| The first part of this thesis (Chapters 1--4) addresses accelerated algorithms for coarsening systems---we review unconditionally stable algorithms for the study of coarsening systems with a conserved or non-conserved scalar order parameter. These algorithms allow us to take arbitrarily large time-steps constrained only by desired accuracy. For conserved coarsening systems, these accelerated algorithms provide maximally-fast numerical algorithms---we can actually use the natural time-step Deltat = At2/3s . To study the accuracy we compare the scaling structure obtained from our maximally-fast conserved systems directly against the standard fixed-time-step Euler algorithm, and find that the error is time-independent in the scaling regime and scales as A ---this is consistent with an approximate bound of the error. Arbitrary accuracy is accessible for these maximally driven coarsening algorithms. These algorithms provide the most efficient and accurate means to reach the scaling regime for large systems. For non-conserved systems, however, with these accelerated algorithms, only effectively finite time-steps are accessible. The maximal time-step obtained by these algorithms is about four times the time-step of the Euler algorithm.; The second part of this thesis (primarily Chapter 5) applies these accelerated algorithms to the study of universality classes of scaled correlations in coarsening systems. Specifically, we study the universality classes found by introducing asymmetric bulk mobilities. We also develop accelerated algorithms for the study of systems with anisotropic surface tension. |
