi })_{i,j,k}, (\overline{v}_\phi +\overline{c}_{f\phi })_{i,j,k+1}\right\} \\ \end{aligned} \end{aligned}$$

For simplicity, by setting

$$\begin{aligned}\begin{gathered} {\left\{ \begin{array}{ll} \mathbf {f}^{n}_{i}=\mathbf {f}\left( \overline{\mathbf {U}}_{i,j,k}\right) , \ \ \mathbf {g}^{n}_{j}=\mathbf {g} \left( \overline{\mathbf {U}}_{i,j,k}\right) , \ \ \mathbf {h}^{n}_{k}=\mathbf {h} \left( \overline{\mathbf {U}}_{i,j,k}\right) \\ \mathbf {U}^{n}_{i}=\overline{\mathbf {U}}_{i,j,k},\ \ \mathbf {U}^{n}_{j}=\overline{\mathbf {U}}_{i,j,k}, \ \ \mathbf {U}^{n}_{k}=\overline{\mathbf {U}}_{i,j,k}\\ \end{array}\right. } \end{gathered}\end{aligned}$$

then the first order local Lax-Friedrichs scheme with a forward Euler time-integration method reads

$$\begin{aligned} \begin{aligned} \overline{\mathbf {U}}^{n+1}_{i,j,k}=~&\overline{\mathbf {U}}^{n}_{i,j,k} - \frac{3 r_{ip}^2 \varDelta t}{2\left( r_{ip}^3-r_{im}^3\right) } \left[ \mathbf {f}^{n}_{i+1}+\mathbf {f}^{n}_{i}-\lambda _{ip}(\mathbf {U}^{n}_{i+1}-\mathbf {U}^{n}_{i})\right] \\&+\frac{3 r_{im}^2 \varDelta t}{2\left( r_{ip}^3-r_{im}^3\right) }\left[ \mathbf {f}^{n}_{i}+\mathbf {f}^{n}_{i-1}-\lambda _{im}(\mathbf {U}^{n}_{i}-\mathbf {U}^{n}_{i-1})\right] \\&- \frac{\varDelta t \sin \theta _{jp} }{2r_i^\theta \left( \cos \theta _{jm}-\cos \theta _{jp}\right) } \left[ \mathbf {g}^{n}_{j+1}+\mathbf {g}^{n}_{j}-\lambda _{jp}(\mathbf {U}^{n}_{j+1}-\mathbf {U}^{n}_{j})\right] \\&+ \frac{\varDelta t \sin \theta _{jm} }{2r_i^\theta \left( \cos \theta _{jm}-\cos \theta _{jp}\right) } \left[ \mathbf {g}^{n}_{j}+\mathbf {g}^{n}_{j-1}-\lambda _{jm}(\mathbf {U}^{n}_{j}-\mathbf {U}^{n}_{j-1})\right] \\&- \frac{\varDelta t\varDelta \theta (j)}{2r_i^\phi \left( \cos \theta _{jm}-\cos \theta _{jp} \right) \varDelta \phi (k)} \left[ \mathbf {h}^{n}_{k+1}+\mathbf {h}^{n}_{k}-\lambda _{kp}(\mathbf {U}^{n}_{k+1}-\mathbf {U}^{n}_{k})\right] \\&+\frac{\varDelta t\varDelta \theta (j)}{2r_i^\phi \left( \cos \theta _{jm}-\cos \theta _{jp} \right) \varDelta \phi (k)}\left[ \mathbf {h}^{n}_{k}+\mathbf {h}^{n}_{k-1}-\lambda _{km}(\mathbf {U}^{n}_{k}-\mathbf {U}^{n}_{k-1})\right] \end{aligned} \end{aligned}$$

(A.39)

where $\varDelta t$ is the time marching step to be determined. Now, one can rewrite (A.39) as follows

