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In this thesis, we develop an explicit multi-rate time stepping method for solving parabolic equations on a one dimensional adaptively refined mesh. Parabolic equations are characterized by their stiffness and as a result are usually solved using implicit time stepping schemes [16]. However, implicit schemes have the disadvantage that they can be expensive in higher dimensions or complicated to implement on adaptive or otherwise non-uniform meshes. Moreover, for coupled systems of parabolic equations, it can be difficult to achieve the expected order of accuracy without using sophisticated operator splitting techniques. For these reasons, we seek to exploit the properties of stabilized explicit methods for parabolic equations. In particular, we use the Runge-Kutta-Chebyshev (RKC) methods, a family of explicit Runge-Kutta methods, with numerical stability regions that extend far into the left half plane [12, 15, 21, 22, 26, 27, 28]. A central goal of this thesis is to use a second order RKC scheme to numerically solve parabolic equations on a one dimensional adaptively refined finite volume mesh. To make our implementation efficient, we design a time stepping algorithm in which time step sizes are chosen to respect the local mesh widths. This time stepping process requires communication between the RKC stages on different refinement levels. By linearly interpolating in time between the stage values, we obtain the ghost cell values for the finite volume scheme on each level. To our knowledge, this approach to adaptively refining in time, commonly referred to as a “multi-rate v time stepping” strategy, combined with RKC time stepping method has not been previously implemented. We develop our multi-rate algorithm on a one dimensional statically refined mesh using the second order finite volume scheme to numerically solve the heat or diffusion equation on each grid stored in a hierarchy of meshes. Using the “method of manufactured solutions”, we demonstrate that our method is second order accurate, and for our test problem, the multi-rate scheme requires only about 20% of the computational work required by the uniformly refined mesh at the same resolution. The algorithm we develop manages the time stepping between the refinement levels only, and so extends directly to higher dimensional problems. Future work in this direction includes applying the new multi-rate RKC time stepping scheme to biological pattern formations or crystal growth in the 2D ForestClaw code [7] on parallel machines.

Multi-Rate Runge-Kutta-Chebyshev Time Stepping for Parabolic Equations on Adaptively Refined Meshes