The oldest and simplest difference scheme is the explicit Euler method. Usually, it is not symplectic for general Hamiltonian systems. It is interesting to ask: Under what conditions of Hamiltonians, the explicit Euler method becomes symplectic? In this paper, we give the class of Hamiltonians for which systems the explicit Euler method is symplectic. In fact, in these cases, the explicit Euler method is really the phase flow of the systems, therefore symplectic. Most of important Hamiltonian systems can be decomposed as the summation of these simple systems. Then composition of the Euler method acting on these systems yields a symplectic method, also explicit. These systems are called symplectically separable. Classical separable Hamiltonian systems are symplectically separable. Especially, we prove that any polynomial Hamiltonian is symplectically separable.

In this paper, spectral and pseudospectral methods are applied to both time and space variables for parabolic equations. Spectral and pseudospectral schemes are given, and error estimates are obtained for approximate solutions.

In this paper, the unitarily invariant norm $\|\cdot\|$ on $\mathbb{C}^{m\times n}$ is used. We first discuss the problem under what case, a rectangular matrix $A$ has minimum condition number $K (A)=\| A \| \ \|A^+\|$, where $A^+$ designates the Moore-Penrose inverse of $A$; and under what condition, a square matrix $A$ has minimum condition number for its eigenproblem? Then we consider the second problem, i.e., optimum of $K (A)=\|A\| \ \|A^{-1}\|_2$ in error estimation. 

We construct and analyze a family of semi-discretized difference schemes with two parameters for the Korteweg-de Vries (KdV) equation. The scheme possesses the first four near-conserved quantities for periodic boundary conditions. The existence and the convergence of its global solution in Sobolev space $\boldsymbol{L}_{\infty} (0, T; \boldsymbol{H}^3)$ are proved and the scheme is also stable about initial values. Furthermore, the scheme conserves exactly the first two conserved quantities in the special case. 

An iterative nonoverlapping domain decomposition procedure is proposed and analyzed for linear elliptic problems. At the interface of two subdomains, one subdomain problem requires that Dirichlet data be passed to it from the previous iteration level, while the other subdomain problem requires that Neumann data be passed to it. This procedure is suitable for parallel processing. A convergence analysis is established. Standard and mixed finite element methods are employed to give discrete versions of this domain decomposition algorithm. Numerical experiments are conducted to show the effectiveness of the method.

In this paper we introduce a Petrov-Galerkin approximation model to the solution of linear and semi-linear elliptic boundary value problems in which piecewise quadratic polynomial space and piecewise linear polynomial space are used as the shape function space and the test function space, respectively. We prove that the approximation order of the standard quadratic finite element can be attained in this Petrov-Galerkin model. Based on the so-called "contractivity" of the interpolation operator, we further prove that the defect iterative sequence of the linear finite element solution converge to the proposed Petrov-Galerkin approximate solution.

Steklov eigenvalue problem of Helmholtz equation is considered in the present paper. Steklov eigenvalue problem is reduced to a new variational formula on the boundary of a given domain, in which the self-adjoint property of the original differential operator is kept and the calculating of hyper-singular integral is avoided. A numerical example showing the efficiency of this method and an optimal error estimate are given.

In this paper, a new class of high resolution schemes satisfying the "condition A" (SCA) and the "condition B" (SCB) for hyperbolic systems of conservation laws in one and two dimensions are constructed. Moreover, the results of the numerical experiments by using these schemes are given for the system of Euler equations in one and two dimensions.