ROS3P : an accurate third-order Rosenbrock solver designed for parabolic problems

In this note we present a new Rosenbrock solver which is third--order accurate for nonlinear parabolic problems. Since Rosenbrock methods suffer from order reductions when they are applied to partial differential equations, additional order conditions have to be satisfied. Although these conditions have been known for a longer time, from the practical point of view only little has been done to construct new methods. {sc Steinebach cite{St95 modified the well--known solver RODAS of {sc Hairer and {sc Wanner cite{HaWa96 to preserve its classical order four for special problem classes including linear parabolic equations. His solver RODASP, however, drops down to order three for nonlinear parabolic problems. Our motivation here was to derive an efficient third--order Rosenbrock solver for the nonlinear situation. Such a method exists with three stages and two function evaluations only. A comparison with other third--order methods shows the substantial potential of our new method.