A stiffly stable semi-discrete scheme for the damped wave equation on the half-line using SBP and SAT techniques

TL;DR

This paper develops a stable semi-discrete numerical scheme for the damped wave equation on the half-line, ensuring uniform stability through SBP and SAT techniques, even with stiff source terms or small spatial steps.

Contribution

It introduces a novel boundary treatment that guarantees stability of the scheme for the damped wave equation on the half-line, extending SBP and SAT methods to characteristic boundary conditions.

Findings

The scheme is proven to be uniformly stable regardless of stiffness.

Stability estimates are rigorously established for the semi-discrete scheme.

The approach builds on and extends previous continuous stability analyses.

Abstract

This paper investigates the stability of both the semi-discrete and the implicit central scheme for the linear damped wave equation on the half-line, where the spatial boundary is characteristic for the limiting equation. The proposed schemes incorporate a discrete boundary condition designed to guarantee the uniform stability of the IBVP, regardless of the stiffness of the source term or the spatial step size. Stability estimates for the semi-discrete scheme are established using the summation-by-parts (SBP) and simultaneous-approximation-term (SAT) penalty techniques, building on the continuous framework analyzed by Xin and Xu (2000).