Locally implicit DGTD methods for electromagnetics

The use of unstructured meshes (based on triangles in two space dimensions and tetrahedra in three space dimensions) is an important feature of DGTD methods studied in the team for dealing with complex geometries and heterogeneous propagation media. Moreover, discontinuous Galerkin discretization methods are naturally adapted to local, conforming as well as non-conforming, refinement of the underlying mesh, Most of the existing DGTD methods rely on explicit time integration schemes and lead to block diagonal mass matrices which is often recognized as one of the main advantages with regards to continuous finite element methods. However, explicit DGTD methods are also constrained by a stability condition that can be very restrictive on highly refined meshes and when the local approximation relies on high order polynomial interpolation. There are basically three strategies that can be considered to mitigate this computational efficiency problem. A first approach is to use an unconditionally stable implicit time integration scheme to overcome the restrictive constraint on the time step for locally refined meshes. A second approach is to use a local time stepping strategy combined with an explicit time integration scheme. In the third approach which is the focus of our activities on this topic, the time step size restriction is overcome by using a hybrid explicit-implicit procedure. In that case, one blends a time implicit and a time explicit schemes where only the solution variables defined on the smallest elements are treated implicitly.

Propagation of a standing wave in a cubic PEC cavity. Examples of two locally refined meshes (implicit treatment: red regions). From top to bottom: interior skeletons of 3D meshes and cross sections of the 3D meshes resulting (on the left: 635 vertices and 2,968 tetraedra; on the right: 7,759 vertices and 40,616 tetraedra).

Propagation of a standing wave in a cubic PEC cavity: numerical convergence and maximum error in L₂-norm as a function of final CPU time for the locally implicit and fully explicit DGTD-P_k methods (top - bottom, respectively).

Related publications

L. Moya
Temporal convergence of a locally implicit discontinuous Galerkin method for Maxwell's equations
ESAIM: Mathematical Modelling and Numerical Analysis, Vol. 46, pp 1225-1246 (2012)
Preprint available as INRIA RR-7533 on Hyper Article Online

L. Moya, S. Descombes and S. Lanteri
Locally implicit time integration strategies in a discontinuous Galerkin method for Maxwell's equations
J. Sci. Comp., Vol. 56, No. 1, pp. 190-218 (2013)
Available as INRIA RR-7983 on Hyper Article Online

L. Moya
Locally implicit discontinuous Galerkin time-domain methods for electromagnetic wave propagation in biological tissues
Doctoral thesis, University of Nice - Sophia Antipolis (2013)