Structure-preserving finite difference schemes for nonlinear wave equations with dynamic boundary conditions

TL;DR

This paper develops structure-preserving finite difference schemes for nonlinear wave equations with dynamic boundary conditions, ensuring numerical stability and energy conservation, and provides theoretical analysis of their solutions.

Contribution

It introduces a discrete variational derivative approach to derive structure-preserving schemes applicable to a broad class of nonlinear wave equations.

Findings

Existence and uniqueness of solutions for the proposed scheme.

Error estimates based on the inherited energy structure.

The schemes preserve the energy structure of the continuous problem.

Abstract

In this article we discuss the numerical analysis for the finite difference scheme of the one-dimensional nonlinear wave equations with dynamic boundary conditions. From the viewpoint of the discrete variational derivative method we propose the derivation of the structure-preserving finite difference schemes of the problem which covers a variety of equations as widely as possible. Next, we focus our attention on the semilinear wave equation, and show the existence and uniqueness of solution for the scheme and error estimates with the help of the inherited energy structure.