ROS 3 P - 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 nonlin-ear parabolic problems. Since Rosenbrock methods suuer from order reductions when they are applied to partial diierential equations, additional order conditions have to be satissed. 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. Steinebach 12] modiied the well{known solver RODAS of Hairer and Wanner 1] 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 eecient 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.