Multisymplectic variational integrators and space/time symplecticity

TL;DR

This paper investigates the properties of multisymplectic variational integrators for PDEs, focusing on their space and time symplecticity, and explores their application to Lie group symmetries and a geometrically exact beam model.

Contribution

It analyzes the discrete evolution maps in multisymplectic schemes, linking Noether theorems to boundary conditions, and extends the study to Lie group integrators with practical numerical examples.

Findings

Discrete Noether theorems relate to boundary conditions.

Spatial and temporal maps preserve multisymplectic structure.

Application to a geometrically exact beam model demonstrates effectiveness.

Abstract

Multisymplectic variational integrators are structure preserving numerical schemes especially designed for PDEs derived from covariant spacetime Hamilton principles. The goal of this paper is to study the properties of the temporal and spatial discrete evolution maps obtained from a multisymplectic numerical scheme. Our study focuses on a 1+1 dimensional spacetime discretized by triangles, but our approach carries over naturally to more general cases. In the case of Lie group symmetries, we explore the links between the discrete Noether theorems associated to the multisymplectic spacetime discretization and to the temporal and spatial discrete evolution maps, and emphasize the role of boundary conditions. We also consider in detail the case of multisymplectic integrators on Lie groups. Our results are illustrated with the numerical example of a geometrically exact beam model.