Discrete conservation laws for finite element discretisations of multisymplectic PDEs

TL;DR

This paper introduces a novel finite element discretisation method for multisymplectic Hamiltonian PDEs that preserves energy conservation laws and demonstrates its effectiveness through numerical experiments.

Contribution

It develops an arbitrary order space-time finite element method combining continuous and discontinuous discretisations that conserves energy in multisymplectic PDEs.

Findings

The method preserves local and global energy conservation.

Existence and uniqueness of discrete solutions are established.

Numerical experiments confirm the method's accuracy and conservation properties.

Abstract

In this work we propose a new, arbitrary order space-time finite element discretisation for Hamiltonian PDEs in multisymplectic formulation. We show that the new method which is obtained by using both continuous and discontinuous discretisations in space, admits a local and global conservation law of energy. We also show existence and uniqueness of solutions of the discrete equations. Further, we illustrate the error behaviour and the conservation properties of the proposed discretisation in extensive numerical experiments on the linear and nonlinear wave equation and on the nonlinear Schr\"odinger equation.