On semi-implicit schemes for the incompressible Euler equations via the vanishing viscosity limit

TL;DR

This paper introduces a novel, unconditionally stable numerical scheme for the incompressible Euler equations based on the vanishing viscosity limit, with rigorous error analysis and extensive numerical validation.

Contribution

It develops a new semi-implicit method with a unique integration by parts technique, reducing regularity requirements and establishing convergence in the inviscid limit.

Findings

Unconditionally stable with energy dissipation and bounded norms

First-order convergence with error estimates for various regularities

Numerical experiments validate the method on benchmark problems

Abstract

A new type of systematic approach to study the incompressible Euler equations numerically via the vanishing viscosity limit is proposed in this work. We show the new strategy is unconditionally stable that the $L^2$-energy dissipates and $H^s$-norm is uniformly bounded in time without any restriction on the time step. Moreover, first-order convergence of the proposed method is established including both low regularity and high regularity error estimates. The proposed method is extended to full discretization with a newly developed iterative Fourier spectral scheme. Another main contributions of this work is to propose a new integration by parts technique to lower the regularity requirement from $H^4$ to $H^3$ in order to perform the $L^2$-error estimate. To our best knowledge, this is one of the very first work to study incompressible Euler equations by designing stable numerical schemes via the inviscid limit with rigorous analysis. Furthermore, we will present both low and high regularity errors from numerical experiments and demonstrate the dynamics in several benchmark examples.