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C. Wintersteiger:
"Mapped Tent Pitching Schemes for Hyperbolic Systems";
Supervisor, Reviewer: J. Schöberl, M. J. Gander; Institut für Analysis und Scientific Computing, 2020; oral examination: 2020-11-24.

This thesis introduces Mapped Tent Pitching (MTP) methods for hyperbolic systems. These hyperbolic system, like the Maxwell equations or the Euler equations, have a welldefined speed of propagation, which can be used to partition the spacetime domain using tent-shaped elements. These spacetime elements, denoted as tents, are generated with a tent pitching algorithm and mapped to spacetime cylinders, which allows to discretize space and time independently. Tent pitched meshes adapt to varying speeds of propagation and different sized spatial mesh leading to a naturally built in local time stepping. The spatial discretization using a high order discontinuous Galerkin method leads to a system of ordinary differential equations, which can be solved by implicit or explicit time stepping methods. Although locally implicit MTP methods based on implicit RungeKutta schemes for the temporal discretization show high order convergence, the memory is a limiting factor for these methods. Fully explicit methods have a low memory consumption, but they are limited to first order when using standard methods for the temporal discretization. To overcome this convergence order reduction, we construct suitable explicit time stepping schemes to propagate hyperbolic solutions within these tent-shaped spacetime elements. These structure aware time stepping schemes recover the high order convergence for linear and nonlinear problems. To demonstrate the optimal convergence rates, we apply these MTP methods using structure aware times stepping schemes to various linear and nonlinear hyperbolic systems. Further we report the discrete stability properties of these methods applied to linear hyperbolic equations.

http://dx.doi.org/10.34726/hss.2020.83746