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Positivity-preserving cell-centered Lagrangian schemes for multi-material compressible flows: from first-order to high-orders. II: The two-dimensional case. (English) Zbl 1351.76128
Summary: This paper is the second part of a series of two. It follows [ibid. 312, 385–415 (2016; Zbl 1351.76127)], in which the positivity-preservation property of methods solving one-dimensional Lagrangian gas dynamics equations, from first-order to high-orders of accuracy, was addressed. This article aims at extending this analysis to the two-dimensional case. This study is performed on a general first-order cell-centered finite volume formulation based on polygonal meshes defined either by straight line edges, conical edges, or any high-order curvilinear edges. Such formulation covers the numerical methods introduced in [G. Carré et al., ibid. 228, No. 14, 5160–5183 (2009; Zbl 1168.76029); the third author et al., SIAM J. Sci. Comput. 29, No. 4, 1781–1824 (2007; Zbl 1251.76028); B. Boutin et al., ESAIM, Proc. 32, 31–55 (2011; Zbl 1235.76079); the first author, “Cell-centered discontinuous Galerkin discretization for two-dimensional Lagrangian hydrodynamics”, Comput. Fluids 64, 64–73 (2012; doi:10.1016/j.compfluid.2012.05.001); the first author et al., J. Comput. Phys. 276, 188–234 (2014; Zbl 1349.76278)]. This positivity study is then extended to high-orders of accuracy. Through this new procedure, scheme robustness is highly improved and hence new problems can be tackled. Numerical results are provided to demonstrate the effectiveness of these methods. It is important to point out that even if this paper is concerned with purely Lagrangian schemes, the theory developed is of fundamental importance for any methods relying on a purely Lagrangian step, as ALE methods or non-direct Euler schemes.

MSC:
76M12 Finite volume methods applied to problems in fluid mechanics
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
76N15 Gas dynamics (general theory)
Software:
CAVEAT; FIVER
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[1] Abgrall, R.; Loubère, R.; Ovadia, J., A Lagrangian discontinuous Galerkin-type method on unstructured meshes to solve hydrodynamics problems, Int. J. Numer. Methods Fluids, 44, 645-663, (2004) · Zbl 1067.76591
[2] Adessio, F. L.; Baumgardner, J. R.; Dukowicz, J. K.; Johnson, N. L.; Kashiwa, B. A.; Rauenzahn, R. M.; Zemach, C., CAVEAT: a computer code for fluid dynamics problems with large distortion and internal slip, (1992), Los Alamos National Laboratory, Technical Report LA-10613-MS, Rev. 1, UC-905
[3] Batten, P.; Clarke, N.; Lambert, C.; Causon, D. M., On the choice of wavespeeds for the HLLC Riemann solver, SIAM J. Sci. Comput., 18, 1553-1570, (1997) · Zbl 0992.65088
[4] Berthon, C.; Dubroca, B.; Sangam, A., A local entropy minimum principle for deriving entropy preserving schemes, SIAM J. Numer. Anal., 50, 2, 468-491, (2012) · Zbl 1423.76291
[5] Boutin, B.; Deriaz, E.; Hoch, P.; Navaro, P., Extension of ALE methodology to unstructured conical meshes, ESAIM Proc., 32, 31-55, (2011) · Zbl 1235.76079
[6] Carré, G.; Delpino, S.; Després, B.; Labourasse, E., A cell-centered Lagrangian hydrodynamics scheme in arbitrary dimension, J. Comput. Phys., 228, 5160-5183, (2009) · Zbl 1168.76029
[7] Cheng, J.; Shu, C.-W., A high order ENO conservative Lagrangian type scheme for the compressible Euler equations, J. Comput. Phys., 227, 2, 1567-1596, (2007) · Zbl 1126.76035
[8] Cheng, J.; Shu, C.-W., A third-order conservative Lagrangian type scheme on curvilinear meshes for the compressible Euler equations, Commun. Comput. Phys., 4, 1008-1024, (2008) · Zbl 1364.76111
[9] Cheng, J.; Shu, C.-W., Positivity-preserving Lagrangian scheme for multi-material compressible flow, J. Comput. Phys., 257, 143-168, (2014) · Zbl 1349.76439
[10] Cheng, J.; Shu, C.-W., Second order symmetry-preserving conservative Lagrangian scheme for compressible Euler equations in two-dimensional cylindrical coordinates, J. Comput. Phys., 272, 245-265, (2014) · Zbl 1349.65367
[11] Cockburn, B.; Lin, S.-Y.; Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems, J. Comput. Phys., 84, 90-113, (1989) · Zbl 0677.65093
[12] Cockburn, B.; Shu, C.-W., The Runge-Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems, J. Comput. Phys., 141, 199-224, (1998) · Zbl 0920.65059
[13] Després, B., Lois de conservation euleriennes, lagrangiennes et méthodes numériques, Mathématiques et Applications, (2010), Springer · Zbl 1348.35002
[14] Després, B.; Mazeran, C., Lagrangian gas dynamics in two dimensions and Lagrangian systems, Arch. Ration. Mech. Anal., 178, 327-372, (2005) · Zbl 1096.76046
[15] Einfeldt, B.; Munz, C.-D.; Roe, P. L.; Sjögreen, B. J., On Godunov-type methods near low densities, J. Comput. Phys., 92, 273-295, (1991) · Zbl 0709.76102
[16] 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
[17] 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
[18] Germain, P., Mécanique, vol. I, (1986), Ellipses
