A revised robust HLLC(Harten-Lax-Van Leer-Contact) solver for multi-component Mie-Grüneisen mixture model is established. In Mie-Grüneisen mixture model, flux can be divided into conservative part and non-conservative part. Original HLLC scheme can well adapt to conservative part. As the original HLLC solver is directly extended to non-conservative part, oscillation is hard to be controlled. In original scheme, moving speed of discontinuous section refers to velocity at left or right side of the gird. It is replaced by average velocity within the grid. After the revision, the HLLC scheme is deduced again and extended to 2D problem. Numerical tests show that Mie-Grüneisen mixture model is robust and accurate with the help of modified HLLC solver.

Taking vorticity equation as a controlling equation, we simulate three vortexes developing progress with initial condition of Gaussian probability vorticity distribution. The wavelet expression of vorticity contains scaling coefficient term and wavelet coefficient tenn. The latter can be divided into a significant wavelet coefficient term and a trivial wavelet coefficient term reference to a given critical value. The critical value contains practical flow information. The vorticity is reconstructed only with scaling coefficients and significant wavelet coefficients. It saves computing time and capture most part of entrophy. Numerical results show that scaling coefficients together with significant wavelet coefficients is less than 10% of total wavelet coefficients, but they contains more than 99% of total enstrophy.

The transmission coefficient for one-dimensional potential with arbitrary shape is calculated using the recurrence formulae of one-dimensional multi-stepped potentials. Resonance tunneling phenomena are studied with the curves of transmission coefficients varying with the energy of incident particles. In particular, the resonance tunneling phenomena in the semiconductor material GaAlAs/GaAs/GaAlAs are discussed in detail. The calculated results show that the resonance tunneling phenomenon for the potential, which has an arbitrary shape well between double symmetric square barriers could also exist.

Self-consistent-field (SCF) method using signle electron wave functions expanded in terms of Gaussian functions is reported for atoms in two dimensions. The energy expressions are derived for closed-shell and open-shell configurations comprising, s, p, and d electrons and Hartree-Fock-Roothaan two-dimensions equation for closed-shell atoms. Finally, main results of SCF numerical computations are given and discussed.

A mathematical model using stream function, vorticity and cemperature as basic varibles is developed to represent the heat transfer and natural convection in the solidification prooess of castings. The solutions of the governing equations are carried out by means of a semi-implicit finite difference scheme based on fixed grid methodology. Consideration is given to the latent heat evolution and velocity conditiion in the liquid-solid phase zone(inte-rface).A specific enthalpy method associated with the alloy equilibrium phase diagram is employed to account for the latent heat of phase change. A new revision approach according to solid fraction is proposed for the treatment of the zero velocity condition as liquid region turns to solid. Moreover, the possibility of applying the present model to the problems of irregular geometries is discussed. In the last part of the paper, a tpyical freezing problem is solved, and the influence of natural convection on the solidification of castings is analysed in detail.