Preconditioning Newton-Krylor Methods for Variably Saturated Flow
Carol S. Woodward, Jim E. Jones · University of North Texas Digital Library (University of North Texas) · 2000
In this paper, we compare the effectiveness of three preconditioning strategies in simulations of variably saturated flow. Using Richards' equation as our model, we solve the nonlinear system using a Newton-Krylov method. Since Krylov solvers can stagnate, resulting in slow convergence, we investigate different strategies of preconditioning the Jacobian system. Our work uses a multigrid method to solve the preconditioning systems, with three different approximations to the Jacobian matrix. One approximation lags the nonlinearities, the second results from discarding selected off-diagonal contributions, and the third matrix considered is the full Jacobian. Results indicate that although the Jacobian is more accurate, its usage as a preconditioning matrix should be limited, as it requires much more storage than the simpler approximations. Also, simply lagging the nonlinearities gives a preconditioning matrix that is almost as effective as the full Jacobian but much easier to compute.