Splitting schemes for the semi-linear wave equation with dynamic boundary conditions

TL;DR

This paper develops and analyzes first- and second-order splitting schemes for semi-linear wave equations with dynamic boundary conditions, enabling efficient numerical solutions with proven convergence properties.

Contribution

It introduces novel bulk-surface splitting schemes for semi-linear wave equations with kinetic and acoustic boundary conditions, demonstrating their convergence through numerical experiments.

Findings

Splitting schemes achieve first-order convergence for kinetic boundary conditions.

Lie and Strang splitting schemes reach first- and second-order convergence for acoustic boundary conditions.

Numerical experiments confirm the theoretical convergence rates.

Abstract

This paper introduces novel bulk-surface splitting schemes of first and second order for the wave equation with kinetic and acoustic boundary conditions of semi-linear type. For kinetic boundary conditions, we propose a reinterpretation of the system equations as a coupled system. This means that the bulk and surface dynamics are modeled separately and connected through a coupling constraint. This allows the implementation of splitting schemes, which show first-order convergence in numerical experiments. On the other hand, acoustic boundary conditions naturally separate bulk and surface dynamics. Here, Lie and Strang splitting schemes reach first- and second-order convergence, respectively, as we reveal numerically.