Construction and Analysis of Multi-Rate Partitioned Runge-Kutta Methods

摘要

Adaptive mesh refinement (AMR) of hyperbolic systems allows us to refine the spatial grid of an initial value problem (IVP), in order to obtain better accuracy and improved efficiency of the numerical method being used. However, the solutions obtained are still limited to the local Courant-Friedrichs-Lewy (CFL) time-step restrictions of the smallest element within the spatial domain. Therefore, we look to construct a multi-rate time-integration scheme capable of solving an IVP within each spatial sub-domain that is congruent with that sub-domains respective time-step size. The primary objective for this research is to construct a multi-rate method for use with AMR. In this thesis we will focus on constructing a 2nd order, multi-rate partitioned Runge-Kutta (MPRK2) scheme, such that the non-uniform mesh is constructed with the coarse and fine elements at a two-to-one ratio. We will use general 2nd and 4th order finite differences (FD) methods for non-uniform grids to discretize the spatial derivative, and then use this model to compare the MPRK2 time-integrator against three explicit, 2nd order, single-rate time-integrators Adams-Bashforth 2 (AB2), Backward Differentiation Formula 2 (BDF2), and Runge-Kutta 2 (RK2).