# zbMATH — the first resource for mathematics

Positivity-preserving Lagrangian scheme for multi-material compressible flow. (English) Zbl 1349.76439
Summary: Robustness of numerical methods has attracted an increasing interest in the community of computational fluid dynamics. One mathematical aspect of robustness for numerical methods is the positivity-preserving property. At high Mach numbers or for flows near vacuum, solving the conservative Euler equations may generate negative density or internal energy numerically, which may lead to nonlinear instability and crash of the code. This difficulty is particularly profound for high order methods, for multi-material flows and for problems with moving meshes, such as the Lagrangian methods. In this paper, we construct both first order and uniformly high order accurate conservative Lagrangian schemes which preserve positivity of physically positive variables such as density and internal energy in the simulation of compressible multi-material flows with general equations of state (EOS). We first develop a positivity-preserving approximate Riemann solver for the Lagrangian scheme solving compressible Euler equations with both ideal and non-ideal EOS. Then we design a class of high order positivity-preserving and conservative Lagrangian schemes by using the essentially non-oscillatory (ENO) reconstruction, the strong stability preserving (SSP) high order time discretizations and the positivity-preserving scaling limiter which can be proven to maintain conservation and uniformly high order accuracy and is easy to implement. One-dimensional and two-dimensional numerical tests for the positivity-preserving Lagrangian schemes are provided to demonstrate the effectiveness of these methods.

##### 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 76K05 Hypersonic flows 76N99 Compressible fluids and gas dynamics
##### Software:
FIVER; AUSM; HLLE
Full Text:
##### References:
 [1] Batten, P.; Clarke, N.; Lambert, C.; Causon, D., On the choice of wavespeeds for the HLLC Riemann solver, SIAM J. Sci. Comput., 18, 1553-1570, (1997) · Zbl 0992.65088 [2] Bezard, F.; Despres, B., An entropic solver for ideal Lagrangian magnetohydrodynamics, J. Comput. Phys., 154, 65-89, (1999) · Zbl 0952.76053 [3] Caramana, E. J.; Burton, D. E.; Shashkov, M. J.; Whalen, P. P., The construction of compatible hydrodynamics algorithms utilizing conservation of total energy, J. Comput. Phys., 146, 227-262, (1998) · Zbl 0931.76080 [4] Carré, G.; Del Pino, S.; Després, B.; Labourasse, E., A cell-centered Lagrangian hydrodynamics scheme in arbitrary dimension, J. Comput. Phys., 228, 5160-5183, (2009) · Zbl 1168.76029 [5] Cheng, J.; Shu, C.-W., A high order ENO conservative Lagrangian type scheme for the compressible Euler equations, J. Comput. Phys., 227, 1567-1596, (2007) · Zbl 1126.76035 [6] Cheng, J.; Shu, C.-W., A high order accurate conservative remapping method on staggered meshes, Appl. Numer. Math., 58, 1042-1060, (2008) · Zbl 1225.76219 [7] Cheng, J.; Shu, C.-W., A third order conservative Lagrangian type scheme on curvilinear meshes for the compressible Euler equation, Commun. Comput. Phys., 4, 1008-1024, (2008) · Zbl 1364.76111 [8] Dukowicz, J. K.; Meltz, B. J.A., Vorticity errors in multidimensional Lagrangian codes, J. Comput. Phys., 99, 115-134, (1992) · Zbl 0743.76058 [9] 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 [10] Estivalezes, J. L.; Villedieu, P., High-order positivity preserving kinetic schemes for the compressible Euler equations, SIAM J. Numer. Anal., 33, 2050-2067, (1996) · Zbl 0863.35081 [11] Farhat, C.; Gerbeau, J.-F.; Rallu, A., FIVER: A finite volume method based on exact two-phase Riemann problems and sparse grids for multi-material flows with large density jumps, J. Comput. Phys., 231, 6360-6379, (2012) · Zbl 1284.76264 [12] Gallice, G., Positive and entropy stable Godunov-type schemes for gas dynamics and MHD equations in Lagrangian or Eulerian coordinates, Numer. Math., 94, 673-713, (2003) · Zbl 1092.76044 [13] Gressier, J.; Villedieu, P.; Moschetta, J.-M., Positivity of flux vector splitting schemes, J. Comput. Phys., 155, 199-220, (1999) · Zbl 0953.76064 [14] Harten, A.; Lax, P. D.; Van Leer, B., On upstream differencing and Godunov-type schemes for hyperbolic conservation laws, SIAM Rev., 25, 35-61, (1983) · Zbl 0565.65051 [15] Jiang, G.-S.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126, 202-228, (1996) · Zbl 0877.65065 [16] Linde, T.; Roe, P. L., Robust Euler codes, (Thirteenth Computational Fluid Dynamics Conference, (1997), AIAA), Paper 97-2098 [17] Liou, M., A sequel to AUSM: AUSM+, J. Comput. Phys., 129, 364-382, (1996) · Zbl 0870.76049 [18] Liu, W.; Cheng, J.; Shu, C.-W., High order conservative Lagrangian schemes with Lax-Wendroff type time discretization for the compressible Euler equations, J. Comput. Phys., 228, 8872-8891, (2009) · Zbl 1287.76181 [19] Maire, P.-H., A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes, J. Comput. Phys., 228, 2391-2425, (2009) · Zbl 1156.76434 [20] Munz, C. D., On Godunov-type schemes for Lagrangian gas dynamics, SIAM J. Numer. Anal., 31, 17-42, (1994) · Zbl 0796.76057 [21] Perthame, B., Boltzmann type schemes for gas dynamics and the entropy property, SIAM J. Numer. Anal., 27, 1405-1421, (1990) · Zbl 0714.76078 [22] 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 [23] Perthame, B.; Shu, C.-W., On positivity preserving finite volume schemes for Euler equations, Numer. Math., 73, 119-130, (1996) · Zbl 0857.76062 [24] Saurel, R.; Abgrall, R., A simple method for compressible multifluid flows, SIAM J. Sci. Comput., 21, 1115-1145, (1999) · Zbl 0957.76057 [25] Sedov, L. I., Similarity and dimensional methods in mechanics, (1959), Academic Press New York · Zbl 0121.18504 [26] Shyue, K.-M., A fluid-mixture type algorithm for barotropic two-fluid flow problems, J. Comput. Phys., 200, 718-748, (2004) · Zbl 1115.76344 [27] Tang, T.; Xu, K., Gas-kinetic schemes for the compressible Euler equations I: positivity-preserving analysis, Z. Angew. Math. Phys., 50, 258-281, (1999) · Zbl 0958.76080 [28] Toro, E. F., Riemann solvers and numerical methods for fluid dynamics, (1999), Springer-Verlag Berlin · Zbl 0923.76004 [29] 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 [30] 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 [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, 467, 2752-2776, (2011) · Zbl 1222.65107 [32] 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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.