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In this paper we first introduce the definition of contractivity region of Runge-Kutta methods and then examine the general features of the contractivity regions. We find that the intersections of the contractivity region and the axis place is $C^s$ are always either the whole axis plane or a generalized disk introduced by Dahlquist and Jeltsch. We also define the AN-contractivity and show that it is equivalent to the algebraic stability and can be determined locally in a neighborhood of the origion. However, many implicit methods are only r-circle contractive but not AN-contractive. A simple bound for the radius r of the r-circle contractive methods is given.  

This paper suggests a family of stiffly stable linear $k$-step methods with order $k$, for arbitrary $k$. Their stability regions are larger than those of the Gear method. Preliminary numerical test shows that these methods are efficient for stiff systems of ordinary differential equations with characteristic values near the imaginary axis.

Based on the singular decomposition of $2×2$ matrix an algorithm for reducing the matrix norm is presented. Under the optimal choice of the parameters the matrix B after transformation may be considered as "Locally normal", that is, the four corresponding elements of $BB^T-B^TB$ are zero.

Some definitions on stability of interpolating process are given and then the sufficient and necessary conditions are obtained. On this basis, we conclude that the Lagrange interpolation is unstable, whereas several types of piecewise low order polynomial interpolation are stable. For high order approximation with data on isometric nodes, we recommend the Bernstein approximation owing to its high stability. Some ideas on the relationship between stability and convergence of interpolating process are also presented.

This note presents a splitting extrapolation process, which uses successively one-dimensional extrapolation procedure along only one variable with other variables kept fixed. This splitting technique is applied to the numerical cubature of multiple integrals, multidimensional integral equations and the difference method for solving the Poisson equation. For each case, the corresponding error estimates are given. They show the advantage of this method over the isotropic extrapolation along all the variables.

Elliptic boundary-value problems can be reduced to integral equations on the boundary by many different ways. The canonical reduction, suggested by Prof. Feng Kang, is a natural and direct approach of boundary reduction. This paper gives the numerical method for solving harmonic and biharmonic canonical integral equations in interior or exterior circular domains, together with their convergence and error estimates. Using the theory of distributions, the difficulty caused by the singularities of integral kernel is overcome. Results of several numerical calculations verify the theoretical estimates.

In this paper we obtain a Hoffman-Wielandt type theorem and a Bauer-Fike type theorem for singular pencils of matrics. These results delineate the relations between the perturbation of the eigenvalues of a singular diagonalizable pencil $A-λB$ and the variation of the orthogonal projection onto the column space $\mathcal{R} \Bigg( \begin{matrix} A^H \\ B^H  \end{matrix} \Bigg)$.

An upwind difference scheme was given by the author in [5] for the numerical solution of steady-state problems. The present work studies this upwind scheme and its corresponding boundary scheme for the numerical solution of unsteady problems. For interior points the difference equations are approximations of the characteristic relations; for boundary points difference equatons are approximations of the characteristic relations corresponding to the outgoing characteristics and the "non-reflecting" boundary conditions. Calculation of a Riemann problem in a finite computational region yields promising numerical results.  

A splitting-up method, which splits the chemical kinetic terms from the flow terms, is presented to solve time-dependent, one-dimensional, laminar, premixed flame problems. An example for studying the development of an ozone decomposition flame is calculated. A movable boundary technique is adopted, so that the number of grid points can be significantly reduced. Special care is taken to maintain the accuracy of the solution. The results are checked in many ways. All checks show that the present method is satisfactory.

A new version of iterative method for solving Riemann problem of gas dynamics is presented. In practice the new procedure exhibited a good convergence in cases where Riemann solution involves a strong rarefaction wave or two rarefaction waves. In the other cases the new version is identical with Godunov procedure.