Stability of finite difference schemes for hyperbolic initial boundary value problems: numerical boundary layers

TL;DR

This paper develops a unified theory for boundary layer expansions in discretized hyperbolic transport equations, accommodating schemes with multiple time levels and ensuring compatibility with continuous estimates as discretization parameters vanish.

Contribution

It introduces a natural assumption allowing boundary layer expansions for schemes with many time levels, extending previous results limited to two-level schemes.

Findings

Boundary layer expansions can be constructed for multi-time-level schemes.

Discrete semigroup estimates align with continuous estimates in the limit.

The approach covers a broad class of discretization schemes.

Abstract

In this article, we give a unified theory for constructing boundary layer expansions for dis-cretized transport equations with homogeneous Dirichlet boundary conditions. We exhibit a natural assumption on the discretization under which the numerical solution can be written approximately as a two-scale boundary layer expansion. In particular, this expansion yields discrete semigroup estimates that are compatible with the continuous semigroup estimates in the limit where the space and time steps tend to zero. The novelty of our approach is to cover numerical schemes with arbitrarily many time levels, while semigroup estimates were restricted, up to now, to numerical schemes with two time levels only.