$$\begin{aligned} \begin{aligned} \overline{\mathbf {U}}^{n+1}_{i,j,k}=~&\overline{\mathbf {U}}^{n}_{i,j,k} \left( 1-\frac{3 (\lambda _{ip} r_{ip}^2 +\lambda _{im} r_{im}^2)\varDelta t}{2\left( r_{ip}^3-r_{im}^3\right) } -\frac{\varDelta t (\lambda _{jp} \sin \theta _{jp}+ \lambda _{jm}\sin \theta _{jm}) }{2r_i^\theta \left( \cos \theta _{jm}-\cos \theta _{jp}\right) } \right. \nonumber \\&\left. -\frac{\varDelta t\varDelta \theta (j) ( \lambda _{kp}+\lambda _{km}) }{2r_i^\phi \left( \cos \theta _{jm}-\cos \theta _{jp} \right) \varDelta \phi (k)} \right) \\&+\frac{3 (r_{im}^2-r_{ip}^2 )\varDelta t}{2\left( r_{ip}^3-r_{im}^3\right) }\mathbf {f}^{n}_{i} +\frac{\varDelta t ( \sin \theta _{jm}- \sin \theta _{jp}) }{2r_i^\theta \left( \cos \theta _{jm}-\cos \theta _{jp}\right) } \mathbf {g}^{n}_{j} \\&+\frac{3 r_{im}^2 \varDelta t}{2\left( r_{ip}^3-r_{im}^3\right) }\left( \mathbf {f}^{n}_{i-1}+\lambda _{im} \mathbf {U}^{n}_{i-1}\right) - \frac{3 r_{ip}^2 \varDelta t}{2\left( r_{ip}^3-r_{im}^3\right) } \left( \mathbf {f}^{n}_{i+1}-\lambda _{ip} \mathbf {U}^{n}_{i+1}\right) \end{aligned} \end{aligned}$$

$$\begin{aligned}&+\frac{\varDelta t \sin \theta _{jm} }{2r_i^\theta \left( \cos \theta _{jm}-\cos \theta _{jp}\right) } \left( \mathbf {g}^{n}_{j-1}+\lambda _{jm} \mathbf {U}^{n}_{j-1}\right) \nonumber \\&-\frac{\varDelta t \sin \theta _{jp} }{2r_i^\theta \left( \cos \theta _{jm}-\cos \theta _{jp}\right) } \left( \mathbf {g}^{n}_{j+1}-\lambda _{jp} \mathbf {U}^{n}_{j+1}\right) \nonumber \\&+\frac{\varDelta t\varDelta \theta (j)}{2r_i^\phi \left( \cos \theta _{jm}-\cos \theta _{jp} \right) \varDelta \phi (k)} \left( \mathbf {h}^{n}_{k-1}+\lambda _{km}\mathbf {U}^{n}_{k-1}\right) \nonumber \\&-\frac{\varDelta t\varDelta \theta (j)}{2r_i^\phi \left( \cos \theta _{jm}-\cos \theta _{jp} \right) \varDelta \phi (k)}\left( \mathbf {h}^{n}_{k+1}-\lambda _{kp} \mathbf {U}^{n}_{k+1}\right) \end{aligned}$$

(A.40)

with $\varDelta \phi (k)=\phi _{kp}-\phi _{km}$, $\varDelta \theta (j)=\theta _{jp}-\theta _{jm}$. For density $\rho$, it follows from (A.40) that

$$\begin{aligned} \begin{aligned} \overline{\rho }^{n+1}_{i,j,k}=&\overline{\rho }^{n}_{i,j,k} \left( 1-\frac{3 (\lambda _{ip} r_{ip}^2 +\lambda _{im} r_{im}^2)\varDelta t}{2\left( r_{ip}^3-r_{im}^3\right) } -\frac{\varDelta t (\lambda _{jp} \sin \theta _{jp}+ \lambda _{jm} \sin \theta _{jm}) }{2r_i^\theta \left( \cos \theta _{jm}-\cos \theta _{jp}\right) }\right. \\&\left. -\frac{\varDelta t\varDelta \theta (j) ( \lambda _{kp}+\lambda _{km}) }{2r_i^\phi \left( \cos \theta _{jm}-\cos \theta _{jp} \right) \varDelta \phi (k)} \right) \\&+\frac{3 (r_{im}^2-r_{ip}^2 )\varDelta t}{2\left( r_{ip}^3-r_{im}^3\right) }(\rho v_r)^{n}_{i} +\frac{\varDelta t ( \sin \theta _{jm}- \sin \theta _{jp}) }{2r_i^\theta \left( \cos \theta _{jm}-\cos \theta _{jp}\right) } (v_\theta \rho )^{n}_{j}\\&+\frac{3 r_{im}^2 \varDelta t}{2\left( r_{ip}^3-r_{im}^3\right) }\left( (v_r \rho )^{n}_{i-1}+\lambda _{im} \rho ^{n}_{i-1}\right) - \frac{3 r_{ip}^2 \varDelta t}{2\left( r_{ip}^3-r_{im}^3\right) } \left( (v_r \rho )^{n}_{i+1}-\lambda _{ip} \rho ^{n}_{i+1}\right) \\&+\frac{\varDelta t \sin \theta _{jm} }{2r_i^\theta \left( \cos \theta _{jm}-\cos \theta _{jp}\right) } \left( (v_\theta \rho )^{n}_{j-1}+\lambda _{jm} \rho ^{n}_{j-1}\right) -\frac{\varDelta t \sin \theta _{jp} }{2r_i^\theta \left( \cos \theta _{jm}-\cos \theta _{jp}\right) } \\&\quad \left( (v_\theta \rho )^{n}_{j+1}-\lambda _{jp} \rho ^{n}_{j+1}\right) \\&+\frac{\varDelta t\varDelta \theta (j)}{2r_i^\phi \left( \cos \theta _{jm}-\cos \theta _{jp} \right) \varDelta \phi (k)} \left( (v_\phi \rho )^{n}_{k-1}+\lambda _{km} \rho ^{n}_{k-1}\right) \\&-\frac{\varDelta t\varDelta \theta (j)}{2r_i^\phi \left( \cos \theta _{jm}-\cos \theta _{jp} \right) \varDelta \phi (k)}\left( (v_\phi \rho )^{n}_{k+1}-\lambda _{kp} \rho ^{n}_{k+1}\right) \end{aligned} \end{aligned}$$

