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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.

References

  1. Abgrall, R., Karni, S.: Computations of compressible multifluids. J. Comput. Phys. 169, 594–623 (2001)

    Article  MathSciNet  Google Scholar 

  2. Abgrall, R., Karni, S.: Ghost-fluids for the poor: a single fluid algorithm for multifluids. In: Hyperbolic Problems: Theory, Numerics, Applications. Birkhäuser, Basel (2001)

  3. Abgrall, R., Saurel, R.: Discrete equations for physical and numerical compressible multiphase mixtures. J. Comput. Phys. 186, 361–396 (2003)

    Article  MathSciNet  Google Scholar 

  4. Allaire, G., Clerc, S., Kokh, S.: A five-equation model for the simulation of interfaces between compressible fluids. J. Comput. Phys. 181, 577–616 (2002)

    Article  MathSciNet  Google Scholar 

  5. Baer, M.R., Nunziato, J.W.: A two-phase mixture theory for the deflagration-to-detonation transition (ddt) in reactive granular materials. Int. J. Multiphas. Flow 12, 861–889 (1986)

    Article  Google Scholar 

  6. Castro Díaz, M. J., Kurganov, A., Morales de Luna, T.: Path-conservative central-upwind schemes for nonconservative hyperbolic systems. ESAIM Math. Model. Numer. Anal. 53, 959–985 (2019)

  7. Cheng, J., Zhang, F., Liu, T.G.: A quasi-conservative discontinuous Galerkin method for solving five equation model of compressible two-medium flows. J. Sci. Comput. 85, 12–35 (2020)

    Article  MathSciNet  Google Scholar 

  8. Chertock, A., Cui, S.M., Kurganov, A., Özcan, C.N., Admor, E.T.: Well-balanced schemes for the Euler equations with gravitation: conservative formulation using global fluxes. J. Comput. Phys. 358, 36–52 (2018)

  9. Chertock, A., Karni, S., Kurganov, A.: Interface tracking method for compressible multifluids. M2AN Math. Model. Numer. Anal. 42, 991–1019 (2008)

    Article  MathSciNet  Google Scholar 

  10. Chu, S., Kurganov, A., Na, M.Y.: Fifth-order a-weno finite-difference schemes based on the path-conservative central-upwind method (in preparation)

  11. Don, W.S., Li, D.-M., Gao, Z., Wang, B.-S.: A characteristic-wise alternative WENO-Z finite difference scheme for solving the compressible multicomponent non-reactive flows in the overestimated quasi-conservative form. J. Sci. Comput. 82, 24 (2020)

  12. Fedkiw, R.P., Aslam, T., Merriman, B., Osher, S.: A non-oscillatory Eulerian approach to interfaces in multimaterial flows (the ghost fluid method). J. Comput. Phys. 152, 457–492 (1999)

    Article  MathSciNet  Google Scholar 

  13. Gottlieb, S., Ketcheson, D., Shu, C.-W.: Strong Stability Preserving Runge–Kutta and Multistep Time Discretizations. World Scientific Publishing CoWorld Scientific, Hackensack (2011)

    Book  Google Scholar 

  14. Gottlieb, S., Shu, C.-W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89–112 (2001)

    Article  MathSciNet  Google Scholar 

  15. Haas, J.-F., Sturtevant, B.: Interaction of weak shock waves with cylindrical and spherical gas inhomogeneities. J. Fluid Mech. 181, 41–76 (1987)

    Article  Google Scholar 

  16. He, Z.W., Li, L., Zhang, Y.S., Tian, B.L.: Consistent implementation of characteristic flux-split based finite difference method for compressible multi-material gas flows. Comput. Fluids 168, 190–200 (2018)

    Article  MathSciNet  Google Scholar 

  17. Hesthaven, J.S.: Numerical Methods for Conservation Laws, vol. 18 of Computational Science and Engineering. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA (2018)

  18. Jiang, Y., Shu, C.-W., Zhang, M.P.: An alternative formulation of finite difference weighted ENO schemes with Lax–Wendroff time discretization for conservation laws. SIAM J. Sci. Comput. 35, A1137–A1160 (2013)

    Article  MathSciNet  Google Scholar 

  19. Johnsen, E., Colonius, T.: Implementation of WENO schemes in compressible multicomponent flow problems. J. Comput. Phys. 219, 715–732 (2006)

    Article  MathSciNet  Google Scholar 

  20. Karni, S.: Hybrid multifluid algorithms. SIAM J. Sci. Comput. 17, 1019–1039 (1996)

    Article  MathSciNet  Google Scholar 

  21. Kurganov, A., Lin, C.-T.: On the reduction of numerical dissipation in central-upwind schemes. Commun. Comput. Phys. 2, 141–163 (2007)

