In this paper, we are devoted to a study of the effect of the treatment of Neumann boundary conditions for one-dimensional advection-diffusion equation by the analysis method, and for tow-dimensional Poisson equation by the numerical test method. The reults show that the scheme is first-order accurate in space (i.e. O(h)can be derived first-order accurate difference solution or second-order solution, and the O(h²) or O(h³) scheme is only derived second-order solution.

We take 15 M_☉ model of Weaver et al. as initial model,the collapse process of stellar core between the onset of collapse and core bounce has been calculated by using the eouation of state of us (EOSⅡ).With the simplified nuclear statistical equilibrium system mentioned above, the treatment of the thermal effects of neutronization, nuclear disintegration, nuclear combination and nuclear excitation has been attemped.The results of us are essential agreed with Van Riper's results.calculated by using the ILPR's equation of state for the same initial model.As the center density of core reaches 1014gcm-3, the maximum velocity of fall and the Ms (mass at sonic point)in model D is greater than that of the typical model of Van Riper by 2.7% and 7% respectively. In model D the neutrino trapping density is less than that of the typical model of Van Riper by 3% (at mass layer 0.15M_☉) but the lepton fraction in model D is greater than that of typical model of Van Riper by 8%(at 0.15M_☉) and by 10% (at 0.5M_☉).

In this paper, the properties of system consisting of various nuclear species and free nucleons are described by a nuclear statistical equilibrium system consisting of a typical heavy nucleus, a particles, free neutrons and protons.starting from a semiempirical expression for the mass of the heavy nucleus and the entropy (energy) of their excited states, an equation of state is developed by a restricted minimization of the total free energy of a fixed volume of matter at given temperature and electron fraction.his scheme gave good agreement with the equation of El Eid & Hillebrandt at density 10⁹-10¹²g cm^-3 and with lamb et al (ILPR's equation of state at density 10¹²-10¹⁴g cm^-3,ye=3,σ/k=1.5k.The imput physics such as the rate of electron capture on matter has been treated also more carefully.