(A.41)

Since

$$\begin{aligned} \lambda _{i}=&\ \mathrm{max}\left\{ (\overline{v}_r+\overline{c}_{fr})_{i-1,j,k}, (\overline{v}_r+\overline{c}_{fr})_{i,j,k},(\overline{v}_r+\overline{c}_{fr})_{i+1,j,k}\right\} \nonumber \\ =&\ \mathrm{max}(\lambda _{im}, \lambda _{ip})>\max ( (v_r)_{i+1}, (v_r)_{i}, (v_r)_{i-1}) \nonumber \\ \lambda _{j}=&\ \mathrm{max}\left\{ (\overline{v}_\theta +\overline{c}_{f\theta })_{i,j-1,k}, (\overline{v}_\theta +\overline{c}_{f\theta })_{i,j,k},(\overline{v}_\theta +\overline{c}_{f\theta })_{i,j+1,k}\right\} \nonumber \\ =&\ \mathrm{max}(\lambda _{jm}, \lambda _{jp})>\max ((v_{\theta })_{j+1}, (v_{\theta })_{j}, (v_{\theta })_{j-1}) \nonumber \\ \lambda _{k}=&\ \mathrm{max}\left\{ (\overline{v}_\phi +\overline{c}_{f\phi })_{i,j,k-1}, (\overline{v}_\phi +\overline{c}_{f\phi })_{i,j,k},\overline{v}_\phi +\overline{c}_{f\phi })_{i,j,k+1}\right\} \nonumber \\ =&\ \mathrm{max}(\lambda _{km}, \lambda _{kp})>\max ((v_{\phi })_{k+1}, (v_{\phi })_{k}, (v_{\phi })_{k-1}) \end{aligned}$$

(A.42)

there follow

$$\begin{aligned}&{\left\{ \begin{array}{ll} (\rho v_r)^{n}_{i-1}+\lambda _{im} \rho ^{n}_{i-1}>0\\ (\rho v_r)^{n}_{i }-\lambda _{im} \rho ^{n}_{i }<0\\ (\rho v_r)^{n}_{i+1}-\lambda _{ip} \rho ^{n}_{i+1}<0 \end{array}\right. } \quad {\left\{ \begin{array}{ll} (\rho v_{\theta })^{n}_{j-1}+\lambda _{jm} \rho ^{n}_{j-1}>0\\ (\rho v_{\theta })^{n}_{j }-\lambda _{jm} \rho ^{n}_{j }<0\\ (\rho v_{\theta })^{n}_{j+1}-\lambda _{jp} \rho ^{n}_{j+1}<0 \end{array}\right. } \nonumber \\&{\left\{ \begin{array}{ll} (\rho v_{\phi })^{n}_{k-1}+\lambda _{km} \rho ^{n}_{k-1}>0\\ (\rho v_{\phi })^{n}_{k }-\lambda _{km} \rho ^{n}_{k }<0\\ (\rho v_{\phi })^{n}_{k+1}-\lambda _{kp} \rho ^{n}_{k+1}<0 \end{array}\right. } \end{aligned}$$

(A.43)

