  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 non-stiff 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 reduction.
  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 framework developed by the authors
[34]. 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. Numerical experiments with two and three
dimensional test problems illustrate the potential of the new schemes to speed
up complex applications.
