On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes

• Published in: Journal of computational physics 2010-11, Vol.229 (23), p.8918-8934
• Main Authors: Zhang, Xiangxiong, Shu, Chi-Wang
• Format: Article
• Language: English
• Subjects: Compressible Euler equations; Construction; Density; Discontinuous Galerkin method; Discretization; Essentially non-oscillatory scheme; Euler equations; Finite volume scheme; Galerkin methods; Gas dynamics; High order accuracy; Hyperbolic conservation laws; Mathematical analysis; Mathematical models; Positivity preserving; Preserving; Weighted essentially non-oscillatory scheme
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2010.08.016

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 Perthame and Shu (1996) [20] and Zhang and Shu (2010) [26], 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 Zhang and Shu (2010) [26], 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.