Alternating Direction Method For Solving A Class Of Parabolic Equations With Mixed Derivatives

In the research field of scientific computing,it is difficult to obtain the exact solutions of many complicated partial differential equations,thus the theory of numerical methods is quite useful.The numerical simulation of the problems from practical engineering project belongs to the forefront of scientific computing research field nowadays,where the exploration of numerical solutions to parabolic equations is a subject with certain difficulties.Finite element method,finite difference method and finite volume method are the most important numerical solutions of the partial differential equation.And finite difference method for its easy operation,and flexibility,is often chosen to discrete parabolic equation.The explicit difference scheme has a small amount of calculation but poor stability.And stability of implicit difference scheme is better,but in the numerical process of concrete,usually together with high dimensional or nonlinear equations,where the calculating amount is quite large.In this paper,we consider the following initial-boundary value problem of nonlinear parabolic equations with mixed derivatives:(?)A is a nonlinear elliptic differential operator.For this problem,we establish a fini te difference scheme and alternating direction format.The main idea of this method is:decompose the nonlinear elliptic equations of differential operator: 1 2A(28)A(10)A,1A is a symmetric linear constant coefficient elliptic operator,using implicit scheme approxim ation;2A with the explicit approximation.The article is divided into five main contents,the first part is to introduce the background and research meaning of the parabolic equations,and the current commonly theory and method used to solve this kind of nonlinear parabolic equations.The second part,we introduce the related difference scheme of one-dimensional parabolic equation.The third part,we explain the related difference scheme of two-dimensional parabolic equation.The fourth part,we introduce the method for the linear parabolic equation,and provide the numerical experiment.The fifth part,we introduce the method for the given nonlinear parabolic equation,and provide the numerical experiment.Finally,we summarize and prospect.