Symplectic local time-stepping in non-dissipative DGTD methods applied to wave propagation problems

Abstract

Abstract. The Discontinuous Galerkin Time Domain (DGTD) methods are now popular for the solution of wave propagation problems. Able to deal with unstructured, possibly locally-refined meshes, they handle easily complex geometries and remain fully explicit with easy parallelization and extension to high orders of accuracy. Non-dissipative versions exist, where some discrete electromagnetic energy is exactly conserved. However, the stability limit of the methods, related to the smallest elements in the mesh, calls for the construction of local-time stepping algorithms. These schemes have already been developed for N -body mechanical problems and are known as symplectic schemes. They are applied here to DGTD methods on wave propagation problems.Mathematics Subject Classification. 65L05, 65L20, 65M12, 65M60, 78M10.

A well‐balanced Runge–Kutta discontinuous Galerkin method for the shallow‐water equations with flooding and drying

“…To exploit the local character of DG methods, the basis functions have support localized at a single mesh cell. On a given mesh cell, the local basis functions are Legendre polynomials in one space dimension and a particular set of modal basis functions constructed using barycentric coordinates in two space dimensions (see [29] for some properties of these modal basis functions). Let − → W h ∈ R N denote the component vector of W h with respect to the basis functions; here, N denotes the total number of degrees of freedom, i.e., the dimension of…”

Recent Achievements on a DGTD Method for Time-Domain Electromagnetics

“…A more viable approach consists in applying an implicit time integration scheme locally in the refined regions of the mesh, while preserving an explicit time scheme in the complementary part. Such an hybrid explicit-implicit DGTD method has been proposed by Piperno in [6]. In this method, the elements of the mesh are assumed to be partitioned into two subsets, S i and S e , on the basis of an appropriate geometrical criterion.…”

“…In this method, the elements of the mesh are assumed to be partitioned into two subsets, S i and S e , on the basis of an appropriate geometrical criterion. Then, the elements of S i are handled using a Crank-Nicolson scheme while those of S e are time advanced using a variant of the classical Leap-Frog scheme known as the Verlet method (see [6] for more details). We have recently completed a stability analysis of this method and subsequently implemented it for the solution of the 2D and 3D time domain Maxwell equations discretized in space by a high order conforming DGTD-P p method on unstructured simplicial meshes [7].…”

Multi-level explicit local time-stepping methods for second-order wave equations

“…1, corresponds to the standard LF method applied to (18) with time-step ⌧ = t/p. For simplicity, we henceforth assume that A is globally defined.…”

“…It was analyzed in [14,15] and later extended to elastodynamics [16] and Maxwell's equations [17]. By combining a symplectic integrator with a DG discretization of Maxwell's equations in first-order form, Piperno [18] proposed a second-order explicit local time-stepping scheme, which also conserves a discrete energy. Starting from the standard LF method, the authors proposed energy conserving fully explicit LTS integrators of arbitrarily high accuracy for the wave equation [1]; that approach was extended to Maxwell's equations in [19].…”