Efficient and robust solvers for Monge-Ampere equations

Efficient and robust multigrid method solvers have been developed for Monge-Ampere equations (MAEs), which are second-order nonlinear partial differential equations. The two principal MAEs studied in this dissertation come from atmospheric dynamics.;Five specific compact discretizations are investigated for a simple MAE. By doing truncation error analysis and local Fourier analysis, the optimal discretization is chosen. It is shown that there is no fourth-order compact discretization for MAEs. To solve MAEs to fourth-order accuracy, tau-extrapolation is applied.;The semigeostrophic model (for atmospheric flows with an approximate balance between the wind and mass fields) includes a MAE. For this problem we reproduced results of a previous study by Fulton. In addition, alternate discretizations are applied, and fourth-order accuracy is obtained by using tau-extrapolation.;The balanced vortex model (for circularly symmetric balanced atmospheric flows) includes a more complicated MAE. The forcing for this problem depends on time; at larger times the forcing is stronger, making the problem more nonlinear and harder to solve. The continuation method is used. The continuation method divides the time into a series of times. At each time the problem is solved on the desired grid, and the final solution is used as the initial approximation for the next time, which will make the method very expensive. The full multigrid method is efficient and cheap, but it cannot solve the problem at a large time. A combination of the continuation and full multigrid methods (C-FMG method) is designed. Two versions of the C-FMG methods have been developed. One is partially adaptive and the other is fully adaptive. Numerical results demonstrate that the fully adaptive C-FMG method is efficient and robust.