Iterative Splitting Methods for Differential Equations by Juergen Geiser

By Juergen Geiser

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Extra info for Iterative Splitting Methods for Differential Equations (Chapman and Hall CRC Numerical Analysis and Scientific Computation Series)

Example text

These algorithms integrate each underlying equation with respect to the last iterated solution. Therefore, the starting solution in each iterative equation is important in order to guarantee fast convergence or a higher-order method. The last iterative solution should at least have a local error of O(τni ) (i-th order in time), where i is the number of iteration steps, to obtain the next higher order. We deal with at least two equations, therefore two operators, but the results can be generalized to n operators (see for example the ideas of waveformrelaxation methods [199]).

For this model, we propose the decomposition of flow and reaction processes, which are given on different time scales, and we are free to accelerate the solving process. 2 Elastic Wave Propagation In the second model, we simulate the propagation of elastic waves produced by earthquakes. Here, the motivation arises from predicting possible earthquake formation in a given scenario of earthquakes about the center. Therefore, the following conditions must be fulfilled: • A realistic model problem has to consider a singular forcing term.

Here the motivation came from applying the algorithms of time and space decomposition methods, see [64], [199]. Here especially waveform relaxation methods are used to solve ordinary and partial differential equations, see [139], [199]. , Parareal, see [65]. , diagonal matrices, see [199]. On the other hand, the methods can accelerate the solver process with parallelized algorithms, see [65]. The second origin is motivated more by solving transport reaction problems. Here, the algorithms are focused on solving each part separately to save on computational time, see [147].

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