Multi-symplectic discretisation of wave map equations

TL;DR

This paper introduces a new multi-symplectic formulation for constrained Hamiltonian PDEs, including wave map equations, and demonstrates an explicit, constraint-preserving discretisation with excellent conservation properties.

Contribution

It develops a novel multi-symplectic framework and a discretisation scheme that generalizes existing algorithms for constrained systems, with proven conservation properties.

Findings

The scheme is explicit and preserves constraints.

Numerical experiments show excellent conservation of invariants.

The formulation generalizes the Shake algorithm for wave map equations.

Abstract

We present a new multi-symplectic formulation of constrained Hamiltonian partial differential equations, and we study the associated local conservation laws. A multi-symplectic discretisation based on this new formulation is exemplified by means of the Euler box scheme. When applied to the wave map equation, this numerical scheme is explicit, preserves the constraint and can be seen as a generalisation of the Shake algorithm for constrained mechanical systems. Furthermore, numerical experiments show excellent conservation properties of the numerical solutions.