Parallelisation, initialisation, and boundary treatments for the diamond scheme

TL;DR

This paper introduces a parallelizable multisymplectic integrator using a diamond mesh for Hamiltonian wave equations, demonstrating improved efficiency, boundary handling, and convergence properties.

Contribution

It develops a novel diamond-shaped mesh scheme with boundary treatments, enhancing parallelization and boundary condition management for multisymplectic integrators.

Findings

Scheme achieves convergence order ≥ number of Runge-Kutta stages.

Enhanced parallelization compared to rectangular mesh methods.

Effective boundary condition treatments demonstrated through numerical tests.

Abstract

We study a class of general purpose linear multisymplectic integrators for Hamiltonian wave equations based on a diamond-shaped mesh. On each diamond, the PDE is discretized by a symplectic Runge--Kutta method. The scheme advances in time by filling in each diamond locally. We demonstrate that this leads to greater efficiency and parallelization and easier treatment of boundary conditions compared to methods based on rectangular meshes. We develop a variety of initial and boundary value treatments and present numerical evidence of their performance. In all cases, the observed order of convergence is equal to or greater than the number of stages of the underlying Runge--Kutta method.