Free-Stream Preserving Finite Difference Schemes for Ideal Magnetohydrodynamics on Curvilinear Meshes

Abstract

In this paper, a high order free-stream preserving finite difference weighted essentially non-oscillatory (WENO) scheme is developed for the ideal magnetohydrodynamic (MHD) equations on curvilinear meshes. Under the constrained transport framework, magnetic potential evolved by a Hamilton–Jacobi (H–J) equation is introduced to control the divergence error. In this work, we use the alternative formulation of WENO scheme (Christlieb et al. in SIAM J Sci Comput 40(4):A2631–A2666, 2018) for the nonlinear hyperbolic conservation law, and design a novel method to solve the magnetic potential. Theoretical derivation and numerical results show that the scheme can preserve free-stream solutions of MHD equations, and reduce error more effectively than the standard finite difference WENO schemes for such problems.

Appendix: WENO method

For the sake of completeness, we give the formula of WENO interpolation for a hyperbolic system and WENO approximation for a H–J equation. Here, we only show the 1D case with fifth order accuracy. For 2D case, it can be obtained in a dimension-by-dimension fashion, in which we fix one index and do the process in the other direction.

1.1 WENO Interpolation for a System

Given the point values $\{{\mathbf{q }}_i\}$, we will approximate the values at half points ${\mathbf{q }}_{i+1/2}^{\pm }$ :

• 1. Compute an average state ${\mathbf{q }}_{i+1/2}$, using either the simple arithmetic mean or a Roe average. Construct the right and left eigenvectors of the Jacobian $\partial \widetilde{{\mathbf{f }}} / \partial {\mathbf{q }}$, and denote their matrices by

  $$\begin{aligned} R_{i+1/2}=R\left( {\mathbf{q }}_{i+1/2}\right) ,\quad R^{-1}_{i+1/2}=R^{-1}\left( {\mathbf{q }}_{i+1/2}\right) . \end{aligned}$$

• 2. Project the conserved quantities ${\mathbf{q }}$ to the local characteristic variables ${\mathbf{v }}$,

  $$\begin{aligned} {\mathbf{v }}_{j} = R^{-1}_{i+1/2} {\mathbf{q }}_{j}, \quad {\text {for}} j = i-2, \ldots , i+3. \end{aligned}$$

• 3. Perform a scalar WENO interpolation on each component of the characteristic variable ${\mathbf{v }}_{j}$ to obtain the corresponding component of ${\mathbf{v }}_{i+1/2}^{\pm }$. Here, the procedure of a fifth-order WENO interpolation to obtain the k-th component $v^{-}_{k,i+1/2}$ is described:

  • (a) Choose one big stencil as $S = \{x_{i-2},\ldots ,x_{i+2}\}$, and three small stencils as $S^{(r)} _{i}= \{x_{i-r},\ldots , x_{i-r+2}\}$, $r=0,1,2$. On each small stencil, the standard interpolation gives

    $$\begin{aligned} v^{(0)}_{k,i+1/2}&= \frac{3}{8}v_{k,i} +\frac{3}{4}v_{k,i+1} -\frac{1}{8}v_{k,i+2},\\ v^{(1)}_{k,i+1/2}&= -\frac{1}{8}v_{k,i-1} +\frac{3}{4}v_{k,i} +\frac{3}{8}v_{k,i+1},\\ v^{(2)}_{k,i+1/2}&= \frac{3}{8}v_{k,i-2} -\frac{5}{4}v_{k,i-1} +\frac{15}{8}v_{k,i}, \\ v^{\text {big}}_{k,i+1/2}&= d_{0} v^{(0)}_{k,i+1/2} +d_{1} v^{(1)}_{k,i+1/2} +d_{2} v^{(2)}_{k,i+1/2}, \end{aligned}$$

    with the linear weights being $d_{0}={5}/{16}$, $d_{1}={5}/{8}$ and $d_{2}={1}/{16}$.

  • (b) Compute nonlinear weights $\omega _r$ from the linear weights $d_r$,

    $$\begin{aligned} \omega _{r}=\frac{\alpha _{r}}{\alpha _{0}+\alpha _{1}+\alpha _{2}}, \quad \alpha _{r}=\frac{d_{r}}{(IS_{r}+\epsilon )^2}, \quad r=0,1,2, \end{aligned}$$

    where $\epsilon = 10^{-6}$ is used to avoid division by zero, and the smoothness indicators are given by

    $$\begin{aligned} IS_{0}&= \frac{13}{12}\left( v_{k,i}-2v_{k,i+1}+v_{k,i+2}\right) ^2 +\frac{1}{4}\left( 3v_{k,i}-4v_{k,i+1}+v_{k,i+2}\right) ^2,\\ IS_{1}&= \frac{13}{12}\left( v_{k,i-1}-2v_{k,i}+v_{k,i+1}\right) ^2 +\frac{1}{4}\left( v_{k,i}-v_{k,i+1}\right) ^2,\\ IS_{2}&= \frac{13}{12}\left( v_{k,i-2}-2v_{k,i-1}+v_{k,i}\right) ^2 +\frac{1}{4}\left( v_{k,i-2}-4v_{k,i-1}+3v_{k,i}\right) ^2. \end{aligned}$$

