DGTD methods using modal basis functions and symplectic local time-stepping

Abstract

The Discontinuous Galerkin Time Domain (DGTD) methods are now widely used for the solution of wave propagation problems. Able to deal with unstructured meshes past complex geometries, they remain fully explicit with easy parallelization and extension to high orders of accuracy. Still, modal or nodal local basis functions have to be chosen carefully to obtain actual numerical accuracy. Concerning time discretization, explicit nondissipative energy-preserving time-schemes exist, but their stability limit remains linked to the smallest element size in the mesh. Symplectic algorithms, based on local-time stepping or local implicit scheme formulations, can lead to dramatic reductions of computational time, which is shown here on two-dimensional acoustics problems.Key-words: waves, acoustics, Maxwell's system, Discontinuous Galerkin methods, mass matrix condition number, symplectic schemes, energy conservation, local time-stepping.Méthodes DGTD avec fonctions de base modales et schémas en temps symplectiques : applications en propagation d'ondes.Résumé : Les méthodes de Galerkine discontinu sont maintenant largement utilisées pour la résolution numérique de problèmes de propagation d'ondes. S'appuyant sur des maillages non-structurés autour des géométries les plus générales, elles restent quasiment complète-ment explicites, facilement parallélisables et d'ordre élevé. Il convient néanmoins d'optimiser le choix des fonctions de base discontinues (modales ou nodales). Pour ce qui est de la discrét-isation en temps, des schémas explicites non-dissipatifs existent, mais leur limite de stabilité reste liée aux plus petits éléments du maillage. Des algorithmes symplectiques, avec pas de temps local ou schéma localement implicite, conduisent à des diminutions considérables du temps de calcul.

Discontinuous Galerkin method for blow-up solutions of nonlinear 1D wave equations

“…This is no longer guaranteed if k 2 since the numerical solution may oscillate between its degrees of freedom. To ensure a discrete maximum principle, one may choose modal functions [41] as a basis of V k h instead of the nodal functions. However, the projection of the nonlinear term |u h | p into V k h would introduce more computational complexity to the system (2.11).…”

DGTD for a Class of Low-Observable Targets: A Comparison With MoM and (2, 2) FDTD

“…Local time-stepping strategies have been efficiently incorporated into the LF stepping procedure [12], [13] to alleviate the computational overload driven by the conditional stability of LF in real problems. Here, we use the LTS algorithm described in [9], [10], to arrange the mesh elements in different tiers, according to the maximum time step allowed for stability, so that different time steps can be used for each tier.…”