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. (C) 2005 Elsevier Ltd. All rights reserved.