The analysis of the finite difference schemes with nonuniform meshes for the problems of partial differential equations is extremely rare even for very simple problems and even for the method of fully heuristic character. In the present work the boundary value problem for quasilinear parabolic system is solved by the finite difference method with nonuniform meshes. By using of the interpolation formulas for the spaces of discrete functions with unequal meshsteps and the method of a priori estimation for the discrete solutions of finite difference schemes with nonuniform meshes, the absolute and relative convergence of the discrete solutions of the finite difference scheme are proved. The limiting vector function is just the unique generalized solution of the original problem for the parabolic system.

In this paper, concerned with the Cauchy problem for 2D nonlinear hyperbolic conservation laws, we construct a class of uniformly second order accurate finite difference schemes, which are based on the E-schemes. By applying the convergence theorem of Coquel-Le Floch [1], the family of approximate solutions defined by the scheme is proven to converge to the unique entropy weak $L^{\infty}$-solution. Furthermore, some numerical experiments on the Cauchy problem for the advection equation and the Riemann problem for the 2D Burgers equation are given and the relatively satisfied result is obtained.

This paper is devoted to study of an iterative procedure for domain decomposition method of second order elliptic problem with mixed boundary conditions (i.e., Dirichlet condition on a part of boundary and Neumann condition on the another part of boundary). For the pure Dirichlet problem, Marini and Quarteroni [3], [4] considered a similar approach, which is extended to more complex problem in this paper.

This paper considers the concave minimization problem with linear constraints, proposes a technique which may avoid the unsuitable Karush-Kuhn-Tucker points, then combines this technique with Frank-Wolfe method and simplex method to form a pivoting method which can determine a strictly local minimizer of the problem in a finite number of iterations. Based on strictly local minimizers, a new cutting plane method is proposed. Under some mild conditions, the new cutting plane method is proved to be finitely terminated at an $\epsilon $-global minimizer of the problem.

In this paper we consider numerical simulation of incompressible viscous flow in an infinite slip channel. Local artificial boundary conditions at an artificial boundary are derived by the continuity of velocity and normal stress at the segment artificial boundary. Then the original problem is reduced to a boundary value problem on a bounded computational domain. Numerical example shows that our artificial boundary conditions are very effective.

A shock interaction problem is solved with finite difference methods for a hypersonic flow of air with chemical reactions. If a body has two concave corners, a secondary shock is formed in the shock layer and it meets the main shock later. As the two shocks meet, the flow becomes singular at the interaction point, and a new main shock, a contact discontinuity and an expansion wave appear as a result of interaction between the two shocks. Therefore, the problem is very complicated. Using proper combinations of implicit and explicit finite difference schemes according to the property of the equations and the boundary conditions, we compute the flow behind the interaction point successfully.

In this paper we seek the solutions of the time dependent Ginzburg-Landau model for type-II superconductors such that the associated physical observables are spatially periodic with respect to some lattice whose basic lattice cell is not necessarily rectangular. After appropriately fixing the gauge, the model can be formulated as a system of nonlinear parabolic partial differential equations with quasi-periodic boundary conditions. We first give some results concerning the existence, uniqueness and regularity of solutions and then we propose a semi-implicit finite element scheme solving the system of nonlinear partial differential equations and show the optimal error estimates both in the $L^2$ and energy norm. We also report on some numerical results at the end of the paper.