High order numerical schemes for transport equations on bounded domains

TL;DR

This paper develops high-order finite difference schemes for transport equations on bounded domains, achieving optimal convergence rates by combining inverse Lax-Wendroff procedures with boundary layer analysis.

Contribution

It introduces a novel approach using inverse Lax-Wendroff boundary treatment to attain high-order accuracy for transport equations with nonzero boundary data.

Findings

Optimal convergence rates in maximum norm achieved

Effective boundary treatment via inverse Lax-Wendroff method

Validated with Lax-Wendroff and O3 schemes

Abstract

This article is an account of the NABUCO project achieved during the summer camp CEMRACS 2019 devoted to geophysical fluids and gravity flows. The goal is to construct finite difference approximations of the transport equation with nonzero incoming boundary data that achieve the best possible convergence rate in the maximum norm. We construct, implement and analyze the so-called inverse Lax-Wendroff procedure at the incoming boundary. Optimal convergence rates are obtained by combining sharp stability estimates for extrapolation boundary conditions with numerical boundary layer expansions. We illustrate the results with the Lax-Wendroff and O3 schemes.