In this note new Rosenbrock methods for ODEs, DAEs, PDEs and PDAEs of index 1 are presented. These solvers are of order 3, have 3 or 4 internal stages, and fulfil certain order conditions to obtain a better convergence if inexact Jacobians and approximations of ∂f ∂t are used. A comparison with other Rosenbrock solvers shows the advantages of the new methods. (2000): 34A09, 65L80. AMS subject classification

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In this paper we present an analysis of a numerical method for a degenerate partial differential equation, called the Black-Scholes equation, governing American and European option pricing. The method is based on a fitted finite volume spatial discretization and an implicit time stepping technique. The analysis is performed within the framework of the vertical method of lines, where the spatial discretization is formulated as a Petrov-Galerkin finite element method with each basis function of the trial space being determined by a set of two-point boundary value problems. We establish the stability and an error bound for the solutions of the fully discretized system. Numerical results are presented to validate the theoretical results. Mathematics Subject Classification (2000)65 M 60 · 65 K 10 · 90 C 33

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