A Fourth Order Accurate Alternating Group Explicit Iterative Method For Diffusion-Convection Equations

In this paper, a fourth order finite exponential difference method of two layers and three points is presented for solving the one dimensional convection-diffusion equations. In order to reduce computational efforts, an efficient parallel iterative explicitmethod based on this difference scheme is established. The convergence theory of the iteration algorithm is reported briefly. The optimum acceleration parameter that is used to have an optimum number of iterations to acheieve convergence is given. Further the fourth-order accurate block alternating group iterative method for two-dimensional diffusion equations is constructed and the convergence of the method is proved. Three examples for 1D problems is presented to illustrate prac-ticality and usefulness of this parallel iteration method.The paper is composed of 5 chapters as shown below:In Chapterâ… , we mainly introduce the present situation and previous research results about the finite difference method and the parallel algorithms for the partial differential equations.Chapterâ…¡is divided into 5 sections.In Section I, we construct the fourth order finite difference method for the diffusion-reaction equations using the fourth-order compact difference approximation formulaand the exponent transformation u=ve^bt. In Sectionâ…¡, we construct the fourth order finite difference method of two layers and throe points for the convection-diffusionequations based on the finite difference scheme of the diffusion-reaction equations and another exponent transformationThe truncation error is O(Ï„²+h⁴).In Sectionâ…¢, based on our fourth-order implicit difference equationswe split the left hand side coefficient matrix A into A₁ and A₂ so that A₁ and A₂ are block diagonal matreces and each block is a (2×2) matrix. Then transfuring each of A₁ and A₂ in turn to the right hand side of the equation results in a single block diagonal matrix on the left side which is easily solvable. The parallel algorithm isIn Sectionâ…£, the convergence of the alternating group iterative method is proved. In Sectionâ…¤, the optimal acceleration parameter is given to beρ=(?), where a and b are such that, 0＜a≤μ,ν≤b, andμandνare eigenvalues of the matrices A₁ and A₂ respectively.Chapterâ…¢gives the numerical examples. For three specific equations, we give the comparison of numerical solutions and exact solutions at different times, the absolute error and relative error, the comparison of iteration numbers with over-relaxationiteration method (SOR) and the comparison of iteration numbers using different acceleration parameters p. Numerical rusults show that the algorithm has higher accuracy, it can convergence faster than the over-relaxation iteration method(SOR) andρ=(?) is optimal. In the fourth chapter,we further discuss the the fourth-order accurate block alternatinggroup iterative algorithm for the two dimensional diffusion equations using the fourth-order compact difference approximation formula of the second derivative. The convergence of the algorithm is also proved.In the fifth chapter, we make a conclusion and discuss the future prospects of research to do.