$L^2$-stability of explicit schemes for incompressible Euler equations

TL;DR

This paper investigates the numerical stability of explicit schemes for solving the incompressible Euler equations, establishing conditions under which these schemes remain stable based on spatial and temporal discretization parameters.

Contribution

It introduces a novel stability analysis leveraging the skewness property of the non-linear term for explicit schemes in incompressible Euler equations.

Findings

Explicit schemes are stable for small perturbations when t  x^{2r/(2r-1)}.

Stability depends on the skewness property of the non-linear term.

Provides a specific stability condition relating time and space steps.

Abstract

We present an original study on the numerical stabiliy of explicit schemes solving the incompressible Euler equations on an open domain with slipping boundary conditions. Relying on the skewness property of the non-linear term, we demonstrate that some explicit schemes are numerically stable for small perturbations under the condition $\delta t\leq C \delta x^{2r/(2r-1)}$ where $r$ is an integer, $\delta t$ the time step and $\delta x$ the space step.