Convergence of Discontinuous Galerkin Schemes for the Euler Equations via Dissipative Weak Solutions

2022/02/21 by Mária Lukáčová-Medviďová, Mária Lukácová-Medvidová, Lukácová-Medvidová, Mária +2 · 1 citation
Computer Science · Engineering · Mathematics · Physics and Astronomy · #65M12 #65M60 #65M70 #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Model Reduction and Neural Networks #Numerical Analysis (math.NA) #Numerical methods for differential equations #cs.NA #math.NA #msc:65M12 #msc:65M60 #msc:65M70

paper · pdf · doi:10.48550/arxiv.2202.10043

28 pages, 3 Figures

openalex publication_date 2022/02/21 · arxiv created 2022/03/04 · arxiv updated 2022/03/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we present convergence analysis of high-order finite element based methods, in particular, we focus on a discontinuous Galerkin scheme using summation-by-parts operators. To this end, it is crucial that structure preserving properties, such as positivity preservation and entropy inequality hold. We demonstrate how to ensure them and prove the convergence of our multidimensional high-order DG scheme via dissipative weak solutions. In numerical simulations, we verify our theoretical results.

Cited by

• Selection criteria for the compressible Euler equations: numerical study

Related

• Stabilizing Discontinuous Galerkin Methods Using Dafermos' Entropy Rate Criterion: II -- Systems of Conservation Laws and Entropy Inequality Predictors
• Monolithic convex limiting in discontinuous Galerkin discretizations of hyperbolic conservation laws
• Positivity and Maximum Principle Preserving Discontinuous Galerkin Finite Element Schemes for a Coupled Flow and Transport
• Continuously bounds-preserving discontinuous Galerkin methods for hyperbolic conservation laws
• A note on higher-order and nonlinear limiting approaches for continuously bounds-preserving discontinuous Galerkin methods