Considering (A.41), (A.42) and (A.43), one obtains

$$\begin{aligned} \begin{aligned} \overline{\rho }^{n+1}_{i,j,k}&\ge \overline{\rho }^{n}_{i,j,k} \left( 1-\frac{3 ( \lambda _{ip} r_{ip}^2 +\lambda _{im} r_{im}^2)\varDelta t}{2\left( r_{ip}^3-r_{im}^3\right) } -\frac{\varDelta t (\lambda _{jp} \sin \theta _{jp}+ \lambda _{jm} \sin \theta _{jm}) }{2r_i^\theta \left( \cos \theta _{jm}-\cos \theta _{jp}\right) } \right. \\&\left. -\frac{\varDelta t\varDelta \theta (j) ( \lambda _{kp}+\lambda _{km}) }{2r_i^\phi \left( \cos \theta _{jm}-\cos \theta _{jp} \right) \varDelta \phi (k)} \right) +\frac{3 (r_{im}^2-r_{ip}^2 )\varDelta t}{2\left( r_{ip}^3-r_{im}^3\right) }\lambda _{im} (\rho )^{n}_{i} \\&\quad -\frac{\varDelta t}{2r_i^\theta }\left| \frac{( \sin \theta _{jm}- \sin \theta _{jp}) }{\left( \cos \theta _{jm}-\cos \theta _{jp}\right) }\right| \lambda _{jm}( \rho )^{n}_{j}\\&\ge \overline{\rho }^{n}_{i,j,k} \left( 1-\frac{3 ( \lambda _{ip} +\lambda _{im} )r_{ip}^2\varDelta t}{2\left( r_{ip}^3-r_{im}^3\right) } -\frac{\varDelta t}{2r_i^\theta }\left( \frac{ ( \lambda _{jp}\sin \theta _{jp}+ \lambda _{jm}\sin \theta _{jm}) }{\left( \cos \theta _{jm}-\cos \theta _{jp}\right) }\right. \right. \\&\left. +\left| \frac{( \sin \theta _{jm}- \sin \theta _{jp}) }{\left( \cos \theta _{jm}-\cos \theta _{jp}\right) } \right| \lambda _{jm}\right) \left. -\frac{\varDelta t\varDelta \theta (j) ( \lambda _{kp}+\lambda _{km}) }{2r_i^\phi \left( \cos \theta _{jm}-\cos \theta _{jp} \right) \varDelta \phi (k)} \right) \\&\ge \overline{\rho }^{n}_{i,j,k} \left( 1-\frac{3 r_{ip}^2 \varDelta t}{\left( r_{ip}^3-r_{im}^3\right) } \lambda _i -\frac{\varDelta t}{2r_i^\theta }\left( \cot \frac{\varDelta \theta (j)}{2}+\left| \cot \theta _j \right| \right) \lambda _j \right. \\&\left. -\frac{\varDelta t\varDelta \theta (j)}{2r_i^\phi \sin \theta _j \sin \frac{\varDelta \theta (j)}{2} \varDelta \phi (k)} \lambda _k \right) \\&=\overline{\rho }^{n}_{i,j,k} \left( 1-\frac{3 r_{ip}^2 \varDelta t}{\left( r_{ip}^3-r_{im}^3\right) } \lambda _i -\frac{\varDelta t}{2r_i^\theta \varDelta \theta (j)}\left[ \left( \cot \frac{\varDelta \theta (j)}{2}+\left| \cot \theta _j \right| \right) \varDelta \theta (j)\right] \lambda _j \right. \\&\left. -\frac{\varDelta t\varDelta \theta (j)}{2r_i^\phi \sin \theta _j \sin \frac{\varDelta \theta (j)}{2} \varDelta \phi (k)} \lambda _k \right) \\ \end{aligned} \end{aligned}$$

(A.44)

If $\overline{\rho }^{n}_{i,j,k}>0$ and

$$\begin{aligned}&\left( 1-\frac{3 r_{ip}^2 \varDelta t}{\left( r_{ip}^3-r_{im}^3\right) } \lambda _i -\frac{\varDelta t}{2r_i^\theta \varDelta \theta (j)}\left[ \left( \cot \frac{\varDelta \theta (j)}{2}+\left| \cot \theta _j \right| \right) \varDelta \theta (j)\right] \lambda _j \right. \\&\quad \left. -\frac{\varDelta t\varDelta \theta (j)}{2r_i^\phi \sin \theta _j \sin \frac{\varDelta \theta (j)}{2} \varDelta \phi (k)} \lambda _k \right) >0 \end{aligned}$$

