The difference variational bicomplex and multisymplectic systems

TL;DR

This paper develops the difference variational bicomplex framework for systems of difference equations, linking geometric structures to conservation laws and multisymplectic integrators, applicable to both uniform and non-uniform meshes.

Contribution

It constructs the difference variational bicomplex, explores its properties, and connects multisymplecticity with Hamiltonian conditions, extending to adaptive mesh systems.

Findings

Exactness of the bicomplex provides a coordinate-free variational framework.

Connection established between Hamiltonian existence and multisymplecticity.

Difference multimomentum maps yield conservation laws for multisymplectic systems.

Abstract

The difference variational bicomplex, which is the natural setting for systems of difference equations, is constructed and used to examine the geometric and algebraic properties of various systems. Exactness of the bicomplex gives a coordinate-free setting for finite difference variational problems, Euler--Lagrange equations and Noether's theorem. We also examine the connection between the condition for the existence of a Hamiltonian and the multisymplecticity of systems of partial difference equations. Furthermore, we define difference multimomentum maps of multisymplectic systems, which yield their conservation laws. To conclude, we adapt the variational bicomplex to multisymplectic integrators on a mesh that is logically rectangular. By scaling horizontal forms and difference operators according to the local step sizes, all of the results derived earlier can be applied, whether or not the mesh is uniform.