Some Parallelism Numerical Methods For Parabolic Equation

Parabolic equation is one of basic partial equations.In many science fields,many phenomena are described by parabolic equation(s)[1],such as the process of heat conduction and diffusion,the chemical reaction etc.Among modern numerical methods, the finite difference method is the earliest and most perfect method.So the finite difference method for solving parabolic equation is always a focal which peoples care about.As the parallel computer comes into being and develops,some disadvantage disappears in different means.For example,the classical explicit scheme is suit for parallel computing but it's conditional stability.Especially for high -dimension problem, the time step is limited very severely.The classical implicit and Crank-Nicolson scheme is absolutely stable,but they can be solved only by solving linear equations. Obviously they are not suit for parallel computing.So it's worthy to constructing other new difference methods which has better stability,parallelism and high-precision.In the early seventies,Miranker[2]pointed out that organizing the traditional difference method in order to parallel computing is the main method when we approximated the partial equation by finite difference method.Between the seventies,the research was mainly about high-order difference scheme for different equations[3-7]. But since the eighties,the situation changed because of Evans and Abdullah's work[8-12].In the early eighties,Evans and Abdullah proposed the idea which constructed group explicit method by appropriate combination of different Saul'yev asymmetric scheme[13].The group explicit method keeps the stability of numerical computing, and has better parallelism because it can be solved explicitly.Because some terms in the truncation error of different Saul'yev scheme is equal for their absolute value and the sigh is contrast,making use of them alternating in a time layer or different layer may cancel some truncation error and the calculation accuracy can be improved.And these Saul'yev scheme were implicit,but the group scheme can be solved explicitly because of appropriate combination.This is Evans-Abdullah's Group Explicit(GE).This work indicated that it's possible to construct new difference methods which satisfy the above conditions.But when they extended the method to variable coefficient problem, the proving of stability is difficult.Based on this,Zhang Baolin et al.proposed the idea which constructed the segment implicit scheme by using the Saul'yev asymmetric scheme,and set up a variety of explicit-implicit and pure implicit alternating parallel methods by making use of alternate technology[14-16].These methods can keep the stability and parallelism.After that,they extended it to variable coefficient problem and proved its stability by energy method.In the course of numerical experiments,they found that the result of segment or block parallel algorithm is better than the result of the method no splitting. So constructing new method by divide and conquer strategy can not only be used for parallel computing but also improve calculation accuracy.At the same time,there are many research coming into being.For example,Han Zhen studied a kind of pure explicit-implicit segment and block alternating method in detail[17,18].Fenghui et al. constructed the new iteration method for elliptical equation by elimination between difference scheme of different nodes,and the method had same parallelism as Jacobi method and higher convergence rate[19-21].Zhang Zhiyue gave the group explicit scheme for parabolic problem with variable coefficient and proved the stability by energy method[22].Wang Wenqia et al.constructed alternating segment explicit-implicit scheme for different problem,proved their stability and gave numerical experiments[13-26].Recently,Sanjiva K.Lele proposed high-rate compact difference scheme and do Fourier analyzing about error and compare it with traditional scheme[27].[28-32]Mark H.Carpenter et al.proposed some high-order compact difference scheme for different problem and did numerical analysis.Then what will be obtained by combining the high-order difference scheme with alternating group? There were many research coming into being[44-52].It's the new focal that combining the high-rate scheme and the strategy of dividing and conquering.Under Prof.Wang Wenqia's carefully instructing,the author constructs some parallel difference methods for some parabolic problem which contain iteration method and high-rate alternating group scheme.And the stability of these methods are proved and some numerical experiments indicate their applicability.The paper extends the work of the predecessors,and has non-repeatability.The paper is divided into five chapters.In Chapter 1,a new iteration method for 2D convection diffusion problem is constructer by making use of numerical Stencil[19].First,it gives the definition of Stencil for parabolic equation,and obtains the final numerical Stencil after three Stencil elimination. Based on this,the new iteration scheme is constructed.Then the convergence of iteration is proved by analyzing the iteration error and the convergence rate is compared with Jacobi method's.Finally the paper gives numerical experiment to show its applicability.The work about§1 is accepted by《Internatinal Journal of Computer Mathematics》The new idea of Chapter 1 is that numerical Stencil is first applied to 2D parabolic equation,and the high-rate convergence and parallelism iteration is constructed.The proving of stability is obtained and numerical experiments shows its applicability.Chapter 2 mainly uses the idea in[22,24]and gives the alternating group explicit for convection diffusion equation with variable coefficient.The alternating group explicit scheme is constructed by combining Crank-Nicolson different scheme and the idea of alternating group.First it gives four asymmetric difference scheme based on Crank-Nicolson difference scheme,and constructs the alternating group explicit scheme by combining these asymmetric difference scheme.Then its stability is proved by energy method and numerical experiment indicates its validity.The work about§2 has been submitted to《International Journal of Computer Mathematics》The new idea of Chapter 2 is that the alternating group explicit scheme is constructed for convection diffusion equation with variable coefficient by combining the idea of alternating group with Crank-Nicolson scheme.The following chapters mainly uses the idea in[14,23-26,46-48]and introduces the high-rate difference scheme[27-32].Based on this,some group methods are constructed for convection diffusion equation.These scheme are all absolutely stable and have parallelism. The rate of local truncation error can reach O(h~4).In Chapter 3,the high-rate explicit and implicit difference scheme are given first. Then four asymmetric schemes are constructed based on implicit scheme and the alternating segment explicit-implicit scheme is constructed by appropriate combination of above difference schemes.The absolutely stability is proved by Kellogg lemma in [33-34]and the local truncation error is obtained by derivation.Finally numerical experiments indicates the applicability and the rate of local truncation error can reach O(h~4).The work about§3 is accepted by《Chinese Journal of Computational Physics》 Chapter 4 gives the alternating group explicit scheme for the same equation.First, it gives high-rate Crank-Nicolson difference schemes.Based on this,eight asymmetric difference schemes are constructed in order to construct the alternating group explicit scheme.Then the absolute stability is proved by the same way,and the truncation error can reach O(τh) because some part of truncation error can be cancelled by using different scheme alternately between two time layers.Finally numerical experiment shows the method is applicable.The work about§4 is published in《Journal of Shan-Dong University》(Natural Science).Based on Chapter 4,Chapter 5 introduces the idea of alternating segment Crank-Nicolson scheme.It combines the eight asymmetric scheme with high-rate Crank-Nicolson scheme and constructs the high-rate alternating segment Crank-Nicolson scheme.It has absolute stability and parallelism.In addition,the truncation error of the points is much less because more parts are canceled by only using Crank-Nicolson on different time layer.It is proved that the result of alternating segment Crank-Nicolson is better than alternating group explicit scheme in numerical experiments. The work about§5 has been submitted to《Applied Mathematics and Mechanics》The new idea of the last three chapter is:1.)It introduces high-rate difference scheme in order to construct new alternating group scheme for the first time,2.) these scheme has better stability and parallelism,especially its error precision can reach O(h~4).