Rosenbrock methods with an explicit first stage

Abstract

Traditional Rosenbrock methods suffer from order reduction when applied to partial differential equations with non-homogeneous boundary conditions and source terms. The paper studies a family of Rosenbrock schemes with an explicit first stage. This structure allows one to construct algorithms with high stage orders, and which do not suffer from order reduction. The paper discusses additional order conditions needed for linear stability, for using inexact Jacobians, and implementation aspects. Second- and third-order practical schemes are constructed, and their application to one- and two-dimensional partial differential equations test problems confirm the theoretical findings.