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978-3-8439-1584-7, Reihe Mathematik

Karen Kuhn Stability and applications of higher-order multirate Rosenbrock and Peer methods

121 Seiten, Dissertation Technische Universität Darmstadt (2014), Softcover, A5

The movement of the planets in the solar system, the behavior of electrical circuits, fluid structure interaction problems, all these physical phenomena fall into the class of coupled physical problems with different time scales. Because such problems can become arbitrarily complex, there is an ongoing research for solving strategies which need as less time and memory for a suitable solution as possible.

Often, coupled time-dependent physical problems can be described mathematically by ordinary differential equation systems, where each equation corresponds to a certain component with a specific activity. Conventional solving strategies (singlerate methods) apply the same time step size to all components simultaneously. In contrast, the idea of multirate methods is to treat the more active components with smaller time steps than the less active ones, i.e., different time step sizes are used for different components. Since the components are coupled, interpolation methods are needed.

This work consists of two parts. The first part employs the original form of the multirate time stepping strategy developed by Savcenco et al in 2007, i.e., in the context of Rosenbrock methods. The main new contribution is the stability analysis for different interpolation methods and the corresponding multirate methods. For interpolation, Rosenbrock continuous extensions, the Hermite interpolation, and a monotone Hermite interpolation are considered. For integration, the focus is on ROS2, ROS3P, ROS3PL, and RODAS.

In literature, so far considered multirate methods lack stability, particularly for stiff and strongly coupled problems. In this work we present a multirate ROS3PL method, for which numerical results indicate, that the method is unconditionally stable. Furthermore, the efficiency of the multirate Rosenbrock methods is tested for several test cases.

The main disadvantage of singlerate and multirate Rosenbrock methods is the order reduction phenomenon for higher-order methods. Since linear-implicit two-step Peer methods do not suffer from order reduction, the second part of this work deals with the construction of multirate Peer methods. The methods PEER3, PEER4, PEER5, and PEER6 are considered and combined with their continuous extensions. We give first ideas for the realization of a general multirate Peer method and analyze its stability. Finally we verify the overcome of order reduction for certain multirate Peer methods for a stiff test equation.