then $\overline{\rho }^{n+1}_{i,j,k}>0$ holds. If setting $\varDelta t=\mathrm{CFL} \frac{ \mathrm{min}\left( \varDelta r(i), r_i^\theta \varDelta \theta (j), r_i^\phi \sin \theta _j \varDelta \phi (k) \right) }{ \mathrm{max} \left( \lambda _i, \lambda _j, \lambda _k\right) } (\equiv \mathrm{AA})$, inequality (A.44) can be further enhanced by

$$\begin{aligned} 1-\left( \frac{3 r_{ip}^2\varDelta r(i)}{\left( r_{ip}^3-r_{im}^3\right) } +\frac{1}{2}\left[ \left( \cot \frac{\varDelta \theta (j)}{2}+\left| \cot \theta _j \right| \right) \varDelta \theta (j)\right] +\frac{ \varDelta \theta (j)}{2\sin \frac{\varDelta \theta (j)}{2} } \right) \mathrm{CFL}>0 \end{aligned}$$

(A.45)

Or equivalently $\varDelta t=\mathrm{min}\left( \mathrm{AA}, \mathrm{BB} \right)$ with

$$ \mathrm{BB} = \frac{1}{\mathrm{max}\left( \frac{3 r_{ip}^2 }{\left( r_{ip}^3-r_{im}^3\right) } \lambda _i +\frac{1}{2r_i^\theta \varDelta \theta (j)}\left[ \left( \cot \frac{\varDelta \theta (j)}{2}+\left| \cot \theta _j \right| \right) \varDelta \theta (j)\right] \lambda _j +\frac{\varDelta \theta (j)}{2r_i^\phi \sin \theta _j \sin \frac{\varDelta \theta (j)}{2} \varDelta \phi (k)}\lambda _k \right) } $$

Certainly, this constraint of $\varDelta t$ will affect the original CFL number. Since $\frac{\pi }{4} \le \theta _j \le \frac{3 \pi }{4}$ in the grid mesh of computational region, $\left| \cot \theta _j \right| \le 1$ does hold. Obviously, $\frac{\varDelta \theta (j)}{2} \cot \frac{\varDelta \theta (j)}{2}\approx 1$ and $\frac{\frac{\varDelta \theta (j)}{2}}{\sin \frac{\varDelta \theta (j)}{2}}\approx 1$ for small $\varDelta \theta (j)$. In these cases, one has the following approximations

$$\begin{aligned} \begin{aligned}&\frac{3 r_{ip}^2 }{\left( r_{ip}^3-r_{im}^3\right) }=\frac{1}{\varDelta r(i)} \frac{3 r_{ip}^2 }{\left( r_{ip}^2+r_{ip} r_{im}+r_{im}^2\right) }\approx \frac{1}{\varDelta r(i)} \\&\frac{1}{2r_i^\theta \varDelta \theta (j)}\left[ \left( \cot \frac{\varDelta \theta (j)}{2}+\left| \cot \theta _j \right| \right) \varDelta \theta (j)\right] \approx \frac{1}{r_i^\theta \varDelta \theta (j)} \frac{2+\varDelta \theta (j)}{2}\\&\frac{\varDelta \theta (j)}{2r_i^\phi \sin \theta _j \sin \frac{\varDelta \theta (j)}{2} \varDelta \phi (k)} \approx \frac{\frac{\varDelta \theta (j)}{2}}{\sin \frac{\varDelta \theta (j)}{2}} \frac{1}{r_i^\phi \sin \theta _j \varDelta \phi (k)}\approx \frac{1}{r_i^\phi \sin \theta _j\varDelta \phi (k)} \end{aligned} \end{aligned}$$

From (A.45), it follows that $1- (3+\frac{\varDelta \theta (j)}{2}) \mathrm{CFL}>0$. That is, the original CFL is reduced by a factor $2+\frac{2+\varDelta \theta (j)}{2}\approx 3$, which is due to the geometrical factor in spherical coordinates.

