A study of the exponential method of lines for a class of parabolic problems

Abstract

In this work, new techniques to improve the performance of exponential integrators are proposed. These new techniques are then combined to form an efficient algorithm for the exponential method of lines. Some of the theoretical aspects of this method are also studied within the framework of semigroup theory and its applications to the Cauchy problem. Unlike straight forward implementation of exponential integrators, the proposed algorithm avoids constructing a new Krylov subspace in each internal stage and for each time step by extracting an initial approximation from an already available Krylov subspace. If the initial approximation turns out to be unsatisfactory, then a new Krylov subspace is generated for its refinement. This is accomplished by developing a block version of the thick restarting Lanczos algorithm. Results and techniques to detect and prevent loss of orthogonality among the Lanczos blocks are extended from their counterparts for standard Lanczos algorithm. Moreover, a-posteriori error estimates for restarting procedure as well as sequential processing of right-hand sides are also derived. Our improvements are numerically validated for a set of test problems.