On well-posed boundary conditions and energy stable finite volume method for the linear shallow water wave equation

TL;DR

This paper develops well-posed boundary conditions and an energy stable finite volume method for the linear shallow water wave equation, ensuring stability across all flow regimes through rigorous analysis and numerical verification.

Contribution

It introduces a novel energy stable finite volume scheme with well-posed boundary conditions for the linear shallow water wave equation, applicable to all flow regimes.

Findings

Proved stability via discrete energy estimates.

Validated the method with numerical experiments.

Established well-posed boundary conditions for all flow regimes.

Abstract

We derive and analyse well-posed boundary conditions for the linear shallow water wave equation. The analysis is based on the energy method and it identifies the number, location and form of the boundary conditions so that the initial boundary value problem is well-posed. A finite volume method is developed based on the summation-by-parts framework with the boundary conditions implemented weakly using penalties. Stability is proven by deriving a discrete energy estimate analogous to the continuous estimate. The continuous and discrete analysis covers all flow regimes. Numerical experiments are presented verifying the analysis.