The main purpose of this paper is to develop and simplify the general conditions for an $s$-stage explicit canonical difference scheme of $q$-th order, while the simplified order conditions for canonical RKN methods which are applied to a special kind of second order ordinary differential equations are also obtained here.  

In this paper, a nonmonotonic trust region method for optimization problems with equality constraints is proposed by introducing a nonsmooth merit function and adopting a correction step. It is proved that all accumulation points of the iterates generated by the proposed algorithm are Kuhn-Tucker points and that the algorithm is $q$-superlinearly convergent.

We extend the SCGS smoothing procedure (Symmetrical Collective Gauss-Seidel relaxation) proposed by S. P. Vanka$^{[4]}$, for multigrid solvers of the steady viscous incompressible Navier-Stokes equations, to corresponding line-wise versions. The resulting relaxation schemes are integrated into the multigrid solver based on second-order upwind differencing presented in [5]. Numerical comparisons on the efficiency of point-wise and line-wise relaxations are presented.

The numerical simulation of the steady incompressible viscous flows in a no-slip channel is considered. A discrete artificial boundary condition on a given segmental artificial boundary is designed by the method of lines. Then the original problem is reduced to a boundary value problem of Navier-Stokes equations on a bounded domain. The numerical examples show that this artificial boundary condition is very effective and more accurate than Dirichlet and Neumann boundary conditions used in engineering literature.

The stability of symplectic algorithms is discussed in this paper. There are following conclusions.
1. Symplectic Runge-Kutta methods and symplectic one-step methods with high order derivative are unconditionally critically stable for Hamiltonian systems. Only some of them are A-stable for non-Hamiltonian systems. The criterion of judging A-stability is given.
2. The hopscotch schemes are conditionally critically stable for Hamiltonian systems. Their stability regions are only a segment on the imaginary axis for non-Hamiltonian systems.
3. All linear symplectic multistep methods are conditionally critically stable except the trapezoidal formula which is unconditionally critically stable for Hamiltonian systems. Only the trapezoidal formula is A-stable, and others only have segments on the imaginary axis as their stability regions for non-Hamiltonian systems.  

The unequal meshsteps are unavoidable in general for scientific and engineering computations especially in large scale computations. The analysis of difference schemes with nonuniform meshes is very rare even by use of fully heuristic methods. For the purpose of the systematic and theoretical study of the finite difference method with nonuniform meshes for the problems of partial differential equations, the general interpolation formulas for the spaces of discrete functions of one index with unequal meshsteps are established in the present work. These formulas give the connected relationships among the norms of various types, such as the sum of powers of discrete values, the discrete maximum modulo, the discrete Hölder and Lipschitz coefficients.