In this paper the general finite difference schemes with intrinsic parallelism for the boundary value problem of the semilinear parabolic system of divergence type with bounded coefficients are constructed, and the existence and uniqueness of the difference solution for the general schemes are proved. And the convergence of the solutions of the difference schemes to the generalized solution of the original boundary value problem of the semilinear parabolic system is obtained. The multi-dimensional problems are also studied.

In this paper, the partial projection finite element method is applied to the time-dependent problem – the damped vibrating Timoshenko beam model. It is proved that this method allows some new combinations of interpolations for stress and displacement fields. When assuming that a smooth solution exists, we obtain optimal convergence rates with constants independent of the beam thickness. 

In this paper, following the paper [7], we analyze the "sharp" estimate of the rate of entropy dissipation of the fully discrete MUSCL type Godunov schemes by the general compact theory introduced by Coquel-LeFloch [1, 2], and find: because of small viscosity of the above schemes, in the vicinity of shock wave, the estimate of the above schemes is more easily obtained, but for rarefaction wave, we must impose a "sharp" condition on limiter function in order to keep its entropy dissipation and its convergence.

An efficient iteration-by-subdomain method (known as the Schwarz alternating algorithm) for incompressible viscous/inviscid coupled model is presented. Appropriate spectral collocation approximations are proposed. The convergence analysis shows that the iterative algorithms converge with a rate independent of the polynomial degree used.

The object of this paper is to establish the relation between stability and convergence of the numerical methods for the evolution equation $u_t-Au-f(u)=g(t)$ on Banach space $V$, and to prove the long-time error estimates for the approximation solutions. At first, we give the definition of long-time stability, and then prove the fact that stability and compatibility imply the uniform convergence on the infinite time region. Thus, we establish a general frame in order to prove the long-time convergence. This frame includes finite element methods and finite difference methods of the evolution equations, especially the semilinear parabolic and hyperbolic partial differential equations. As applications of these results we prove the estimates obtained by Larsson [5] and Sanz-serna and Stuart [6].

Correction methods for the steady semi-periodic motion of incompressible fluid are investigated. The idea is similar to the influence matrix to solve the lack of vorticity boundary conditions. For any given boundary condition of the vorticity, the coupled vorticity-stream function formulation is solved. Then solve the governing equations with the correction boundary conditions to improve the solution. These equations are numerically solved by Fourier series truncation and finite difference method. The two numerical techniques are employed to treat the non-linear terms. The first method for small Reynolds number $R=0-50$ has the same results as that in M. Anwar and S.C.R. Dennis' report. The second one for $R>50$ obtains the reliable results.

Some optimization problems in mathematical programming can be translated to a variant variational inequality of the following form: Find a vector $\u^*$,such that $$Q(u^*)∈Ω,(v-Q(u^*))^Tu^* ≥ 0, ∀_v∈Ω.$$. This paper presents a simple iterative method for solving this class of variational inequalities. The method can be viewed as an extension of the Goldstein's projection method. Some results of preliminary numerical experiments are given to indicate its applications.

We study the Fredholm integro-differential equation $$D^{2s}_xσ(x) + \int_{-∞}^{+∞}  k(x  y)σ(y)dy = g(x)$$ by the wavelet method. Here $σ(x)$ is the unknown fun tion to be found, $k(y)$ is a convolution kernel and $g(x)$ is a given function. Following the idea in [7], the equation is discretized with respect to two different wavelet bases. We then have two different linear systems. One of them is a Toeplitz-Hankel system of the form $(H_n + T_n)x = b$ where $T_n$ is a Toeplitz matrix and $H_n$ is a Hankel matrix. The other one is a system $(B_n + C_n)y = d$ with condition number $k = O(1)$ after a diagonal scaling. By using the preconditioned conjugate gradient (PCG) method with the fast wavelet transform (FWT) and the fast iterative Toeplitz solver, we can solve the systems in $O(n$ ${\rm log}$ $n)$ operations.

This paper deals with the stability analysis of numerical methods for the solution of delay differential equations. We focus on the behaviour of three $θ$-methods in the solution of the linear test equation $u'(t)=A(t)u(t)+B(t)u(τ(t))$ with $τ(t)$ and $A(t), B(t)$ continuous matrix functions. The stability regions for the three $θ$-methods are determined.