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Hybrid Multifluid Algorithms Based on the Path-Conservative Central-Upwind Scheme

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Abstract

We develop new hybrid numerical algorithms for compressible multicomponent fluids problem. The fluid components are assumed to be immiscible and are separated by material interface. We track the location of the interface using the level set approach and replace the energy equation in the original model with the corresponding pressure equation in its neighborhoods. In these neighboring areas we solve the resulting nonconservative system using a path-conservative central-upwind scheme, while in the rest of the computational domain, a central-upwind scheme is used to numerically solve the original conservative system. We first develop a finite-volume method of the second order and then extend it to the fifth order via the finite-difference alternative WENO (A-WENO) framework. In order to reduce oscillations, we switch from A-WENO back to second-order central-upwind scheme in certain nonsmooth parts of the computational solution. We illustrate the performance of the new hybrid methods on a number of one- and two-dimensional examples including the shock–bubble interaction tests.

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Data and Software Availability

The data that support the findings of this study and FORTRAN codes developed by the authors and used to obtain all of the presented numerical results are available from the corresponding author upon reasonable request.

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Funding

The work of A. Chertock was supported in part by NSF Grant DMS-1818684. The work of A. Kurganov was supported in part by NSFC Grants 11771201 and 1201101343, and by the fund of the Guangdong Provincial Key Laboratory of Computational Science and Material Design (No. 2019B030301001).

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Authors and Affiliations

  1. Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA

    Alina Chertock

  2. Department of Mathematics, Southern University of Science and Technology, Shenzhen, 518055, China

    Shaoshuai Chu

  3. Department of Mathematics, SUSTech International Center for Mathematics and Guangdong Provincial Key Laboratory of Computational Science and Material Design, Southern University of Science and Technology, Shenzhen, 518055, China

    Alexander Kurganov

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  1. Alina Chertock
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  2. Shaoshuai Chu
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Correspondence to Shaoshuai Chu.

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Appendix: Semi-discrete Central-Upwind (CU) Scheme

Appendix: Semi-discrete Central-Upwind (CU) Scheme

In this section, we briefly describe the semi-discrete CU scheme for the homogeneous 2-D systems (1.1)–(1.5). The 2-D semi-discrete CU scheme from [8, 21] admits the following flux form:

$$\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}\,\bar{{\varvec{U}}}_{j,k}=-\frac{{\varvec{H}}^x_{{j+\frac{1}{2}},k}-{\varvec{H}}^x_{{j-\frac{1}{2}},k}}{\Delta x}-\frac{{\varvec{H}}^y_{j,{k+\frac{1}{2}}}-{\varvec{H}}^y_{j,{k-\frac{1}{2}}}}{\Delta y}, \end{aligned}$$

where the numerical fluxes are

$$\begin{aligned} {\varvec{H}}^x_{{j+\frac{1}{2}},k}= & {} \frac{a^+_{{j+\frac{1}{2}},k}{\varvec{F}}({\varvec{U}}^\mathrm{E}_{j,k})-a^-_{{j+\frac{1}{2}},k}{\varvec{F}}({\varvec{U}}^\mathrm{W}_{j+1,k})}{a^+_{{j+\frac{1}{2}},k}-a^-_{{j+\frac{1}{2}},k}}+{a^+_{{j+\frac{1}{2}},k}a^-_{{j+\frac{1}{2}},k}}\nonumber \\&\times \left[ \frac{{\varvec{U}}^\mathrm{W}_{j+1,k}-{\varvec{U}}^\mathrm{E}_{j,k}}{a^+_{{j+\frac{1}{2}},k}-a^-_{{j+\frac{1}{2}},k}}-{\varvec{Q}}^x_{{j+\frac{1}{2}},k}\right] ,\nonumber \\ {\varvec{H}}^y_{j,{k+\frac{1}{2}}}= & {} \frac{b^+_{j,{k+\frac{1}{2}}}{\varvec{G}}({\varvec{U}}^\mathrm{N}_{j,k})-b^-_{j,{k+\frac{1}{2}}}{\varvec{G}}({\varvec{U}}^\mathrm{S}_{j,k+1})}{b^+_{j,{k+\frac{1}{2}}}-b^-_{j,{k+\frac{1}{2}}}}+{b^+_{j,{k+\frac{1}{2}}}b^-_{j,{k+\frac{1}{2}}}}\nonumber \\&\times \left[ \frac{{\varvec{U}}^\mathrm{N}_{j,k+1}-{\varvec{U}}^\mathrm{S}_{j,k}}{b^+_{j,{k+\frac{1}{2}}}-b^-_{j,{k+\frac{1}{2}}}}-{\varvec{Q}}^y_{j,{k+\frac{1}{2}}}\right] . \end{aligned}$$
(A.1)

