Energy conservation issues in the numerical solution of the semilinear wave equation

TL;DR

This paper examines how energy conservation can be maintained in the numerical solutions of the nonlinear wave equation, emphasizing the use of energy-conserving methods like HBVMs to preserve Hamiltonian properties.

Contribution

It demonstrates that energy conservation properties of the continuous problem can be transferred to semi-discrete and fully discrete numerical solutions using HBVMs methods.

Findings

Energy conservation is preserved in semi-discrete solutions.

HBVMs methods effectively maintain energy in fully discretized problems.

Results extend to other Hamiltonian PDEs like nonlinear Schrödinger equation.

Abstract

In this paper we discuss energy conservation issues related to the numerical solution of the nonlinear wave equation. As is well known, this problem can be cast as a Hamiltonian system that may be autonomous or not, depending on the specific boundary conditions at hand. We relate the conservation properties of the original problem to those of its semi-discrete version obtained by the method of lines. Subsequently, we show that the very same properties can be transferred to the solutions of the fully discretized problem, obtained by using energy-conserving methods in the HBVMs (Hamiltonian Boundary Value Methods) class. Similar arguments hold true for different types of Hamiltonian Partial Differential Equations, e.g., the nonlinear Schr\"odinger equation.