Design, Implementation and Performance of Exponential Integrators for High Performance Computing Applications
Exponential integrators have received renewed interest in recent years as a means to approximate stiff systems of ODEs, but are not currently widely used in high performance computing. There have been only limited performance studies comparing them to currently used methods, little work investigating how to optimize their design for computational efficiency, and almost no work on implementing and studying their performance on parallel computers. We present here a detailed performance breakdown and comparison of Krylov-based exponential integrators to each other and to Newton-Krylov implicit solvers, the currently most widely used class of methods for large-scale stiff problems. Our results show exponential integrators perform favorably compared to implicit integrators across a
number of different problems. We then introduce a new class of exponential integrators called exponential propagation iterative methods of Runge-Kutta type (EPIRK). Based on our performance analysis we consider some strategies for utilizing their structural features to construct schemes with improved computational eciency and demonstrate their effectiveness with some numerical experiments. We also describe a parallel implementation of a suite of exponential integrators and give some performance results which show encouraging performance of the methods on problems scaled up to thousands of processors when compared to CVODE, a production-grade parallel implementation of a Newton-Krylov implicit
integrators popularly used for high performance computing applications today. We conclude with consideration of possible future research directions.