Finite Difference Domain Decomposition Parallel Implementation For Elliptic Equations Based On MPI

With the rapid development of high-speed network and multi-core pro-cessor technology, the performance of cluster system is increasing day by day. Since high cost performance and good scalability, it’s becoming popular paral-lel computing platform. It is very important to high performance computing. MPI (message passing interface)is a typical parallel programming model for distributed storage systems. It has became one of the most important parallel programming environment among the cluster system. We often solve partial differential equations in scientific computing and engineering calculation. In order to make the computation of complex engineering on acceptable level, we have to shorten the calculation time and improve calculation efficiency. At present, there are two ways to solve the problem:one is to improve the numer-ical method by using a variety of convergence technology to reduce computing time; the other one is to develop parallel algorithms and make parallel com-puting a reality, so that we can reduce the computation time and make some huge computation possible. Domain decomposition algorithm as an important branch of parallel algorithm has been the research focus in this field. Among them, based on the finite difference domain decomposition algorithm has be-come one of the important methods of numerical solution of partial differential equations.According to many physical problems in different regions have different performance, it is obviously difficult to achieve satisfactory results by making a uniform step treatment. Traditional Jacobi finite difference domain decomposi-tion parallel method discrete on the same step lenghth without fast algorithm, when the step length is very small, it needs a lot of iterations and takes lots of time. In this paper, we give a stability of the difference scheme of vari-able step size of the two-dimensional variable coefficient elliptic equation and get the same order of convergence. By using SOR iteration, we overcome the shortcomings of the traditional Jacobi iteration method. With MPI parallel program development environment, we use overlapping communication and computing to shield the network latency and improve the parallel performance of the program. Based on MPI communication MPI_COMM_WORLD, we establish a two-dimensional Cartesian topology and all types of MPI message-passing bases on the topology mechanism to improve the scalability.So that we implement the parallelization of ellipse partial differential equations numeri-cal calculation. Combined with a numerical example, we show that the new algorithm has lower time complexity.space complexity and better parallelism by contrasting with traditional Jacobi iteration.