Here, \({\varvec{U}}^\mathrm{E}\), \({\varvec{U}}^\mathrm{W}\), \({\varvec{U}}^\mathrm{N}\), \({\varvec{U}}^\mathrm{S}\) are the approximate point values of \({\varvec{U}}\), which are computed as follows. First, from the available cell averages \(\,\bar{{\varvec{U}}}_{j,k}=\big (\,\bar{\rho }_{j,k},\,\bar{(\rho u)}_{j,k},\,\bar{(\rho v)}_{j,k}, \,\bar{E}_{j,k},\,\bar{(\rho \phi )}_{j,k}\big )^\top \), we compute cell centered values of the velocities u and v and pressure p:

$$\begin{aligned} \begin{aligned}&u_{j,k}=\frac{\,\bar{(\rho u)}_{j,k}}{\,\bar{\rho }_{j,k}},\quad v_{j,k}=\frac{\,\bar{(\rho v)}_{j,k}}{\,\bar{\rho }_{j,k}},\\&p_{j,k}=(\gamma _{j,k}-1)\Bigg [\,\bar{E}_{j,k}-\frac{\big (\,\bar{(\rho u)}_{j,k}\big )^2+\big (\,\bar{(\rho v)}_{j,k}\big )^2}{2\,\bar{\rho }_{j,k}}\Bigg ]-\gamma _{j,k}(p_\infty )_{j,k}, \end{aligned} \end{aligned}$$
(A.2)

and construct a piecewise linear reconstruction applied to the primitive variables \({\varvec{V}}=(\rho ,u,v,p,\phi )^\top \):

$$\begin{aligned} \widetilde{{\varvec{V}}}(x,y)=\,\bar{{\varvec{V}}}_{j,k}+({\varvec{V}}_x)_{j,k}(x-x_j)+({\varvec{V}}_y)_{j,k}(y-y_k),\quad (x,y)\in C_{j,k}. \end{aligned}$$

We then obtain

$$\begin{aligned} \begin{aligned}&{\varvec{V}}^\mathrm{E}_{j,k}=\,\bar{{\varvec{V}}}_{j,k}+\frac{\Delta x}{2}({\varvec{V}}_x)_{j,k},\quad {\varvec{V}}^\mathrm{W}_{j,k}=\,\bar{{\varvec{V}}}_{j,k}-\frac{\Delta x}{2}({\varvec{V}}_x)_{j,k},\\&{\varvec{V}}^\mathrm{N}_{j,k}=\,\bar{{\varvec{V}}}_{j,k}+\frac{\Delta y}{2}({\varvec{V}}_y)_{j,k},\quad {\varvec{V}}^\mathrm{S}_{j,k}=\,\bar{{\varvec{V}}}_{j,k}-\frac{\Delta y}{2}({\varvec{V}}_y)_{j,k}, \end{aligned} \end{aligned}$$
(A.3)

which are the values of \(\widetilde{{\varvec{V}}}\) at midpoints of the edges of the cell \(C_{j,k}\). As in the 1-D case, the numerical derivatives \(({\varvec{V}}_x)_{j,k}\) and \(({\varvec{V}}_y)_{j,k}\) are computed using the generalized minmod limiter as

$$\begin{aligned} \begin{aligned}&({\varvec{V}}_x)_{j,k}=\mathrm{minmod}\left( \theta \,\frac{\,\bar{{\varvec{V}}}_{j+1,k}-\,\bar{{\varvec{V}}}_{j,k}}{\Delta x},\, \frac{\,\bar{{\varvec{V}}}_{j+1,k}-\,\bar{{\varvec{V}}}_{j-1,k}}{2\Delta x},\,\theta \,\frac{\,\bar{{\varvec{V}}}_{j,k}-\,\bar{{\varvec{V}}}_{j-1,k}}{\Delta x}\right) ,\\&({\varvec{V}}_y)_{j,k}=\mathrm{minmod}\left( \theta \,\frac{\,\bar{{\varvec{V}}}_{j,k+1}-\,\bar{{\varvec{V}}}_{j,k}}{\Delta y},\, \frac{\,\bar{{\varvec{V}}}_{j,k+1}-\,\bar{{\varvec{V}}}_{j,k-1}}{2\Delta y},\,\theta \,\frac{\,\bar{{\varvec{V}}}_{j,k}-\,\bar{{\varvec{V}}}_{j,k-1}}{\Delta y}\right) , \end{aligned} \end{aligned}$$

where the minmod function is defined by (2.13) and, as in the 1-D case, applied to the vector quantity \({\varvec{V}}\) in a component-wise manner.

