An evaluation of temporally adaptive transformation approaches for solving Richards' equation

Developing robust and ecient numerical solution methods for Richards' equation (RE) continues to be a challenge for certain problems. We consider such a problem here: in®ltration into unsaturated porous media initially at static conditions for uniform and non-uniform pore size media. For ponded boundary conditions, a sharp in®ltration front results, which propagates through the media. We evaluate the resultant solution method for robustness and eciency using combinations of variable transformation and adaptive time-stepping methods. Transformation methods introduce a change of variable that results in a smoother solution, which is more amenable to ecient numerical solution. We use adaptive time-stepping methods to adjust the time-step size, and in some cases the order of the solution method, to meet a constraint on nonlinear solution convergence properties or a solution error criterion. Results for three test problems showed that adaptive time-stepping methods provided robust solutions; in most cases transforming the dependent variable led to more ecient solutions than untransformed approaches, especially as the pore-size uniformity increased; and the higher-order adaptive time integration method was robust and the most ecient method evaluated. Ó 1999 Elsevier Science Ltd. All rights reserved