The Difference Scheme Of Convection Diffusion Equation, The Uniform Convergence

Numerical boundary problems are generally acknowledged to be a hard task forPDE. There are many methods to deal with it, for example, finite difference method,finite element method etc. It is well known that if the standard finite difference methodor finite element method is used in regions where layers occur, then unphysicaloscillations arise. Green’s function method for the numerical solution of sharp boundaryproblem has high efficiency to avoid numerical oscillations and improve efficiency ofalgorithm compute. However, in several variables PDE, used as standard finitedifference method and finite element method are hard to derive in an analytical scheme.Green function can reduce the solution procedure to specific boundary problems, and itcan get the exact expression. Green’s function can deal with several variables PDE.Delta function’s some special properties can solve some outstanding boundary, thusmaking the solving process is simple and effective.For one dimensional convection-diffusion equation, used convection part anddiffusion part as operator M, an integration by parts yields uniformly convergentdifference schemes that it is interaction Green function with convection diffusionequation and the error analysis for the scheme is also given. Then the classics algorithmof chasing method is used to the numerical solution of equation, and the numericalexperiments prove the schemes’ effective. Then the stability analysis for differencescheme is proved to be exponential stability.For the case of two dimensional convection-diffusion equation, the integralequation is obtained by the case one dimensional. The five nine point difference schemeare derived by Green’s function and Green’s equation. The source function is appliedtwo dim lagrange interpolation method and it derive the calculation templates of thedifference scheme and the error analysis for the scheme is also given. Then the classicsalgorithm of BICG and BICGSTAB are used to the numerical solution of equation, andthe numerical experiments prove the schemes’ effective. Since this approach has thefollowing advantages: religious theory, less interactive, less error and simple, it can beextended to the n-dimensional case or applied to other aspects.