×

Parametrized positivity preserving flux limiters for the high order finite difference WENO scheme solving compressible Euler equations. (English) Zbl 1383.76365

Summary: In this paper, we develop parametrized positivity satisfying flux limiters for the high order finite difference Runge-Kutta weighted essentially non-oscillatory scheme solving compressible Euler equations to maintain positive density and pressure. Negative density and pressure, which often leads to simulation blow-ups or nonphysical solutions, emerges from many high resolution computations in some extreme cases. The methodology we propose in this paper is a nontrivial generalization of the parametrized maximum principle preserving flux limiters for high order finite difference schemes solving scalar hyperbolic conservation laws. To preserve the maximum principle, the high order flux is limited towards a first order monotone flux, where the limiting procedures are designed by decoupling linear maximum principle constraints. High order schemes with such flux limiters are shown to preserve the high order accuracy via local truncation error analysis and by extensive numerical experiments with mild CFL constraints. The parametrized flux limiting approach is generalized to the Euler system to preserve the positivity of density and pressure of numerical solutions via decoupling some nonlinear constraints. Compared with existing high order positivity preserving approaches, our proposed algorithm is positivity preserving by the design; it is computationally efficient and maintains high order spatial and temporal accuracy in our extensive numerical tests. Numerical tests are performed to demonstrate the efficiency and effectiveness of the proposed new algorithm.

MSC:

76M20 Finite difference methods applied to problems in fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
76Nxx Compressible fluids and gas dynamics
35Q31 Euler equations

Software:

