Invited Lecture
A Practical Guide to Exponential Integration
University of California Merced, USA
Abstract
Over the past several decades exponential integration has become a more prominent topic in numerical analysis and scientific computing. It is still, however, a new-kid-on-the-block compared to the extensively studied and used explicit and implicit temporal integrators. Thus, no broad recognition exists in the scientific community on how to answer questions like when one should use an exponential method, what kind of exponential schemes are appropriate for specific problems or what an efficient implementation of an exponential integrator looks like. In this talk, we will attempt to address some of these issues and present considerations that can guide the construction, analysis and application of exponential methods. We will discuss how the structure of the problem is connected to the choice of an exponential integrator and present an Exponential Propagation Iterative (EPI) framework that enables construction of different classes of methods. The relationship between the exponential scheme and the algorithms chosen to calculate the exponential-like functions of the Jacobian or its approximations will be outlined. We will also present a novel approach to derivation of exponential schemes using stiffness-resilient framework and highlight its advantages. Performance of exponential methods will be illustrated with examples from several fields such as plasma physics and weather and climate modeling.