Fourth-order parallel rosenbrock formulae for stiff systems

Rosenbrock methods are frequently used for the numerical solution of stiff initial value problems. Such linearly implicit methods are characterized by a relatively easy implementation together with excellent linear stability properties. In this paper, we consider modified Rosenbrock methods with s external linearly implicit stages each of which contains p additional linearly implicit internal stages. The internal stages are already parallel so that they can be solved for independently of each other and, consequently, the processors need to exchange their results only after the completion of each of the s external stages. We focus on the design of fourth-order methods with three external stages. Using embedded third-order methods, a variable step size implementation is compared with well-known Rosenbrock codes for performance on the Robertson problem.