UC Merced

Stiff systems of ordinary differential equations (ODEs) play an essential role in the temporal integration of many scientific and engineering problems. With the increase in complexity and size of these problems, it is crucial to have efficient time integration methods. In this dissertation, we focus on the construction, analysis, and applications of exponential time integrators. We address the efficiency of these methods in three main directions. First, we construct new efficient exponential multi-step methods. These methods are carefully derived to combine the accuracy of high-order schemes with the efficiency of low-order methods. We validate and apply the new schemes to a simplified atmospheric model and show that they can capture more details in the solution with a limited computational cost. Secondly, we develop a new ansatz for deriving partitioned implicit-exponential integrators. These methods are applied to problems where the forcing term of the system is comprised of stiff additive terms. We propose a new way of analyzing the stability properties of partitioned methods. Our results show that the increased accuracy and stability of our schemes offer improved efficiency compared to state-of-the-art methods. Finally, we introduce a new theoretical framework for deriving stiffly resilient exponential methods. This framework addresses the complexity of deriving exponential schemes and enforces advantageous properties. We establish an analytical solution for the order conditions used to derive methods. We derive a new class of exponential integrators and new schemes with an order of convergence higher than currently available methods. We also evaluate the performance of exponential time integration for the reduced magnetohydrodynamics equations. Our initial results focus on a simplified model capturing the essence of the full equations. We show that exponential integrators are a valuable alternative to the schemes currently used for this problem.