SFB TRR 352 Mathematics of Many-Body Quantum Systems and Their Collective Phenomena
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06.02.2025

Marlis Hochbruck: On the error analysis of full discretizations of Friedrichs' systems

In this talk, we address the full discretization of Friedrichs’ systems with a two-field structure, such as Maxwell’s equations or the acoustic wave equation in div-grad form.

We follow a method of lines approach, where we first discretize in space via the discontinuous Galerkin method. Subsequently, we consider different second-order schemes for time integration, namely the Crank–Nicolson scheme, the leapfrog scheme, and a general class of local time integration methods which comprises a locally implicit and a new local time-stepping scheme. We show the stability of the fully discrete schemes (subject to an appropriate CFL condition where necessary) and error bounds that are optimal in space and time and robust under mesh refinement. These bounds are derived within a unified error analysis based on the fact that all schemes can be interpreted as perturbations of the Crank–Nicolson scheme.

We conclude with numerical experiments in which we compare different local time integration methods for Maxwell's equations.

This is joint work with Malik Scheifinger, KIT

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 258734477 – SFB 1173