Exponential Runge–Kutta time integration for PDEs

Abstract

This dissertation focuses on the development of adaptive time-stepping and high-order parallel stages exponential Runge–Kutta methods for discretizing stiff partial differential equations (PDEs). The design of exponential Runge–Kutta methods relies heavily on the existing stiff order conditions available in the literature, primarily up to order 5. It is well-known that constructing higher-order efficient methods that strictly satisfy all the stiff order conditions is challenging. Typically, methods up to order 5 have been derived by relaxing one or more order conditions, depending on the desired accuracy level. Our approach will be based on a comprehensive investigation of these conditions. We will derive novel and efficient exponential Runge–Kutta schemes of orders up to 5, which not only fulfill the stiff order conditions in a strict sense but also support the implementation of variable step sizes. Furthermore, we develop the first-ever sixth-order exponential Runge–Kutta schemes by leveraging the exponential B-series theory. Notably, all the newly derived schemes allow the efficient computation of multiple stages, either simultaneously or in parallel. To establish the convergence properties of the proposed methods, we perform an analysis within an abstract Banach space in the context of semigroup theory. Our numerical experiments are given on parabolic PDEs to confirm the accuracy and efficiency of the newly constructed methods.