................................................................................................................. v CHAPTER

Finite element and finite difference methods have been widely used, among other methods, to numerically solve the Fokker-Planck equation for investigating the time history of the probability density function of linear and nonlinear 2d and 3d problems, and also the application to 4d problems has been addressed. However, due to the enormous increase of the computational costs, different strategie...

Based on multiplicative calculus, the finite difference schemes for the numerical solution of multiplicative differential equations and Volterra differential equations are presented. Sample problems were solved using these new approaches.

In this notes, we summarize numerical methods for solving Stokes equations on rectangular grid, and solve it by multigrid vcycle method with distributive Gauss-Seidel relaxation as smoothing. The numerical methods we concerned are MAC scheme, nonconforming rotate bilinear FEM and nonconforming rotate bilinear FVM. 1. PROBLEM STATEMENT We consider Stokes equation (1.1) 8 >< >: μ ~ u +rp =~ f in ...

1 STATEMENT OF THE PROBLEM Our goal is to introduce how derivatives can be approximated by using difference quotients. Suppose we have an interval [a,b] ⊂ R. Let a = x0 < x1 < ·· · < xN−1 < xN = b be a partition. We call {x1, . . . , xN−1} the interior points, and {x0, xN } the boundary. Given a function f : [a,b] → R, we want to approximate the derivative f ′ using our partition. 2 DIFFERENCE ...

we focus on the use of two stable and accurate explicit finite difference schemes in order to approximate the solution of stochastic partial differential equations of it¨o type, in particular, parabolic equations. the main properties of these deterministic difference methods, i.e., convergence, consistency, and stability, are separately developed for the stochastic cases.