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978-3-8439-0502-2, Reihe Mathematik
Fast Numerical Algorithms for Advection-Diffusion Equations and Applications in Particle Dynamics
109 Seiten, Dissertation Technische Universität Kaiserslautern (2012), Softcover, A5
The scope of the work is twofold. On the one hand, we develop a fast solver for particle orientation in turbulent carrier fluids. This kind of software is highly specialized for the one purpose it serves. We evaluate the possibility to employ the Finite Volume Method (FVM) and adopt the concept of geodesic grids, which were previously used in the field of meteorology, in order to discretized the spherical domain. As it turns out, adaptivity is an important tool in this context to obtain low CPU runtime and maintain high quality of the solution simultaneously. In order to taylor a custom scheme, we introduce a new refinement indicator based on the ideas of goal based error estimators. In addition, we choose a suitable time stepping procedure with favorable stability properties relying on the analysis of the spectrum of the semi-discrete system. The high performance of the new solver is also supported by concepts from computational science, such as the gather and scatter paradigm and last but not least by properly chosen heuristics. Overall, the new solver proves to be beneficial over the already available approaches in certain scenarios.
On the other hand, we develop a new kind of schemes for a variety of problems with a common structure through expansion of the class of two-step Runge-Kutta (TSRK) methods to partitioned TSRK, which can handle partitioned ODE systems adequately by treating each partition with a possibly different, specifically tailored method. We perform an analysis of order conditions, needed to obtain schemes with high stage and global orders. After the analysis, we give two practical schemes with optimized coefficients and show in various numerical benchmarks that the schemes indeed behave as predicted by the theory. Moreover, we note that the high stage order of the new schemes does not only greatly simplify the analysis, but also prevents them of suffering the order reduction phenomenon. The new schemes need significantly less stage evaluations in order to provide high global order of convergence, if compared to implicit-explicit Runge-Kutta methods.