Enhanced Euler's method to a free boundary porous media flow problem

We study an ODE-based iterative method, the residual velocity method, for steady state free boundary problems. The convergence analysis of the method, as well as the numerical implementation based on Euler's method were provided by Donaldson and Wetton (J Appl Math 71 (2006), 877897). In this article, we develop an enhanced Euler's method which is nearly as simple as the modified Euler's method but can achieve a rapid convergence rate similar to the fourth-order Runge-Kutta method. Numerical results are also provided to verify the validity of our method. (c) 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012