Optimally zero-stable superconvergent IMEX Peer methods

Rüdiger Weiner (Martin Luther University Halle-Wittenberg), Behnam Soleimani, Jens Lang, Moritz Schneider

Many systems of ODEs are of the form $$y'=f(t,y)+g(t,y)$$ with stiff part $f$ and nonstiff part $g$, for instance MOL discretizations of diffusion-advection-reaction equations. This kind of problems can be treated efficiently by implicit-explicit (IMEX) methods. In IMEX methods the stiff part is solved by an implicit method, the nonstiff part is solved by an explicit method.

In this talk we consider $s$-stage IMEX peer methods of order $p=s$ for variable and of order $p=s+1$ for constant step sizes. They are combinations of $s$-stage superconvergent implicit and explicit peer methods. Due to their high stage order no order reduction appears. This is in contrast to one-step IMEX methods. On the other hand compared with multistep methods there is no order bound for A-stability of the implicit part.

We construct methods of order $p=s+1$ for $s=3,4,5$ where we compute the free parameters numerically to give good stability with respect to a general linear test problem frequently used in the literature. Numerical tests and comparison with two-step IMEX Runge-Kutta methods confirm the high potential of the superconvergent IMEX peer methods.