Assessment of initial solution estimates and adaptive vs. heuristic time stepping for variably saturated flow

Abstract

The performance of improved (higher order extrapolation) initial estimates and a poste(heuristic) and a priori(adaptive) techniques for time step adaptation in the solution of the nonlinear Richards ' equation governing flow in unsaturated porous media is evaluated. The so-called heuristic technique uses the convergence behavior of the iterative scheme (Picard linearization in this case) to estimate the next time step (e.g., increase the step size if convergence was fast, decrease it if convergence was slow). The a priori technique adapts the time step on the basis of an approximation of the local time truncation error obtained from finite difference estimates of the first and second order time derivatives. The local error estimate is used to predict the largest step size that satisfies a preset error tolerance. The sample problem used to assess these various schemes is characterized by non-uniform (in time) boundary conditions and sharp gradients in the soil hydraulic properties and is thus a challenging test case for numerical simulators. The influence of chord slope approximations to the derivatives of the hydraulic functions in the presence of steep gradients is also included in the assessment of the time stepping techniques. It is found that higher order initial solution estimates improve the convergence of the Picard linearization scheme when using the a posteriori technique. When simulating strong nonlinearities. the a priori technique diminishes computational performance the a posteriori technique seems to be more appropriate to guide the simulation under such circumstances.