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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)
CAVEAT; FIVER
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