The built-in “anti-diffusion” terms \({\varvec{Q}}^x_{{j+\frac{1}{2}},k}\) and \({\varvec{Q}}^y_{j,{k+\frac{1}{2}}}\) in (A.1) are given by (see [8])

$$\begin{aligned} \begin{aligned}&{\varvec{Q}}^x_{{j+\frac{1}{2}},k}=\mathrm{{minmod}}\left( \frac{{\varvec{U}}^*_{{j+\frac{1}{2}},k}-{\varvec{U}}^\mathrm{E}_{j,k}}{a^+_{{j+\frac{1}{2}},k}-a^-_{{j+\frac{1}{2}},k}}, \frac{{\varvec{U}}^\mathrm{W}_{j+1,k}-{\varvec{U}}^*_{{j+\frac{1}{2}},k}}{a^+_{{j+\frac{1}{2}},k}-a^-_{{j+\frac{1}{2}},k}}\right) ,\\&{\varvec{Q}}^y_{j,{k+\frac{1}{2}}}=\mathrm{{minmod}}\left( \frac{{\varvec{U}}^*_{j,{k+\frac{1}{2}}}-{\varvec{U}}^\mathrm{N}_{j,k}}{b^+_{j,{k+\frac{1}{2}}}-b^-_{j,{k+\frac{1}{2}}}}, \frac{{\varvec{U}}^\mathrm{S}_{j,k+1}-{\varvec{U}}^*_{j,{k+\frac{1}{2}}}}{b^+_{j,{k+\frac{1}{2}}}-b^-_{j,{k+\frac{1}{2}}}}\right) , \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned}&{\varvec{U}}^*_{{j+\frac{1}{2}},k}=\frac{a^+_{{j+\frac{1}{2}},k}{\varvec{U}}^\mathrm{W}_{j+1,k}-a^-_{{j+\frac{1}{2}},k}{\varvec{U}}^\mathrm{E}_{j,k}-\left\{ {\varvec{F}}\big ({\varvec{U}}^\mathrm{W}_{j+1,k}\big )- {\varvec{F}}\big ({\varvec{U}}^\mathrm{E}_{j,k}\big )\right\} }{a^+_{{j+\frac{1}{2}},k}-a^-_{{j+\frac{1}{2}},k}},\\&{\varvec{U}}^*_{j,{k+\frac{1}{2}}}=\frac{b^+_{j,{k+\frac{1}{2}}}{\varvec{U}}^\mathrm{S}_{j,k+1}-b^-_{j,{k+\frac{1}{2}}}{\varvec{U}}^\mathrm{N}_{j,k}-\left\{ {\varvec{G}}\big ({\varvec{U}}^\mathrm{S}_{j,k+1}\big )- {\varvec{G}}\big ({\varvec{U}}^\mathrm{N}_{j,k}\big )\right\} }{b^+_{j,{k+\frac{1}{2}}}-b^-_{j,{k+\frac{1}{2}}}}. \end{aligned} \end{aligned}$$

Finally, \(a^\pm _{{j+\frac{1}{2}},k}\) and \(b^\pm _{j,{k+\frac{1}{2}}}\) are the one-sided local propagation speeds in the x- and y-directions, respectively. They are obtained using the largest/smallest eigenvalues of the Jacobians \(\frac{\partial {\varvec{F}}}{\partial {\varvec{U}}}\) and \(\frac{\partial {\varvec{G}}}{\partial {\varvec{U}}}\). For the reactive Euler systems (1.1)–(1.5), these speeds can be estimated by

$$\begin{aligned}&a^+_{{j+\frac{1}{2}},k}=\max \Big (u^\mathrm{E}_{j,k}+c^E_{j,k},\,u^\mathrm{W}_{j+1,k}+c^W_{j+1,k},\,0\Big ),\quad \\&a^-_{{j+\frac{1}{2}},k}=\min \Big (u^\mathrm{E}_{j,k}-c^E_{j,k},\,u^\mathrm{W}_{j+1,k}-c^W_{j+1,k},\,0\Big ),\\&b^+_{j,{k+\frac{1}{2}}}=\max \Big (u^\mathrm{N}_{j,k}+c^N_{j,k},\,u^\mathrm{S}_{j,k+1}+c^S_{j,k+1},\,0\Big ),\quad \\&b^-_{j,{k+\frac{1}{2}}}=\min \Big (u^\mathrm{N}_{j,k}-c^N_{j,k},\,u^\mathrm{S}_{j,k+1}-c^S_{j,k+1},\,0\Big ), \end{aligned}$$

where \(c=\sqrt{\gamma (p+p_\infty )/\rho }\).

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Chertock, A., Chu, S. & Kurganov, A. Hybrid Multifluid Algorithms Based on the Path-Conservative Central-Upwind Scheme. J Sci Comput 89, 48 (2021). https://doi.org/10.1007/s10915-021-01656-z

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