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

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.

Exponential collocation methods based on continuous finite element approximations for efficiently solving the cubic Schrödinger equation

“…In order to obtain fully discrete schemes, the differential operator should be dealt with in an appropriate way and there are many different choices to achieve this purpose. In the numerical experiments of this paper, we consider the pseudospectral method (see [12, 13, 70, 71]) as the discretization in space. More details concerning the spatial semi‐discretization and the implementation of the matrix exponential will be given in Section 7. Remark As is known, all the existing energy‐preserving methods are implicit, and so is this novel kind of methods proposed in this paper.…”

“…Concerning the semi‐discretization in space, we consider the spectral collocation method as did in [12, 13, 70, 71]. Denote by ℳ = {− M , …, M − 1} d and consider the trigonometric polynomial as an ansatz for the solution u of the NSE (1.1), where u M ( t , x ) is evaluated at the collocation points x k , k ∈ ℳ.…”

Long-Time Oscillatory Energy Conservation of Total Energy-Preserving Methods for Highly Oscillatory Hamiltonian Systems

“…However, it is noted that, until now, the technique of modulated Fourier expansions has not been well applied to the long-term analysis for energy-preserving method in the literature. Very recently, the authors of [35] studied long-time momentum and actions behaviour of energy-preserving methods for semilinear wave equations.…”