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On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes. (English) Zbl 1282.76128
Summary: We construct uniformly high order accurate discontinuous Galerkin (DG) schemes which preserve positivity of density and pressure for Euler equations of compressible gas dynamics. The same framework also applies to high order accurate finite volume (e.g. essentially non-oscillatory (ENO) or weighted ENO (WENO)) schemes. Motivated by B. Perthame and C.-W. Shu [Numer. Math. 73, No. 1, 119–130 (1996; (Zbl 0857.76062)] and the authors’ paper [J. Comput. Phys. 229, No. 9, 3091–3120 (2010; Zbl 1187.65096)], a general framework, for arbitrary order of accuracy, is established to construct a positivity preserving limiter for the finite volume and DG methods with first order Euler forward time discretization solving one-dimensional compressible Euler equations. The limiter can be proven to maintain high order accuracy and is easy to implement. Strong stability preserving (SSP) high order time discretizations will keep the positivity property. Following the idea in [Zbl 1187.65096], we extend this framework to higher dimensions on rectangular meshes in a straightforward way. Numerical tests for the third order DG method are reported to demonstrate the effectiveness of the methods.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
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