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

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.

On the convergence of residual distribution schemes for the compressible Euler equations via dissipative weak solutions

“…Proof. The proof follows analogous steps as presented in [30,36]. It uses the fact that the RD schemes with the MOOD approach lead to consistent and stable approximation of the Euler equations.…”

“…At all, we demonstrated that our final implemented scheme is high-order, structure preserving and consistent. These are exactly the properties which are needed to prove a convergence result in the spirit of [28,29,37,36]. We start with the weak convergence theorem similar to [36].…”

“…These are exactly the properties which are needed to prove a convergence result in the spirit of [28,29,37,36]. We start with the weak convergence theorem similar to [36].…”

“…Due to the weak-strong uniqueness principle, it can be proven that the numerical solutions converge strongly to the strong solution on its lifespan. In [29,37,30] these results were extended to several finite volume (FV) methods and quite recently, in [36], a convergence analysis via DW solutions of a particular discontinuous Galerkin scheme has been performed. In the current work, we will extend those investigations and prove the convergence of residual distribution (RD) methods to the Euler equations via DW solutions.…”

Comparison to control oscillations in high-order Finite Volume schemes via physical constraint limiters, neural networks and polynomial annihilation

“…Extensions to multiphase flows are as well planned. Finally, our high-order FV blending schemes can be also the starting point of a convergence analysis for the Euler equations via dissipative measure-valued solutions [17,34] which is already work in progress.…”