High Efficiency Differential Format. Diffusion Equation And The Convection-diffusion Equation And Its Parallel Algorithm

In this paper we report a class of high-precise,high-stability difference schemes in the product difference space and it applies to the one-order one-dimensional constant coefficient parabolic equation.The last we give a high-efficiency parallel algorithm. In this chapter every type scheme is given numerical compute,and the results validate the theory analysis.There are four parts in this paper:The first, we give some basic definitions and some lemmas about combination difference algorithm,we analyze the relation of node distributing and type of difference schemes, then bring forward the undetermined coefficient principle in the product difference space.The second, a semi-explicit differencing scheme of seven points for solving the parabolic equations of one-dimensions is presented,The truncation error isO(τ~2 + h~4) ,and the stability condition is 0 < r≤0.4647.The third, a semi-explicit different scheme of five points for solving the diffusion convection equation is presented in this paper. It can be calculated explicitly forinitial-boundary value problems, The truncation error is O(τ~2 +h~2) .and thestability is well.The last, we give a high-efficiency parallel algorithm, the schemetruncation error is O(τ+ h) and the stability condition is r≤h/ε< 2/|α| .the schemeis given numerical compute,and the results validate the theory analysis.