The Research On Characteristic Finite Difference Method Based On Shape-Preserving Interpolation By Functional Analysis

It is well known that the convection-diffusion equation is the governing equation of fluid dynamics. Using multistep characteristic difference scheme to solve convection-diffusion equation not only could make the scheme very stable in computing but also could increase the accuracy of the scheme, so it has comprehensive application in the field of numerical simulation of hydrodynamics. In addition, shape-preserving interpolation has the advantage that preserving the monotonicity of the function meanwhile it could achieve relatively high accuracy. Putting the above theories together, this paper mainly presents a method of multistep characteristics difference method based on cubic Hermite shape-preserving interpolation for solving convection-diffusion equtions and analyze the error of the scheme by functional analysis methods.The paper is organized into three major parts.The first part concerns about characteristic difference method based on cubic Hermite shape-preserving interpolation for one-dimentional linear and nonlinear convection-diffusion eqution. We discrete the convection term along characteristic line and discrete diffusion term by second order divided center difference. At the same time, cubic Hermite shape-preserving interpolation is exploited in the interpolation part. Finally, discrete L₂-norm error estimate is obtained.The second part focus on multistep characteristic difference method based on cubic Hermite shape-preserving interpolation for one-dimentional linear and nonlinear convection-diffusion eqution and the discrete L₂-norm error estimate of the method is obtained. Numerical examples show that applying multistep characteristics difference method in solving problems could...