The Study Of The Difference Schemes Of Convection-Diffusion Equations By Functional Analysis

The characteristics difference scheme of convection-diffusion equation has virtues that it is very stable to compute and the efficiency is very high. So it has a comprehensive application in numerical value simulation of hydrody-namics.The spline interpolation is very smooth and has a high accuracy.So it has a comprehensive application in the engineering practice. The paper combine the virtue of the characteristics difference method and the spline in-terplation.We obtain characteristics difference method based on cubic spline interpolation for convection-diffusion equtions and analyze the error estimate of exact solution and difference solution by functional analysis methods.The paper has three parts.The first part is characteristics difference method based on cubic natural spline interpolation for one-dimentional convection-diffusion eqution. we disrete the convection term along characteristics and the diffusion term is replaced by the second order centered difference quotients.We use cubic natural spline interpolation in the interpolation part and obtains discrete L₂-norm error estimate. The numerical examples show that to a large extent the scheme cancels the effect of the error in the interpolation.The secend part is characteristics difference method based on cubic natural spline interpolation for two-dimentional convection-diffusion eqution. We extend the first part to the two-dimensional convection-diffusion equation.The last part presents a higher order compact difference scheme and use cubic periodic spline interplation in the interpolation. The numerical example shows that the scheme is very good.