| This thesis is on numerical methods for Fokker-Planck equations, especially those equations with degenerate diffusion coefficients. Emphasis is focused on the two-dimensional case which corresponds to second order stochastic differential equations.;First, Gauss-Galerkin/finite difference methods are proposed. To solve two dimensional Fokker-Planck equations which involve two spatial variables and one time variable, the idea of variable splitting is adopted. More specifically, finite difference methods are utilized in one of the spatial variables and Gauss-Galerkin methods are employed in the other. As a consequence, the Gauss-Galerkin approach is used only in one dimension at each time step.;After two-dimensional Fokker-Planck equations are discretized to semi-discrete equations by finite difference methods in one direction, the convergence of the difference approximation is established based on energy type estimates. Using theory of measures and moments, the Gauss-Galerkin approximation for the semi-discrete equations is shown to converge when one-dimensional Gauss-Galerkin approximation is used in the second direction. Combining the above results, the convergence of the Gauss-Galerkin/finite difference approximation is established.;Computer implementation and intensive numerical tests of Gauss-Galerkin/finite difference methods are then carried out. The methods appear very efficient and very accurate. |
