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

  • Authors:

  • Xiangxiong Zhang;Chi-Wang Shu

  • Affiliations:

  • Department of Mathematics, Brown University, Providence, RI 02912, United States;Division of Applied Mathematics, Brown University, Providence, RI 02912, United States

  • Venue:
  • Journal of Computational Physics
  • Year:
  • 2010

Quantified Score

Hi-index 31.52

Visualization

Abstract

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.