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

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.