ISBN 978-3-8439-4832-6

978-3-8439-4832-6, Reihe Mathematik

Moritz Schneider
Super-convergent Implicit-Explicit Peer Methods

161 Seiten, Dissertation Technische Universität Darmstadt (2021), Softcover, A5

Zusammenfassung / Abstract

Peer methods are two-step time integrators, where a certain number of intermediate stages is computed in each time step. Hence, they belong to the class of general linear methods. Their distinct feature is that all stages have the same accuracy, which motivates the name and prevents order reduction. Thus, they are well suited for the solution of stiff differential equations. If the problem consists of stiff and non-stiff parts, we can apply an implicit method to the full equation and use extrapolation on the non-stiff terms to obtain an implicit-explicit (IMEX) solver that combines the favorable stability properties of implicit methods with the lower computational costs of explicit schemes.

We discuss three questions concerning IMEX Peer methods:

1. Can we improve the standard results concerning the order of convergence for constant time steps and construct super-convergent methods based on extrapolation?

2. How do these schemes perform when variable time stepping is needed and how can we guarantee the additional order of convergence in this case?

3. Are IMEX Peer methods suitable for the application to hyperbolic balance laws?

For constant time steps, we show that super-convergence is available for implicit, explicit, and IMEX methods and construct new super-convergent schemes with A-stable implicit part and optimized stability regions. Further, we prove that these methods keep their ordinary order of convergence for any step size sequence. Moreover, super-convergence can be attained in this case by satisfying additional conditions and we derive corresponding schemes as well. In the last part, we show that our IMEX methods are well-balanced and asymptotic preserving and, hence, ideally suited for the numerical solution of hyperbolic balance laws. Finally, our numerical experiments verify the theoretical results and show that the new IMEX Peer methods designed in this thesis are competitive time integrators, both for constant and variable time steps.