Finite Difference Methods For Several Evolution Equation

This dissertation focuses on the numerical solutions to several evolutionary partialdifferential equations. By using characteristics difference method, second orderupwind alternating direction method, some numerical schemes are proposed. The mainresults of this paper are outlined as follows:1. In chapter two, we present two difference schemes for two-phase, compressibleflow in porous media model, it is governed by a system of nonlinear partial differentialequations including pressure equation and concentration equation. Especially for theconcentration equation, it is a convection-diffusion equation. Firstly, we combinecharacteristics difference method and bilinear interpolation method to solve it. Thismethod can overcome oscillation and faults likely to occur in the traditional methods.A five points implicit difference scheme is proposed for the pressure equation. Thenthe convergence analysis based on Max-norm estimates is proved. Secondly,characteristics difference method is available, however, it suffers from complexitybrought about by the computation of boundary condition in solvingconvection-diffusion problem. Upwind scheme can overcome this difficulty, generalupwind scheme have once order accuracy. This paper, we present a second orderupwind alternating direction schemes, which have many advantages such as lesscomputational complexity, better stability, high order accuracy and so on. A stabilizingcorrections scheme is presented for the pressure equation. Their convergence is provedby using the energy method.2. In chapter three, we propose a high accurate difference scheme for parabolicequation. It's convergence is proved by using the energy method, the precision of thisscheme isΔt~2+h~4. General alternating direction scheme onl.y have accuracyΔt~α--h~β,whereα,β≤2.By means of numerical computing, we get the conclusion that thescheme in this chapter has higher precision than the other schemes.3. In chapter four, Barbour's two-host model for schistosomiasis japonica isdiscussed, it is governed by two nonlinear partial difference equations. We presentalternating direction schemes for it and its convergence is proved by using the energymethod.