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Numerical solution methods for pricing American options are considered. We propose a second-order accurate Runge-Kutta scheme for the time discretization of the Black-Scholes partial differential equation with an early exercise constraint. We reformulate the algorithm introduced by Brennan and Schwartz into a simple form using LU decomposition and a modified backward substitution with a projection. In addition, we describe a direct solution method given by Elliott and Ockendon and we consider the similarity of these two direct algorithms. Numerical experiments demonstrate that the Runge-Kutta scheme produces smaller errors and less oscillations to numerical solutions than the Crank-Nicolson method. Experiments also show that the Brennan and Schwartz algorithm is much faster than the projected SOR method.