Stability conditions for the numerical solution of convection-dominated problems with skew-symmetric discretizations

TL;DR

This paper establishes near-optimal stability conditions for numerical schemes solving convection-dominated problems, linking time and space steps more strongly than the CFL criterion, and verifies these conditions through numerical tests.

Contribution

It introduces new stability conditions that are stronger than CFL for convection-dominated problems and demonstrates their applicability to nonlinear equations.

Findings

Stability condition: elta t  elta x^lpha with lpha=rac{2r}{2r-1}

Numerical tests confirm the stability condition's effectiveness

Applicable to nonlinear convection-dominated equations under smoothness assumptions

Abstract

This paper presents original and close to optimal stability conditions linking the time step and the space step, stronger than the CFL criterion: $\delta t\leq C\delta x^\alpha$ with $\alpha=\frac{2r}{2r-1}$, $r$ an integer, for some numerical schemes we produce, when solving convection-dominated problems. We test this condition numerically and prove that it applies to nonlinear equations under smoothness assumptions.