Finite Difference Parallel Scheme Bansed On MPI Implementation For Several Partial Differential Equations

Many applications in applied mathematics and engineering involve numerical solutions of partial differential equations(PDEs).The methods of numerical solution for PDEs mainly includes finite element method(FEM)and finite difference method(FDM).Nowadays,the problem of parallel FDM be used in parallel machine became more and more important with the rapid development of distributed computing platform.In this paper,the main idea is that Group Explicit(GE)method for several partial difference equations,and construct MPI(Message Passing Interface)parallel communication environment for partial difference equations.In the introduction of this article,first,the paper analyzed the significance and the status of parallel finite difference method and the development trend at home and abroad.Then introduced the development trend and research significance of MPI parallel technology.The first chapter briefly introduces the parallel computing principle and the configuration process of MPI.In the second chapter,we explore the parallel numerical algorithm of parabolic equation,first,for the second order parabolic equation,a group explicit method is constructed,the method is constructed by Saul'yev asymmetric difference schemes,and the stability of the method is briefly analyzed in chapter two.Then,the paper focuses on the numerical calculation of the GE method in MPI parallel environment.Two different MPI parallel algorithms are constructed,one is blocking communication(wait communication)mode,and the other is non-blocking communication(non-wait communication)mode.These two parallel algorithms both better than serial algorithm to calculate numerical solutions for parabolic equation,and the non-blocking communication is higher computational than the other.In the third chapter,the four-point scheme for solving the high-order parabolic equation is established by using the Saul'yev asymmetric scheme.It's one explicit difference scheme,so the computational domain can be divided into a number of large area,each sub-region compute themselves,and the parallel scheme is unconditionally stable.Then,the paper focuses on the numerical calculation of four-point scheme in MPI parallel environment.Blocking and non-blocking communication mode is constructed for fourth order parabolic equation.These two parallel patterns both better than serial algorithm,and the non-blocking communication mode is higher computational than the other,because the wait time in non-blocking communication mode less than blocking communication mode.In order to improve the efficiency of the MPI parallel model,the paper gives computing time of two message passing models in difference processes.In chapter four,a numerical parallel algorithm for nonlinear partial differential equations is explored.Take the Burgers equation as an example.First,through the Hopf-Cole transformation,it can be converted to linear equations,and then construct a finite parallel difference scheme,and then construct the MPI parallel algorithm for difference scheme.The parallel model is applied to large-scale numerical operations,and the results of parallel computing efficiency are obtained.