Accurate simulation of water resource management problems
requires the solution of large problems with many spatial zones.
In the case of variably saturated flow problems, we need to
develop scalable and highly efficient algorithms for solving
the large nonlinear systems of equations that arise from the
discretization of Richards' equation.
One approach to solving these systems is to apply a Newton method requiring a linear Jacobian system solve at each iteration. Although Newton's method can have very fast convergence properties, it can run slowly if the linear system solver is not efficient. In this work, we solve the linear systems using a multigrid preconditioned Krylov method. Properly designed multigrid solvers are optimally efficient in that the work grows linearly with problem size while the convergence rate is constant.
In this talk, we compare various strategies for solving the Jacobian system with multigrid. One approach is to base the multigrid preconditioner on the full Jacobian matrix, which is non-symmetric. To save storage, and simplify the preconditioner computation, we also consider multigrid preconditioners based on symmetric approximations to the full Jacobian matrix. We compare the efficiency of these various multigrid preconditioning strategies within the context of a Newton-Krylov method to solve the nonlinearities.
This work was performed under the auspices of the U.S. Department of Energy by the Lawrence Livermore National Laboratory under Contract W-7405-Eng-48.