Energy conserving methods for Hamiltonian PDEs based on spectral space decomposition

TL;DR

This paper investigates energy conservation in numerical solutions of nonlinear wave equations using spectral Fourier methods and energy-preserving Runge-Kutta integrators, enhancing the accuracy and stability of simulations.

Contribution

It introduces a framework combining spectral Fourier discretization with HBVMs to ensure energy conservation in Hamiltonian PDEs.

Findings

Spectral Fourier methods effectively discretize nonlinear wave equations.

Energy-preserving Runge-Kutta methods maintain energy invariants.

The approach improves numerical stability and accuracy.

Abstract

In this paper we discuss energy conservation issues related to the numerical solution of the nonlinear wave equation, when a Fourier expansion is considered for the space discretization. The obtained semi-discrete problem is then solved in time by means of energy-conserving Runge-Kutta methods in the HBVMs class.