[19] Gurtin, M. E.; Fried, E.; Anand, L., The mechanics and thermodynamics of continua, (2009), Cambridge University Press
[20] Harten, A.; Engquist, B.; Osher, S.; Chakravarthy, S., Uniformly high order essentially non-oscillatory schemes, III, J. Comput. Phys., 71, 231-303, (1987) · Zbl 0652.65067
[21] Horn, R. A.; Johnson, C. R., Matrix analysis, (1985), Cambridge University Press · Zbl 0576.15001
[22] Jiang, G.-S.; Shu, C.-W., Efficient implementation of weighted ENO schemes, J. Comput. Phys., 126, 202-228, (1996) · Zbl 0877.65065
[23] Kamm, J. R.; Timmes, F. X., On efficient generation of numerically robust sedov solutions, (2007), Los Alamos National Laboratory, Technical Report LA-UR-07-2849
[24] Kluth, G.; Després, B., Discretization of hyperelasticity on unstructured mesh with a cell-centered Lagrangian scheme, J. Comput. Phys., 229, 24, 9092-9118, (2010) · Zbl 1427.74029
[25] Van Leer, B., Towards the ultimate conservative difference scheme. V—A second-order sequel to Godunov’s method, J. Comput. Phys., 32, 101-136, (1979) · Zbl 1364.65223
[26] LeVeque, R. J., Finite volume methods for hyperbolic problems, Cambridge Texts in Applied Mathematics, vol. 31, (2002) · Zbl 1010.65040
[27] Liu, W.; Cheng, J.; Shu, C.-W., High order conservative Lagrangian scheme with Lax-Wendroff type time discretization for the compressible Euler equations, J. Comput. Phys., 228, 23, 8872-8891, (2009) · Zbl 1287.76181
[28] Liu, X.-D.; Osher, S.; Chan, T., Weighted essentially non-oscillatory schemes, J. Comput. Phys., 115, 200-212, (1994) · Zbl 0811.65076
[29] Loubère, R., Une Méthode particulaire lagrangienne de type Galerkin discontinu. application à la Mécanique des fluides et l’interaction laser/plasma, (2002), Université Bordeaux I, PhD thesis
[30] Maire, P.-H., Contribution to the numerical modeling of inertial confinement fusion, (2011), Bordeaux University, available at
[31] Maire, P.-H.; Abgrall, R.; Breil, J.; Loubère, R.; Rebourcet, B., A nominally second-order cell-centered Lagrangian scheme for simulating elastic-plastic flows on two-dimensional unstructured grids, J. Comput. Phys., 235, 626-665, (2013) · Zbl 1291.74186
[32] Maire, P.-H.; Abgrall, R.; Breil, J.; Ovadia, J., A cell-centered Lagrangian scheme for two-dimensional compressible flow problems, SIAM J. Sci. Comput., 29, 1781-1824, (2007) · Zbl 1251.76028
[33] Maire, P.-H.; Breil, J., A second-order cell-centered Lagrangian scheme for two-dimensional compressible flow problems, Int. J. Numer. Methods Fluids, 56, 1417-1423, (2008) · Zbl 1151.76021
[34] Maire, P.-H.; Nkonga, B., Multi-scale Godunov-type method for cell-centered discrete Lagrangian hydrodynamics, J. Comput. Phys., 228, 799-821, (2009) · Zbl 1156.76039
[35] Munz, C. D., On Godunov-type schemes for Lagrangian gas dynamics, SIAM J. Numer. Anal., 31, 17-42, (1994) · Zbl 0796.76057
[36] Noh, W. F., Errors for calculations of strong shocks using artificial viscosity and an artificial heat flux, J. Comput. Phys., 72, 78-120, (1987) · Zbl 0619.76091
[37] 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
[38] Perthame, B.; Shu, C.-W., On positivity preserving finite volume schemes for Euler equations, Numer. Math., 73, 119-130, (1996) · Zbl 0857.76062
[39] Plohr, B. J.; Sharp, D. H., A conservative Eulerian formulation of the equations for elastic flows, Adv. Appl. Math., 9, 481-499, (1988) · Zbl 0663.73012
[40] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77, 439-471, (1988) · Zbl 0653.65072
[41] Vilar, F., Cell-centered discontinuous Galerkin discretization for two-dimensional Lagrangian hydrodynamics, Comput. Fluids, 64, 64-73, (2012) · Zbl 1365.76129
[42] Vilar, F.; Maire, P.-H.; Abgrall, R., Cell-centered discontinuous Galerkin discretizations for two-dimensional scalar conservation laws on unstructured grids and for one-dimensional Lagrangian hydrodynamics, Comput. Fluids, 46, 1, 498-604, (2011) · Zbl 1433.76093
[43] Vilar, F.; Maire, P.-H.; Abgrall, R., A discontinuous Galerkin discretization for solving the two-dimensional gas dynamics equations written under total Lagrangian formulation on general unstructured grids, J. Comput. Phys., 276, 188-234, (2014) · Zbl 1349.76278
[44] Vilar, F.; Shu, C.-W.; Maire, P.-H., Positivity-preserving cell-centered Lagrangian schemes for multi-material compressible flows: from first-order to high-orders. part I: the one-dimensional case, J. Comput. Phys., 312, 385-415, (2016) · Zbl 1351.76127
[45] 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
[46] 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
[47] Zhang, X.; Shu, C.-W., Positivity preserving high order discontinuous Galerkin schemes for compressible Euler equations with source term, J. Comput. Phys., 230, 1238-1248, (2011) · Zbl 1391.76375
[48] 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
[49] Zhang, X.; Xia, Y.; Shu, C.-W., Maximum-principle-satisfying and positivity-preserving high order discontinuous Galerkin schemes for conservation laws on triangular meshes, J. Sci. Comput., 50, 29-62, (2012) · Zbl 1247.65131
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