Implicit Runge--Kutta Methods and Discontinuous Galerkin Discretizations for Linear Maxwell's Equations

Abstract

In this paper we consider implicit Runge-Kutta methods for the time-integration of linear Maxwell's equations. We first present error bounds for the abstract Cauchy problem which respect the unboundedness of the differential operators using energy techniques. The error bounds hold for algebraically stable and coercive methods such as Gauß and Radau collocation methods. The results for the abstract evolution equation are then combined with a discontinuous Galerkin discretization in space using upwind fluxes. For the case that permeability and permittivity are piecewise constant functions, we show error bounds for the full discretization, where the constants do not deteriorate if the spatial mesh width tends to zero.

Analysis of a Generalized Dispersive Model Coupled to a DGTD Method with Application to Nanophotonics

“…This, as usual, will result in a constraint in the time step in terms of a CFL condition. Let us also mention that the implicit counterpart of RK schemes coupled to a discontinuous Galerkin framework have been recently studied in the context of linear Maxwell's equations in [HP15].…”

Efficient time integration for discontinuous Galerkin approximations of linear wave equations

“…The stopping criteria in Line 17 of Algorithm 1, was introduced in [37], see also [2] for a detailed investigation of residuals of the matrix exponential. Here, δ m is an estimation of the relative error if m ≤ Tol then 20: break 21: end if 22: end for 23: if m ≥ MaxIter and m > Tol then 24: no convergence 25: end if 26: x m = [v 1 , . .…”

“…In all explicit methods, the required number of steps is proportional to 2 l , since the CFL condition requires O(τ ) = O(h). For the implicit methods, the required number of time steps is independent of the spatial discretization, see [22] for a rigorous error analysis for Maxwell's equations. Obviously, implicit Runge-Kutta methods and rational Krylov methods are only more efficient than explicit methods, if a suitable preconditioner is available.…”

Error analysis of implicit Runge–Kutta methods for quasilinear hyperbolic evolution equations

“…In the very recent preprint [14] implicit RungeKutta schemes have been analyzed in Kato's original framework of [11]. For linear Maxwell equations such schemes were studied in [7] including the space discretization with discontinuous Galerkin methods.…”