Positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations with source terms

• Published in: Journal of computational physics Vol. 230; no. 4; pp. 1238 - 1248
• Main Authors: Zhang, Xiangxiong, Shu, Chi-Wang
• Format: Journal Article
• Language: English
• Published: Kidlington Elsevier Inc 20.02.2011; Elsevier
• Subjects: Chemical reactions; Compressible Euler equations with source terms; Computational techniques; Construction; Density; Discontinuous Galerkin method; Equations of state; Essentially non-oscillatory scheme; Euler equations; Exact sciences and technology; Finite volume scheme; Galerkin methods; Gas dynamics; High order accuracy; Hyperbolic conservation laws; Mathematical analysis; Mathematical methods in physics; Physics; Positivity preserving; Preserves; Runge-Kutta method; Weighted essentially non-oscillatory scheme; Hyperbolic conservation laws; Finite volume scheme; Discontinuous Galerkin method; Compressible Euler equations with source terms; Essentially non-oscillatory scheme; Gas dynamics; Positivity preserving; High order accuracy; Weighted essentially non-oscillatory scheme; Third order; Chemical reactions; Euler equations; Euler scheme; Calculation methods; Conservation laws; Galerkin-Petrov method; Equations of state; Calculation; Gravity
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2010.10.036

In [16,17], 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 [16,17] 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.