Transparent numerical boundary conditions for evolution equations: Derivation and stability analysis

TL;DR

This paper systematically develops transparent boundary conditions for finite difference schemes solving evolution equations, ensuring stability and broad applicability across various types of equations including transport, diffusion, and dispersive models.

Contribution

It derives a general framework for transparent boundary conditions applicable to a wide class of discretized evolution equations and proves stability criteria for these boundary conditions.

Findings

Derived transparent boundary conditions for discretized transport, diffusion, and dispersive equations.

Proved stability estimates for schemes with transparent boundary conditions under non-characteristic boundary assumptions.

Characterized schemes satisfying the Uniform Kreiss-Lopatinskii Condition for stability.

Abstract

The aim of this article is to propose a systematic study of transparent boundary conditions for finite difference approximations of evolution equations. We try to keep the discussion at the highest level of generality in order to apply the theory to the broadest class of problems. We deal with two main issues. We first derive transparent numerical boundary conditions, that is, we exhibit the relations satisfied by the solution to the pure Cauchy problem when the initial condition vanishes outside of some domain. Our derivation encompasses discretized transport, diffusion and dispersive equations with arbitrarily wide stencils. The second issue is to prove sharp stability estimates for the initial boundary value problem obtained by enforcing the boundary conditions derived in the first step. We focus here on discretized transport equations. Under the assumption that the numerical boundary is non-characteristic, our main result characterizes the class of numerical schemes for which the corresponding transparent boundary conditions satisfy the so-called Uniform Kreiss-Lopatinskii Condition introduced in [GKS72]. Adapting some previous works to the non-local boundary conditions considered here, our analysis culminates in the derivation of trace and semigroup estimates for such transparent numerical boundary conditions. Several examples and possible extensions are given.