Parallel iterated methods based on variable step-size multistep Runge-Kutta methods of Radau type for stiff problems

In our previous paper [3], the performance of a variable step-size implementation of Parallel Iterated Methods based on Multistep Runge-Kutta methods (PIMRK) is far from satisfactory. This is due to the fact that the underlying parameters of the Multistep Runge-Kutta (MRK) method, and the splitting matrices W that are needed to solve the nonlinear system are designed on a fixed step-size basis. Similar unsatisfactory results based on this method were also noted by Schneider [12], who showed that the method is only suitable when the step-size does not vary too often. In this paper, we design the Variable step-size Multistep Runge-Kutta (VMRK) method as the underlying formula for Parallel Iterated methods. The numerical results show that Parallel Iterated Variable step-size MRK (PIVMRK) methods improve substantially on the PIMRK methods and are usually competitive with Parallel Iterated Runge-Kutta methods (PIRKs).