    MathSciNet  MATH  Google Scholar 

  22. Kurganov, A., Noelle, S., Petrova, G.: Semidiscrete central-upwind schemes for hyperbolic conservation laws and Hamilton–Jacobi equations. SIAM J. Sci. Comput. 23, 707–740 (2001)

    Article  MathSciNet  Google Scholar 

  23. Lei, X., Du, Z., Li, J.: The simulation of compressible multi-fluid flows by a GRP-based energy-splitting method. Comput. Fluids 181, 416–428 (2019)

    Article  MathSciNet  Google Scholar 

  24. Lei, X., Li, J.: An energy-splitting high order numerical method for multi-material flows. arXiv:2010.01515 (2020)

  25. LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems, Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)

  26. Lie, K.-A., Noelle, S.: On the artificial compression method for second-order nonoscillatory central difference schemes for systems of conservation laws. SIAM J. Sci. Comput. 24, 1157–1174 (2003)

    Article  MathSciNet  Google Scholar 

  27. Liu, H.: A numerical study of the performance of alternative weighted ENO methods based on various numerical fluxes for conservation law. Appl. Math. Comput. 296, 182–197 (2017)

    MathSciNet  MATH  Google Scholar 

  28. Liu, H., Qiu, J.: Finite difference Hermite WENO schemes for conservation laws, II: an alternative approach. J. Sci. Comput. 66, 598–624 (2016)

    Article  MathSciNet  Google Scholar 

  29. Mulder, W., Osher, S., Sethian, J.A.: Computing interface motion in compressible gas dynamics. J. Comput. Phys. 100, 209–228 (1992)

    Article  MathSciNet  Google Scholar 

  30. Nessyahu, H., Tadmor, E.: Nonoscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87, 408–463 (1990)

    Article  MathSciNet  Google Scholar 

  31. Nonomura, T., Fujii, K.: Characteristic finite-difference WENO scheme for multicomponent compressible fluid analysis: overestimated quasi-conservative formulation maintaining equilibriums of velocity, pressure, and temperature. J. Comput. Phys. 340, 358–388 (2017)

    Article  MathSciNet  Google Scholar 

  32. Nonomura, T., Morizawa, S., Terashima, H., Obayashi, S., Fujii, K.: Numerical (error) issues on compressible multicomponent flows using a high-order differencing scheme: weighted compact nonlinear scheme. J. Comput. Phys. 231, 3181–3210 (2012)

    Article  MathSciNet  Google Scholar 

  33. Quirk, J., Karni, S.: On the dynamics of a shock–bubble interaction. J. Fluid Mech. 318, 129–163 (1996)

    Article  Google Scholar 

  34. Saurel, R., Abgrall, R.: A multiphase Godunov method for compressible multifluid and multiphase flows. J. Comput. Phys. 150, 425–467 (1999)

    Article  MathSciNet  Google Scholar 

  35. Saurel, R., Abgrall, R.: A simple method for compressible multifluid flows. SIAM J. Sci. Comput. 21, 1115–1145 (1999)

    Article  MathSciNet  Google Scholar 

  36. Shyue, K.M.: An efficient shock-capturing algorithm for compressible multicomponent problems. J. Comput. Phys. 142, 208–242 (1998)

    Article  MathSciNet  Google Scholar 

  37. Sweby, P.K.: High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal. 21, 995–1011 (1984)

    Article  MathSciNet  Google Scholar 

  38. Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction, 3rd edn. Springer, Berlin, Heidelberg (2009)

  39. Wang, B.-S., Don, W.S., Garg, N.K., Kurganov, A.: Fifth-order A-WENO finite-difference schemes based on a new adaptive diffusion central numerical flux. SIAM J. Sci. Comput. 42, A3932–A3956 (2020)

    Article  MathSciNet  Google Scholar 

  40. Wang, B.-S., Li, P., Gao, Z., Don, W.S.: An improved fifth order alternative WENO-Z finite difference scheme for hyperbolic conservation laws. J. Comput. Phys. 374, 469–477 (2018)

    Article  MathSciNet  Google Scholar 

  41. Wang, C.W., Shu, C.-W.: An interface treating technique for compressible multi-medium flow with Runge–Kutta discontinuous Galerkin method. J. Comput. Phys. 229, 8823–8843 (2010)

    Article  MathSciNet  Google Scholar 

Download references

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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