SciCADE 2013
International Conference on Scientific Computation and Differential Equations
September 16-20, 2013, Valladolid (Spain)

Invited Talk

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Implicit Runge-Kutta schemes and discontinuous Galerkin methods for Maxwell's equations

T. Pazur and M. Hochbruck

Abstract
Maxwell's equations can be considered as an abstract initial value problem $ u'(t) = A u(t) + f(t), u(0) = u_0 $ on a suitable Hilbert space $H$. In this talk we first present an error analysis for Gauß and Radau collocation methods applied to this abstract problem. Our error analysis is based on energy technique discussed in [1]. For $s$-stage collocation methods we obtain an order reduction to order $s+1$ instead of the classical order $2s$ and $2s-1$ for Gauß and Radau collocation methods, respectively. Next we discretize Maxwell's equations in space using the discontinuous Galerkin method and then apply a collocation method to integrate the semidiscrete problem in time. We can prove that the full discretization error is of size $\mathcal{O}(h^{p+1/2} + \tau^{s+1})$, where $h$ denotes maximum diameter of the finite elements and $\tau$ denotes the time step. Finally, we illustrate our theoretical results by numerical experiments.

Bibliography
[1] C. Lubich and A. Ostermann, Runge-Kutta approximation of quasi-linear parabolic equations, Mathematics of Computation, Vol. 64, No. 210, pp 601-628, 1995.

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