The Multisymplectic Diamond Scheme

Abstract

We introduce 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, leading to greater efficiency and parallelization and easier treatment of boundary conditions compared to methods based on rectangular meshes.

Parallelization, initialization, and boundary treatments for the diamond scheme

“…The following basic properties of the diamond scheme are established in [10]. The conservation law in Theorem 2 is a discretization of the integral of ω t +κ x = 0 over a single diamond, transferred to the boundary of the diamond using Stokes's theorem and discretized by Gauss quadrature.…”

“…This feature is unique amongst schemes of such broad applicability and motivates its further exploration. In particular, we argued in [10] that the local nature of the Fig. 1: Schematic of the diamond scheme for periodic boundary conditions.…”

“…The diamond scheme is a family of fully discrete numerical methods for firstorder hyperbolic PDEs introduced in [10]. It is based on the diamond grid shown in Figure 1.…”

On structure-preserving discontinuous Galerkin methods for Hamiltonian partial differential equations: Energy conservation and multi-symplecticity

“…For the multisymplectic HPDEs (1.1), numerical methods preserving (1.3) are referred as multi-symplectic integrators. There have been many studies on such methods, with examples including the Preissman box schemes [3,46,1,17], Euler box schemes [32,17], diamond schemes [31], spectral methods [4,8], multi-symplectic (partitioned) Runge-Kutta (RK) methods [35,21,36] and recently, DG methods [40,6]. These methods have been successfully applied to various equations including the Hamiltonian wave equation [3,31], the BBM equation [37,24], the CH equation [17], the KdV equation [46,1], the Schrödinger equation [8,40] and the Dirac equation [20].…”

“…There have been many studies on such methods, with examples including the Preissman box schemes [3,46,1,17], Euler box schemes [32,17], diamond schemes [31], spectral methods [4,8], multi-symplectic (partitioned) Runge-Kutta (RK) methods [35,21,36] and recently, DG methods [40,6]. These methods have been successfully applied to various equations including the Hamiltonian wave equation [3,31], the BBM equation [37,24], the CH equation [17], the KdV equation [46,1], the Schrödinger equation [8,40] and the Dirac equation [20]. Recently, there have been increasing interests in designing local energy conserving numerical schemes for the continuous dynamical systems.…”