An Economical Difference Scheme For Nonlinear Convection-Diffusion Equations

An economical difference scheme is proposed to solve non-linear convection-diffusion equations.The transport term (?)+b(x,y)·▽u is discrctizcd by the methodof characteristies,then the difference systerm are docomposed to several problems of individual variable using alternating direction method.Two kinds of interpolation operators are supplied for the technique of characteristics.The stability and convergence are analysed by energy method.This paper discusses the following non-linear convection-diffusion equation.f∈C~0((?)×(0, T]) , u~0∈C~0((?))are known function.Assumedα(x, y, u)has positive upper and lower bounds .The n-layer approximate solution is known.By solving two one-dimensional problems,obtain the following A.D.I.scheme: The full text is divided into four chapters :The first chapter is about the busic outline of the advectiou-diffusion equation.In chapterⅡ,a formal difference scheme is established by using the method of characteristics and alternating direction implicit method.In chapterⅢ,two options are provided for interpolation methods.One is pieee-wise a bilinear interpolation and the other is the nine points biquadratic interpolation of product type.In chapterⅣ, the stability and convergence is analized by energy method for the schemes established above.The main results of this paper are two theorems:The theory obtained by the case of the piecewise a bilinear interpolation.The theory obtained by the case of the nine points biquadratic interpolation.