Incorporating Krylov Subspace Methods in the ETDRK4 Scheme

INCORPORATING KRYLOV SUBSPACE METHODS IN THE ETDRK4 SCHEME by Jeffrey H. Allen The University of Wisconsin–Milwaukee, 2014 Under the Supervision of Professor Bruce Wade A modification of the (2, 2)-Padé 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)-Padé 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)-Padé 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.