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.

In this paper, we consider an implicit finite difference method for solving fuzzy partial differential equations (FPDEs). We present stability of this method and solve the parabolic equation with this scheme.

An explicit finite difference scheme for one-dimensional Burgers equation is derived from the lattice Boltzmannmethod. The system of the lattice Boltzmann equations for the distribution of the fictitious particles is rewritten as a three-level finite difference equation. The scheme is monotonic and satisfies maximum value principle; therefore, the stability is proved. Numerical solutions have b...

We put forward and analyze an explicit finite difference scheme for the Camassa-Holm shallow water equation that can handle general H1 initial data and thus peakon-antipeakon interactions. Assuming a specified condition restricting the time step in terms of the spatial discretization parameter, we prove that the difference scheme converges strongly in H1 towards a dissipative weak solution of C...

If we consider a finite difference method simply as a set of equations containing a small parameter (the grid spacing), it is evident that the tools of asymptotic analysis can give us useful information about the method. The applicability of this approach for studying consistency, long time behavior and stability is demonstrated. As example, we use a simple lattice Boltzmann scheme for the 1D a...

We investigate the formulation and implementation of new explicit group iterative methods in solving the two-dimensional Poisson equation with Dirichlet boundary conditions. The methods are derived from a fourth order compact nine point finite difference discretization. The methods are compared with the existing second order standard five point formula to show the dramatic improvement in comput...