Adaptive Localized Replay: An efficient integration scheme for accurate simulation of coarsening dynamical systems
Coarsening dynamical systems (CDS) of ordinary differential equations (ODEs) have as a defining feature the presence of discontinuities in the governing equations, due to the removal of vanishing elements from the system. Simple, commonly-used integration strategies for ODEs neglect these discontinuities, resulting in large errors. In contrast, available methods to resolve discontinuities exhibit total running times that scale like O(N2) O ( N 2 ) as the number of elements N?? N ? ? . The resulting need to choose between either speed or accuracy frustrates attempts to gather well-converged statistical data from large ensembles of elements, a common goal in studies of coarsening. In this work, we introduce a numerical scheme, called Adaptive Localized Replay (ALR), which overcomes this dilemma, provided the underlying system has sparse neighbor dependence. By analyzing the system after each timestep, and then selectively re-simulating only those (small) parts of the system that were affected by discontinuities, we attain high-order accuracy while maintaining running times of O(N) O ( N ) . Using a reference implementation of our algorithm, applied to a sample problem from the field of faceted surface evolution, we obtain convergence and runtime results for the ALR method, which compares favorably with existing methods.
coarsening dynamical systems discontinuous odes numerical integration