Mathematics Seminar 02/11/21

Abstract: 

Multirate time integration methods have shown to be more efficient than traditional time integration methods for solving multirate differential equations. Unlike the traditional methods (which use a single time step for integrating the whole system), the idea of multirate integrators is to employ different time steps for different components (e.g., fast and slow) of the system, thereby enhancing computational efficiency while ensuring the overall stability and accuracy. So far, methods of orders up to four have been proposed in the literature, e.g., multirate infinitesimal step (MIS) methods, multirate general-structure additive Runge-Kutta (GARK) methods. Their derivations, however, require solving a complicated set of coupling and decoupling order conditions, which is very difficult to obtain higher orders. To avoid this difficulty, we propose a new approach based on the idea of using backward error analysis for exponential methods. This allows us to construct two new classes of multirate methods of orders up to six in a very elegant way. A rigorous convergence analysis of these methods is also carried out and they allow both fast and slow components to be integrated in explicit-explicit and implicit-explicit manners. Numerical experiments are given to demonstrate the accuracy and efficiency of the newly derived schemes.

The talk is based on joint works with Daniel R. Reynolds (SMU) and Rujeko Chinomona (Temple Uni.).