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

TL;DR

This paper proves that residual distribution schemes for the compressible Euler equations converge to dissipative weak solutions, ensuring stability and structure preservation, thus generalizing classical convergence results to nonlinear hyperbolic systems.

Contribution

It establishes the convergence of residual distribution schemes to dissipative weak solutions, extending the Lax-Richtmyer theorem to nonlinear Euler equations.

Findings

Residual distribution schemes preserve positivity of density and internal energy.

The schemes provide consistent and stable approximations.

Convergence to dissipative weak solutions is proven.

Abstract

In this work, we prove the convergence of residual distribution schemes to dissipative weak solutions of the Euler equations. We need to guarantee that the residual distribution schemes are fulfilling the underlying structure preserving properties such as positivity of density and internal energy. Consequently, the residual distribution schemes lead to a consistent and stable approximation of the Euler equations. Our result can be seen as a generalization of the Lax-Richtmyer equivalence theorem to nonlinear problems that consistency plus stability is equivalent to convergence.