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We propose a very efficient numerical method to solve a nonlinear partial differential problem that is encountered in the pricing of American options. In particular, by using the front-fixing approach originally developed in Wu and Kwok (J Financ Eng 6:83–97, 1997) and Nielsen et al. (J Comput Finance 5:69–97, 2002) in conjunction with a suitable change of the time variable, a (nonlinear) partial differential problem is obtained which can be solved very efficiently by means of a finite difference scheme enhanced by repeated Richardson extrapolation. Numerical results are presented showing that the novel algorithm yields excellent results, and performs significantly better than a finite different method with Bermudan approximation.