Characteristic Compact Finite Volume Method Of The Convection-diffusion Equation

This dissertation mainly talks about the initial boundary value problems of one dimensional convection-diffusion equation:where u satisfies the periodic boundary conditions.The convection-diffusion equation is important in the practical application for it can describe a lot of physical phenomenon. The phenomenon of numerical fluc-tuation and numerical dispersion are always emerges in the traditional methods.Considering that the compact finite difference method and the compact finite vol-ume method have high precision, this dissertation combines the two methods with the characteristic line method, and proposes the characteristic compact finite dif-ference method and the characteristic compact finite volume method which two has higher precision.The second chapter is mainly talking about the convection-diffusion equation.The hyperbolic part is dispersed by Crank-Nicolson characteristic difference which has second-order precision regarding to time. And the interpolation part in the scheme adopts the cubic periodic spline interpolation method. As for the diffusion part, it is dispersed by fourth-order compact difference. Consequently the Crank-Nicolson characteristic compact difference scheme and the norm error estimation of the numerical solution is gived. The numerical example proves the effectiveness of the method.The third chapter furthermore, combines the compact algorithm with the local conservation of the finite volume method, then integrates the function on the con-trolled volume and selects the integral of unknown function on the controlled volume as the unknown quantity. The hyperbolic part is backward differenced by the charac-teristic line direction and the spatial part adopts the compact finite volume method.Then the characteristic compact finite volume scheme is formed. This scheme not only has the merit of keep local conservation just as the finite volume method. but has the character of high precision just as the compact difference method. Finally,the truncation error and the stability of the scheme are gived. The numerical ex-ample proves the performance of the method.