  • (c) The WENO interpolation is defined by

    $$\begin{aligned} v^{-}_{k,i+1/2}=\sum _{r=0}^{2}\omega _{r}v^{(r)}_{k,i+1/2}. \end{aligned}$$

• 4. Finally, project ${\mathbf{v }}_{i+1/2}^{\pm }$ back to the conserved quantities,

  $$\begin{aligned} {\mathbf{q }}_{i+1/2}^{\pm } = R_{i+1/2} {\mathbf{v }}_{i+1/2}^{\pm }. \end{aligned}$$

Note that the process to obtain ${\mathbf{v }}^{+}_{i+1/2}$ is mirror-symmetric to the procedure described above, with respect to the target point $x_{i+1/2}$.

1.2 WENO Approximation for H–J Equations

Given th point values $\{\phi _i\}$, we want to approximate the derivative $\partial _{\xi }\phi$ at $\xi = \xi _{i}$. Introduce $\Delta ^{+} \phi _{k}=\phi _{k+1}-\phi _{k}$ and $\Delta ^{-} \phi _{k}=\phi _{k}-\phi _{k-1}$. Then, the fifth order approximation of $\partial _{\xi }\phi ^{+}_{i}$ can be obtained as follows.

• 1. There are three small stencils $S^{(r)}_{i}=\{\xi _{i-2+r},\ldots ,\xi _{i+1+r}\}$, $r=0,1,2$. On each stencil, we can obtain the approximation to $\partial _\xi \phi _{i}$.

  $$\begin{aligned} \partial _\xi \phi _{i}^{(0)}&= \frac{1}{\Delta \xi } \left( \frac{1}{6}\phi _{i-2}-\phi _{i-1}+\frac{1}{2}\phi _{i}+\frac{1}{3}\phi _{i+1} \right) ,\\ \partial _\xi \phi _{i}^{(1)}&= \frac{1}{\Delta \xi } \left( -\frac{1}{3}\phi _{i-1}-\frac{1}{2}\phi _{i}+\phi _{i+1}-\frac{1}{6}\phi _{i+2} \right) ,\\ \partial _\xi \phi _{i}^{(2)}&= \frac{1}{\Delta \xi } \left( -\frac{11}{6}\phi _{i}+3\phi _{i+1}-\frac{3}{2}\phi _{i+2}+\frac{1}{3}\phi _{i+3} \right) , \\ \partial _\xi \phi _{i} ^{big}&= d_0 \partial _\xi \phi _{i}^{(0)} + d_1 \partial _\xi \phi _{i}^{(1)} + d_2 \partial _\xi \phi _{i}^{(2)} , \end{aligned}$$

  with linear weights $d_0=3/10$, $d_1=6/10$ and $d_2=1/10$.

• 2. Get the nonlinear weights

  $$\begin{aligned} \omega _{r}=\frac{\alpha _{r}}{\alpha _{0}+\alpha _{1}+\alpha _{2}}, \quad \alpha _{r}=\frac{d_r}{\left( \epsilon +IS_{r}\right) ^{2}}, \quad r=0,1,2. \end{aligned}$$

  Here, we take $\epsilon =10^{-6}$. And the smoothness indicators $IS_{r}$ are defined by

  $$\begin{aligned} IS_{0}&=13(b-a)^{2}+3(3b- a)^{2}, \\ IS_{1}&=13(c-b)^{2}+3( c+ b)^{2}, \\ IS_{2}&=13(d-c)^{2}+3( d-3c)^{2} , \end{aligned}$$

  and the parameters have form as

  $$\begin{aligned} a&=\frac{1}{ \Delta \xi } \Delta ^+ \Delta ^- \phi _{i-1}, \quad b =\frac{1}{ \Delta \xi } \Delta ^+ \Delta ^- \phi _{i }, \\ c&=\frac{1}{ \Delta \xi } \Delta ^+ \Delta ^- \phi _{i+1}, \quad d =\frac{1}{ \Delta \xi } \Delta ^+ \Delta ^- \phi _{i+2}. \end{aligned}$$

• 3. Finally, we can obtain the WENO approximation

  $$\begin{aligned} \partial _\xi \phi ^+_{i}=\sum ^2_{r=0}\omega _{r} {\partial _\xi \phi _{i}}^{(r)}. \end{aligned}$$

The process to obtain $\partial _{\xi }\phi ^{-}_{i}$ is mirror-symmetric to that for $\partial _{\xi }\phi ^{+}_{i}$ with respect to the target point $\xi _{i}$.