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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.