SHASTA
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Borges, R., Carmona, M., Costa, B., Don, W.S.: An improved weighted essentially non-oscillatory scheme for hyperbolic conservation laws. J. Comput. Phys. 227, 3191-3211 (2008) · Zbl 1136.65076 · doi:10.1016/j.jcp.2007.11.038
[2] Boris, J.P., Book, D.L.: Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works. J. Comput. Phys. 11, 38-69 (1973) · Zbl 0251.76004 · doi:10.1016/0021-9991(73)90147-2
[3] Chakravarthy, S.R., Osher, S.: High Resolution Applications of the Osher Upwind Scheme for the Euler Equations. AIAA (1983) · Zbl 0526.76074
[4] Einfeldt, B., Munz, C.-D., Roe, P.L., Sjögreen, B.: On Godunov-type methods near low densities. J. Comput. Phys. 92, 273-295 (1991) · Zbl 0709.76102 · doi:10.1016/0021-9991(91)90211-3
[5] Engquist, B., Osher, S.: Stable and entropy satisfying approximations for transonic flow calculations. Math. Comput. 34, 45-75 (1980) · Zbl 0438.76051 · doi:10.1090/S0025-5718-1980-0551290-1
[6] Engquist, B., Sjögreen, B.: The convergence rate of finite difference schemes in the presence of shocks. SIAM J. Numer. Anal. 35, 2464-2485 (1998) · Zbl 0922.76254 · doi:10.1137/S0036142997317584
[7] Fisher, T.C., Carpenter, M.H.: High-order entropy stable finite difference schemes for nonlinear conservation laws: finite domains. J. Comput. Phys. 252, 518-557 (2013) · Zbl 1349.65293 · doi:10.1016/j.jcp.2013.06.014
[8] Fjordholm, U.S., Mishra, S., Tadmor, E.: ENO reconstruction and ENO interpolation are stable. Found. Comput. Math. 13, 139-159 (2013) · Zbl 1273.65120 · doi:10.1007/s10208-012-9117-9
[9] Ha, Y., Gardner, C.L.: Positive scheme numerical simulation of high mach number astrophysical jets. J. Sci. Comput. 34, 247-259 (2008) · Zbl 1133.76029 · doi:10.1007/s10915-007-9165-5
[10] Harten, A., Engquist, B., Osher, S., Chakravarthy, S.R.: Uniformly high order accurate essentially non-oscillatory schemes, III. J. Comput. Phys. 71, 231-303 (1987) · Zbl 0652.65067 · doi:10.1016/0021-9991(87)90031-3
[11] Henrick, A.K., Aslam, T.D., Powers, J.M.: Mapped weighted essentially non-oscillatory schemes: achieving optimal order near critical points. J. Comput. Phys. 207, 542-567 (2005) · Zbl 1072.65114 · doi:10.1016/j.jcp.2005.01.023
[12] Hu, C., Shu, C.-W.: Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys. 150, 97-127 (1999) · Zbl 0926.65090 · doi:10.1006/jcph.1998.6165
[13] Hu, X.Y., Adams, N.A., Shu, C.-W.: Positivity-preserving method for high-order conservative schemes solving compressible Euler equations. J. Comput. Phys. 242, 169-180 (2013) · Zbl 1311.76088 · doi:10.1016/j.jcp.2013.01.024
[14] Jiang, G.-S., Shu, C.-W.: Efficient implementation of weighted ENO schemes. J. Comput. Phys. 126, 202-228 (1996) · Zbl 0877.65065 · doi:10.1006/jcph.1996.0130
[15] Liang, C., Xu, Z.: Parametrized maximum principle preserving flux limiters for high order schemes solving multi-dimensional scalar hyperbolic conservation laws. J. Sci. Comput. 58, 41-60 (2014) · Zbl 1286.65102 · doi:10.1007/s10915-013-9724-x
[16] Linde, T., Roe, P.L.: Robust Euler codes. In: Thirteenth Computational Fluid Dynamics Conference, AIAA Paper-97-2098 (1997) · Zbl 1282.76128
[17] Liu, X.-D., Osher, S.: Nonoscillatory high order accurate self-similar maximum principle satisfying shock capturing schemes I. SIAM J. Numer. Anal. 33, 760-779 (1996) · Zbl 0859.65091 · doi:10.1137/0733038
[18] Perthame, B.: Second-order boltzmann schemes for compressible euler equations in one and two space dimensions. SIAM J. Numer. Anal. 29, 1-19 (1992) · Zbl 0744.76088 · doi:10.1137/0729001
[19] Perthame, B., Shu, C.-W.: On positivity preserving finite volume schemes for Euler equations. Numer. Math. 73, 119-130 (1996) · Zbl 0857.76062 · doi:10.1007/s002110050187
[20] Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In: Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, pp. 325-432 (1998) · Zbl 0927.65111
[21] Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439-471 (1988) · Zbl 0653.65072 · doi:10.1016/0021-9991(88)90177-5
[22] Sweby, P.K.: High resolution schemes using flux limiters for hyperbolic conservation laws. SIAM J. Numer. Anal. 21, 995-1011 (1984) · Zbl 0565.65048 · doi:10.1137/0721062
[23] Van Leer, B.: Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme. J. Comput. Phys. 14, 361-370 (1974) · Zbl 0276.65055 · doi:10.1016/0021-9991(74)90019-9
[24] Wang, C., Zhang, X., Shu, C.-W., Ning, J.: Robust high order discontinuous Galerkin schemes for two-dimensional gaseous detonations. J. Comput. Phys. 231, 653-665 (2012) · Zbl 1243.80011 · doi:10.1016/j.jcp.2011.10.002
[25] Wang, W., Shu, C.-W., Yee, H., Sjögreen, B.: High-order well-balanced schemes and applications to non-equilibrium flow. J. Comput. Phys. 228, 6682-6702 (2009) · Zbl 1261.76024 · doi:10.1016/j.jcp.2009.05.028
[26] Xiong, T., Qiu, J.-M., Xu, Z.: A parametrized maximum principle preserving flux limiter for finite difference RK-WENO schemes with applications in incompressible flows. J. Comput. Phys. 252, 310-331 (2013) · Zbl 1349.76553 · doi:10.1016/j.jcp.2013.06.026
[27] Xiong, T., Qiu, J.-M., Xu, Z., Christlieb, A.: High order maximum principle preserving semi-Lagrangian finite difference WENO schemes for the Vlasov equation. J. Comput. Phys. 273, 618-639 (2014) · Zbl 1351.76193 · doi:10.1016/j.jcp.2014.05.033
[28] Xu, Z.: Parametrized maximum principle preserving flux limiters for high order scheme solving hyperbolic conservation laws: one-dimensional scalar problem. Math. Comput. 83, 2213-2238 (2014) · Zbl 1300.65063 · doi:10.1090/S0025-5718-2013-02788-3
[29] Zhang, X., Shu, C.-W.: On maximum-principle-satisfying high order schemes for scalar conservation laws. J. Comput. Phys. 229, 3091-3120 (2010) · Zbl 1187.65096 · doi:10.1016/j.jcp.2009.12.030
[30] Zhang, X., Shu, C.-W.: On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. J. Comput. Phys. 229, 8918-8934 (2010) · Zbl 1282.76128 · doi:10.1016/j.jcp.2010.08.016
[31] Zhang, X., Shu, C.-W.: Maximum-principle-satisfying and positivity-preserving high-order schemes for conservation laws: survey and new developments. Proc. R. Soc. A Math. Phys. Eng. Sci. 467, 2752-2776 (2011) · Zbl 1222.65107 · doi:10.1098/rspa.2011.0153
[32] Zhang, X., Shu, C.-W.: Positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations with source terms. J. Comput. Phys. 230, 1238-1248 (2011) · Zbl 1391.76375 · doi:10.1016/j.jcp.2010.10.036
[33] Zhang, X., Shu, C.-W.: Positivity-preserving high order finite difference WENO schemes for compressible Euler equations. J. Comput. Phys. 231, 2245-2258 (2012) · Zbl 1426.76493 · doi:10.1016/j.jcp.2011.11.020
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.