A Note on the Convergence of the Godunov Method for Impact Problems

TL;DR

This paper examines a unique numerical pathology in the convergence of the Godunov method for impact problems, revealing issues related to insufficient dissipation leading to extremely weak solutions with no derivatives post-shock.

Contribution

It identifies a new pathology affecting the convergence of the Godunov method in impact problems, highlighting the role of numerical dissipation in weak solution formation.

Findings

Numerical solutions can become extremely weak with no derivatives in the post-shock region.

Insufficient numerical dissipation can cause convergence issues in impact problems.

The pathology is distinct from previously known shock-related phenomena.

Abstract

This paper identifies a new pathology that can be found for numerical simulations of nonlinear conservation law systems. Many of the difficulties already identified in the literature (rarefaction shocks, carbuncle phenomena, slowly moving shocks, wall heating, etc) can be traced to insufficient numerical dissipation, and the current case is no different. However, the details of the case we study here are somewhat unique in that the solution which is found by the numerics is very weak and can fail to have a derivative anywhere in the post-shock region.