High Order Implicit-explicit General Linear Methods with Optimized Stability Regions

Abstract

In the numerical solution of partial differential equations using a method-of-lines approach, the availability of high order spatial discretization schemes motivates the development of sophisticated high order time integration methods. For multiphysics problems with both stiff and nonstiff terms implicit-explicit (IMEX) time stepping methods attempt to combine the lower cost advantage of explicit schemes with the favorable stability properties of implicit schemes. Existing high order IMEX Runge--Kutta or linear multistep methods, however, suffer from accuracy or stability limitations. This work shows that IMEX general linear methods (GLMs) are competitive alternatives to classic IMEX schemes for large problems arising in practice. High order IMEX-GLMs are constructed in the partitioned GLM framework developed earlier by the authors [J. Sci. Comput., 61 (2014), pp. 119--144]. The stability regions of the new schemes are optimized numerically. The resulting IMEX-GLMs have similar stability properties as IMEX Runge--Kutta methods, but they do not suffer from order reduction and are superior in terms of accuracy and efficiency. The new IMEX-GLMs have considerably better stability properties than the IMEX linear multistep methods. Numerical experiments with two- and three-dimensional test problems illustrate the potential of the new schemes to speed up complex applications.