A Nodal Continuous-Discontinuous Galerkin Time-Domain Method for Maxwell's Equations

Abstract

A new nodal hybrid continuous-discontinuous Galerkin time-domain (CDGTD) method for the solution of Maxwell's curl equations is proposed and analyzed. This hybridization is made by clustering small collections of elements with a continuous Galerkin (CG) formalism. These clusters exchange information with their exterior through a discontinuous Galerkin (DG) numerical flux. This scheme shows reduced numerical dispersion error with respect to classical DG formulations for certain orders and numbers of clustered elements. The spectral radius of the clustered semi-discretized operator is smaller than its DG counterpart allowing for larger time steps in explicit time integrators. Additionally, the continuity across the element boundaries allows us a reduction of the number of degrees of freedom of up to about 80% for a low-order three-dimensional implementation. Index Terms-Continuous-discontinuous Galerkin time-domain (CDGTD), continuous Galerkin (CG) method, discontinuous Galerkin (DG) method, discontinuous Galerkin time-domain (DGTD), Maxwell's equations.

Novel adaptive locally implicit scheme of the discontinuous Galerkin time-domain method for distributed transient simulation of antennas and resonance models

“…So that it is unnecessary to consider whether the implicit regions are adjacent, the efficiency of the IMEX method is only affected by the number of DOFs in the implicit region and the time step of the explicit region. The number of DOFs in the matrix A is proportional to the number of elements in implicit regions, and Δ𝑡 is determined by the minimum electrical size of elements in explicit regions, and is proved in [12].…”

“…The explicit DGTD method, on the other hand, is time step size dependent. The Courant-Friedrichs-Lewy (CFL) stability condition demonstrates that the smallest element determines the explicit time step size, rendering the DGTD method inefficient for solving multi-scale and resonance models [11,12,13]. The locally implicit method, also known as the implicit-explicit (IMEX) method [14,15,16,17,18,19], is a popular approach used to re-solve the multi-scale problem.…”

“…The locally implicit method, also known as the implicit-explicit (IMEX) method [14,15,16,17,18,19], is a popular approach used to re-solve the multi-scale problem. Implicit schemes like Crank-Nicholson (CN) scheme [12,20,21,22,23] have unconditional stability, so the IMEX method applies implicit difference schemes to small-size elements to extend the global time step. The IMEX method in [15] has simple iteration and high computational efficiency, so that is widely used in antennas and resonance models simulations.…”