Long-time momentum and actions behaviour of energy-preserving methods for semilinear wave equations via spatial spectral semi-discretizations

TL;DR

This paper investigates the long-time behavior of energy-preserving spectral semi-discretization methods for semilinear wave equations, demonstrating near conservation of modified physical quantities over extended periods.

Contribution

It provides a rigorous analysis of the long-time near conservation of momentum and actions for energy-preserving schemes using modulated Fourier expansions.

Findings

Near conservation of modified momentum over long times

Near conservation of modified actions over long times

Rigorous proof via modulated Fourier expansions

Abstract

As is known that wave equations have physically very important properties which should be respected by numerical schemes in order to predict correctly the solution over a long time period. In this paper, the long-time behaviour of momentum and actions for energy-preserving methods is analysed for semilinear wave equations. A full discretisation of wave equations is derived and analysed by firstly using a spectral semi-discretisation in space and then by applying the adopted average vector field (AAVF) method in time. This numerical scheme can exactly preserve the energy of the semi-discrete system. The main theme of this paper is to analyse another important physical property of the scheme. It is shown that this scheme yields near conservation of a modified momentum and modified actions over long times. Both the results are rigorously proved based on the technique of modulated Fourier expansions in two stages. First a multi-frequency modulated Fourier expansion of the AAVF method is constructed and then two almost-invariants of the modulation system are derived.