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Macroscopic simulation of multiscale systems within the equation-free framework
Advisor(s): Dirk Roose
For many problems in science and engineering, the best available model is given on a microscopic level, while we would like to analyze the system on a much coarser, macroscopic level. To bridge this gap between the scale of the available model and the scale of interest, the so-called equation-free framework was developed. This framework is built around the central idea of a coarse time-stepper, which evolves the macroscopic variables in time, based on short, appropriately initialized simulations with the microscopic model. In this thesis, we study the numerical properties of several key algorithms in the equation-free framework.
First, we study the accuracy and stability of the coarse time-stepper when a lattice Boltzmann model is used as the microscopic simulator. As these numerical properties depend not only on the lattice Boltzmann model parameters but also on the specifics of the coarse time-stepper, the analysis allows to determine the influence of various aspects involved in the construction of the coarse time-stepper in a systematic way. Most importantly, it is shown that an appropriate initialization of the microscopic simulations is crucial to recover the correct macroscopic behavior.
Then, we focus on the class of constrained runs functional iterations that was developed to appropriately initialize the microscopic simulations. In certain cases, the constrained runs iterations fail to converge. The conditions for convergence are determined and a Newton-Krylov variant of the constrained runs scheme is developed to overcome the potential convergence problems.
Finally, we investigate the accuracy and stability of several time integration acceleration methods. It is shown that, under certain conditions, these methods can be used to accelerate a time integrator, irrespective of the exact nature of the underlying model. The different methods are applied to accelerate a lattice Boltzmann model, or, in the equation-free context, the coarse time-stepper for a (noisy) lattice Boltzmann model.Libridoc 1931 / Doctadmin 3E060588 / text.pdf (5.5M) / mailto: twr team