Date of Award

May 2014

Degree Type

Thesis

Degree Name

Master of Science

Department

Mathematics

First Advisor

Bruce Wade

Committee Members

Bruce Wade, Dexuan Xie, Lei Wang

Keywords

Allen-Cahn, ETDRK4, Krylov Subspace Methods, Lanczos Iteration, Matrix Exponential

Abstract

A modification of the (2,2)-Pade algorithm developed by Wade et al. for implementing the exponential time differencing fourth order Runge-Kutta (ETDRK4) method is introduced. The main computational difficulty in implementing the ETDRK4 method is the required approximation to the matrix exponential. Wade et al. use the fourth order (2,2)-Pade approximant in their algorithm and in this thesis we incorporate Krylov subspace methods in an attempt to improve efficiency. A background of Krylov subspace methods is provided and we describe how they are used in approximating the matrix exponential and how to implement them into the ETDRK4 method. The (2,2)-Pade and Krylov subspace algorithms are compared in solving the one and two dimensional Allen-Cahn equation with the ETDRK4 scheme. We find that in two dimensions, the Krylov subspace algorithm is faster, provided we have a spatial discretization that produces a symmetric matrix.

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