In this paper, we propose a new definition of symplectic multistep methods. This definition differs from the old ones in that it is given via the one step method defined directly on $M$ which is corresponding to the $m$ step scheme defined on $M$ while the old definitions are given out by defining a corresponding one step method on $M\times M \times \cdots \times M=M^m$ with a set of new variables. The new definition gives out a step-transition operator $g: M\longrightarrow M$. Under our new definition, the Leap-frog method is symplectic only for linear Hamiltonian systems. The transition operator $g$ will be constructed via continued fractions and rational approximations.

In this paper, we study fully discrete spectral method and long time behavior of solution of generalized Kuramoto-Sivashinsky equation with periodic initial condition. We prove that the large time error estimation for fully discrete solution of spectral method. We prove the existence of approximate attractors $\mathcal{A}_N$, $\mathcal{A}_N^k$ respectively and $d (\mathcal{A}_N, \mathcal{A})\to 0$, $d (\mathcal{A}_N^k, \mathcal{A}) \to 0$.  

The finite element solutions of elliptic eigenvalue equations are shown to have a multi-parameter asymptotic error expansion. Based on this expansion and a splitting extrapolation technique, a parallel algorithm for solving multi-dimensional equations with high order accuracy is developed.

A class of asynchronous matrix multi-splitting multi-parameter relaxation methods, including the asynchronous matrix multisplitting SAOR, SSOR and SGS methods as well as the known asynchronous matrix multisplitting AOR, SOR and GS methods, etc., is proposed for solving the large sparse systems of linear equations by making use of the principle of sufficiently using the delayed information. These new methods can greatly execute the parallel computational efficiency of the MIMD-systems, and are shown to be convergent when the coefficient matrices are $H$-matrices. Moreover, necessary and sufficient conditions ensuring the convergence of these methods are concluded for the case that the coefficient matrices are $L$-matrices.

We consider the linearized incompressible Navier-Stokes (Oseen) equations in a flat channel. A sequence of approximations to the exact boundary condition at an artificial boundary is derived. Then the original problem is reduced to a boundary value problem in a bounded domain, which is well-posed. A finite element approximation on the bounded domain is given, furthermore, the error estimate of the finite element approximation is obtained. Numerical example shows that our artificial boundary conditions are very effective.

In this paper, we have obtained an expression of the bivariate Vandermonde determinant for the Elliptic Type Node Configuration in $R^2$, and discussed the possibility of the corresponding multivariate Lagrange, Hermite and Birkhoff interpolation.

A splitting iteration method is proposed to compute double $X_0$-breaking bifurcation points. The method will reduce the computational work and storage, it converges linearly with an adjustable speed. Numerical computation shows the effectiveness of splitting iteration method.

In this paper, the initial value problem of nonlinear reaction-diffusion equation is considered. The Dufort-Frankel finite difference approximation for the long time scheme is given for the $d$-dimensional reaction-diffusion equation with the two different cases. The global solution and global attractor are discussed for the Dufort-Frankel scheme. Moreover, properties of the solution are studied. The error estimate is presented in a finite time region and in the global time region for some special cases. Finally the numerical results for the equation are investigated for Allen-Cahn equation and some other equations and the homoclinic orbit is simulated numerically.