High-order Implicit Difference Scheme AGEI Method For Solving The Diffusion Equation

Many physical phenomena are modeled by diffusion equations. The diffusion equations usually exist in chemical diffusion, heat conduction,medical science, bio-chemistry and certain biological process. At present, the parallel numerical solu-tion of diffusion equation is concerned with the development of parallel computing. Among modern numerical methods, the finite difference method is more perfect method.And for numerical solution of time-dependent diffusion problems, there are two types of finite difference method, explicit and implicit difference methods. The explicit method is easy to implement on parallel computer, but it has severe time step restriction, and at each time step one has to solve a global system of equations which implementation on parallel computer is not straightforward. So in the early eighties, Evans and Abdullah proposed the alternating group explicit method which is unconditional stability. And the idea can be directly used for parallel computation.In this paper, the idea of an alternating group explicit iterative method can be used to design a kind of new parallel iterative algorithm for solving diffusion equation. The basic idea is to divide the system of difference equations into a set. of subsystem that can be solved individually in parallel at every time step. In the. process of designing the algorithm, we use the matrix equation Au-F to realize. To obtain better truncation error, alternating group explicit iterative method is used to the equations at the n time step. Thus, the solution of the equation can obtain better truncation error at every time step.In the paper, the unconditional stability of the high order difference scheme is proved by Fourier method. And the truncation error can be achieved to O(т+ h4). We design the first, the second and Crank-Nicolson alternating group iterative method. And the property of convergence is proved by matrix theory. In the end of the paper, the result, of numerical experiment of example is obtained. It shows that the method has good applicability.