Parallelisation, initialisation, and boundary treatments for the diamond scheme
For researchers in numerical PDEs, this work offers a more efficient and parallelizable symplectic integrator with improved boundary handling, though it is an incremental improvement over existing rectangular mesh methods.
The paper studies a class of linear multisymplectic integrators for Hamiltonian wave equations using a diamond-shaped mesh, demonstrating greater efficiency, parallelization, and easier boundary treatment compared to rectangular mesh methods. Numerical results show observed order of convergence at least equal to the number of Runge-Kutta stages.
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.