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Positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations with source terms. (English) Zbl 1391.76375
Summary: In [X. Zhang and C.-W. Shu, J. Comput. Phys. 229, No. 23, 8918–8934 (2010; Zbl 1282.76128)] and [X. Zhang et al., J. Sci. Comput. 50, No. 1, 29–62 (2012; Zbl 1247.65131)], we constructed uniformly high order accurate discontinuous Galerkin (DG) schemes which preserve positivity of density and pressure for the Euler equations of compressible gas dynamics with the ideal gas equation of state. The technique also applies to high order accurate finite volume schemes. For the Euler equations with various source terms (e.g., gravity and chemical reactions), it is more difficult to design high order schemes which do not produce negative density or pressure. In this paper, we first show that our framework to construct positivity-preserving high order schemes in [Zhang and Shu, loc. cit.] and [Zhang et al., loc. cit.] can also be applied to Euler equations with a general equation of state. Then we discuss an extension to Euler equations with source terms. Numerical tests of the third order Runge-Kutta DG (RKDG) method for Euler equations with different types of source terms are reported.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76M12 Finite volume methods applied to problems in fluid mechanics
65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35Q35 PDEs in connection with fluid mechanics
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
76N15 Gas dynamics (general theory)
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