It is noted that the above formulations have used the following algebraic identities:

$$\begin{aligned} \begin{aligned}&\sin \theta _{jm}+\sin \theta _{jp}=2 \sin \frac{\theta _{jp}+\theta _{jm}}{2} \cos \frac{\theta _{jp}-\theta _{jm}}{2} =2\sin \theta _j\cos \frac{\varDelta \theta (j)}{2}\\&\sin \theta _{jp}-\sin \theta _{jm}=2 \cos \frac{\theta _{jp}+\theta _{jm}}{2} \sin \frac{\theta _{jp}-\theta _{jm}}{2} =2\cos \theta _j\sin \frac{\varDelta \theta (j)}{2} \end{aligned} \end{aligned}$$

$$\begin{aligned} \begin{aligned}&\cos \theta _{jm}-\cos \theta _{jp}=2 \sin \frac{\theta _{jp}+\theta _{jm}}{2} \sin \frac{\theta _{jp}-\theta _{jm}}{2} =2\sin \theta _j\sin \frac{\varDelta \theta (j)}{2} \\&\frac{\varDelta t ( \sin \theta _{jm}- \sin \theta _{jp}) }{2r_i^\theta \left( \cos \theta _{jm}-\cos \theta _{jp}\right) }=\frac{\varDelta t ( -2\cos \theta _{j} \sin \frac{\varDelta \theta (j)}{2}) }{2r_i^\theta \left( 2\sin \theta _{j}\sin \frac{\varDelta \theta (j)}{2}\right) }=-\frac{\varDelta t }{2r_i^\theta \varDelta \theta (j)} \varDelta \theta (j) \cot \theta _{j} \\&\frac{\varDelta t ( \sin \theta _{jm}+\sin \theta _{jp}) }{2r_i^\theta \left( \cos \theta _{jm}-\cos \theta _{jp}\right) }=\frac{\varDelta t ( 2\sin \theta _{j} \cos \frac{\varDelta \theta (j)}{2}) }{2r_i^\theta \left( 2\sin \theta _{j}\sin \frac{\varDelta \theta (j)}{2}\right) }=\frac{\varDelta t }{2r_i^\theta \varDelta \theta (j)} \varDelta \theta (j) \cot \frac{\varDelta \theta (j)}{2} \\&\frac{ \sin \theta _{jm} }{ 2r_i^\theta \left( \cos \theta _{jm}-\cos \theta _{jp}\right) }= \frac{ \sin \theta _{jm} }{ 4r_i^\theta \sin \theta _j \sin \frac{\varDelta \theta (j)}{2} } \\&\frac{ \sin \theta _{jp} }{ 2r_i^\theta \left( \cos \theta _{jm}-\cos \theta _{jp}\right) }= \frac{ \sin \theta _{jp} }{ 4r_i^\theta \sin \theta _j \sin \frac{\varDelta \theta (j)}{2} } \end{aligned} \end{aligned}$$

In the implementation, the correction steps can be carried out only at the end of each time step [28, 29, 200, 201, 203] instead of each stage of RK methods [25, 27, 194]. Christlieb et al. [29] show that the differences between the two operations are negligible through their numerical results by SSP-RK3 time integration. If there exist negative density or/and pressure in the intermediate stages of the RK method, the speed of sound can be calculated by $\sqrt{\gamma \frac{|p|}{|\rho |}}$ with the absolute values of density and pressure [29, 201]. As commented before [200, 201, 203], the flux limiter can be applied only at the final RK stage such that the implementation complexity is significantly reduced. They also claimed that if the MPP flux limiter is applied at each of the intermediate stage of RK method, due to the influence of the special cancellation of RK, high order temporal accuracy could be lost. However, Christlieb et al. [27] showed that the correction steps carried out only at the end of each time step may result in accumulation of the divergence error, with the low-storage 10-stage SSP-RK4 method for their numerical simulations. Thus, the operation that the correction steps are only implemented at the end of each time step is not recommended for RK methods with large stage numbers.

By the way, Wu [194] studied the high-order accurate density and pressure positivity preserving schemes under the Courant-Friedrichs-Lewy condition for the relativistic magnetohydrodynamics (RMHD) and reveals for the first time that the discrete divergence-free condition is closely connected with the density and pressure positivity-preserving property in theory. His formulation of the convexity, scaling invariance and Lax-Friedrichs splitting property can be hopefully used as the building block for high-order accuracy design of the MPP flux-corrected limiting procedure of MHD system. Moreover, a first order finite volume Lax-Friedrichs type scheme ((69) of Ranocha [143]) update procedure using an explicit Euler step in time, where the flux of the first order finite volume scheme is reformulated by logarithmic mean, is proved to be positivity-preserving. By numerical experiments, Ranocha [143] investigated robustness properties of entropy stable numerical fluxes and positivity preservation for compressible Euler equations via several entropy conservative fluxes enhanced with local Lax-Friedrichs type dissipation operators. This technique, combined with the entropy consistent schemes [38, 39, 115, 191, 192] for MHD equations, is expected to provide us entropy-stable positivity-preserving schemes for MHD system.