# Large Time Step and Overlapping Grids for Conservation Laws

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## Abstract

One focus of this dissertation is to construct a large time step Finite Volume Method for computing numerical solutions to hyperbolic systems of conservation laws. We also consider a method of overlapping spatial grids for which variants have proved to be an important consideration in large scale applications.

In practice we often run into grids which have a fairly large range of cell sizes -- some cells may be relatively large compared to others which may be significantly smaller. For traditional finite volume methods, the smallest spatial cell size dictates the time step size limit when employing explicit time marching. Moreover, if a solution is obtained as a limit from a sequence of approximations which use exceedingly irregular girds, the limit solution may not even be a proper weak solution. The large time step method we propose here addresses both of these problems. We prove approximate solutions obtained are stable, and when convergent will always converge to a weak solution, regardless of relative grid cell sizes.

Overlapping grids arise often in practice in order to discretized very complicated flow domains. One problem when grids overlap is how to identify a single valued approximation. A second issue is how to interface overlapping grids in such a way to obtain a conservative scheme. The method we propose here addresses both these issues. We identify a single valued approximate solution which employs overlapping spatial grids, and we prove its limit is a weak solution. Moreover, we show the method satisfies the maximum principle and is therefore stable.

Chapter one is an introduction to the theory of hyperbolic conservation laws. In chapter two we introduce the finite volume method, approximate Riemann problem solvers, and we establish the Lax-Wendroff Theorem for the multidimensional algorithm. In chapter three we present our large time step method and establish the theoretical results noted above. Numerical examples are also given in this chapter. In the last chapter we present our overlapping grid method. The theoretical results indicated above are proved and several numerical examples are presented.