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

Many physical phenomena can be described by time-dependent partial differential equations (PDEs). When they are discretized in space by the Method of Lines, one obtains a set of coupled ordinary differential equations (ODEs). Each ODE describes the behavior of one spatial component, which typically includes the coupling with some of the other components. Furthermore numerous phenomena in science and technology can be modelled directly by time-dependent ODEs. However many of these problems of interest contain different time scales for different sets of components.

A normal singlerate time integrator solves all of these ODEs with the same time step sizes, which are determined by taking all the components into account. This might produce very small time steps, that also have to be applied to components with much less activity.

The idea of multirate methods is to use different time step sizes for different components, depending on their individual activity. Inherently, there will be a differentiation between active and latent components. The coupling between both can be managed by interpolation and extrapolation.

In the first instance this work is the study of stability and efficiency of several multirate Rosenbrock-methods. Rosenbrock-methods are linear-implicit one-step methods. Several A-stable Rosenbrock-methods are considered. For interpolation continuous extensions are used, but also the standard Hermite interpolation and a monotone version of it are analysed.

The drawback of Rosenbrock-methods is the order reduction phenomenon, which shows up for stiff problems in the higher-order singlerate case as well as in the corresponding multirate case. To avoid order reduction, the use of multi-step methods instead of one-step methods might be beneficial. For example linear-implicit higher-order two-step Peer-methods do not suffer from order reduction when they are applied to stiff problems. Therefore this work provides first steps for the construction of higher-order multirate methods using Peer-methods.

Multi-Physics; Numerical mathematics of differential equations; Multirate Methods.

Karen Kuhn

Dipl.-Math.

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