【作者】 张青洁；

【导师】 王文洽；

【作者基本信息】 山东大学 ， 计算数学， 2009， 博士

【摘要】 色散方程作为重要的数学物理方程之一一直受到业内人士的普遍关注,在非线性波及孤立子理论的物理问题中,也一直占有相当重要的位置。鉴于色散方程在物理领域重要的应用价值,人们已经开始广泛的关注其数值解法的研究,许多专家、学者在这方面也已经做了不少工作,例如文献([1—13])。其中,文献([4])详细讨论了各种差分格式及其相应的稳定性情况。这些差分格式可分为显式格式和隐式格式两大类。我们知道显格式形式简单并适于并行计算,但其稳定性条件通常比较苛刻、不易实现。尽管人们在稳定性条件方面做过一些改进([7,8]),但是这些改进还是极其有限的。比如文献([5])讨论了一类两参数的恒稳显格式,但是参数的选择还是要满足比较复杂的条件的。相对于显格式而言,隐式格式虽然具有稳定性好的优点,然而它却不能直接应用于并行计算。随着信息时代的到来和计算机的蓬勃发展,并行计算以其快速解决大型且复杂的计算问题的特点迅速吸引了很多业内人士的普遍关注([10—37]),因此,怎样找到一个稳定的、适合并行计算的数值求解方法,便成了相关研究人员亟待解决的重要问题。关于交替分组算法的研究是随着并行数值计算在计算机上的广泛应用而逐步深入的,目前,两类主要的并行算法就是:交替分组方法([2,4,9—24])和区域分裂算法([38—49,63,64,66])。前者是无条件稳定的,所以我们通常可以采用比较大的时间步长,而后者是条件稳定的,因此,在使用过程中,我们通常需要选取比较小的时间步长来进行计算。交替分组方法已成为行之有效的并行数值算法之一,它不但是绝对稳定的,而且还具有本质并行的特性。例如,抛物方程的并行差分解法已经在很多文献中被广泛研究([15,23,25,63—67]),关于扩散方程和对流-扩散方程的并行差分算法的研究也已经有了不少成果([15—17,19—21]),近些年来,交替分组方法的研究和应用又逐渐扩展到了三阶色散方程、KdV方程等领域([1—13,22,50])。不过,对于三阶色散偏微分方程而言,这样的并行差分算法并不多见。早在1983年,Evans和Abdullah首先提出了交替分组显式算法(AGE)([15,16]),后来,张宝林提出了交替分段显隐算法(ASEI)([19])。2000年以来,朱少红又将交替分组显式算法(AGE)推广到了三阶色散方程的求解过程中来([11,12])。我们谈到的这些算法都是无条件稳定的,并且可以并行计算(随着计算机的蓬勃发展,并行计算也越来越多的被人们关注),不过,这些算法在空间上的收敛阶都只能接近2阶。众所周知,提高数值解的精度也一直是数值解法研究人员的一个重要的目标和努力方向([50—62]),这也是我们在求解理论问题和实际应用问题过程中都不会改变的追求。综上所述,本文作者在导师的悉心指导和精心培育下,提出了一类求解具有周期边界条件的色散方程的高精度、可并行、绝对稳定的算法。在论文的第一章,作者介绍了色散方程的高精度并行迭代法。在论文的第二、三章中,我们将给出四类Saul’yev型非对称差分格式来求解色散方程。基于这些Saul’yev型格式,我们又分别给出了求解带周期边界条件的色散方程的新的交替六点分组算法、新的高精度的交替显隐算法、高精度交替十二点分组算法以及4阶交替分段Crank-Nicolson算法。这四个新算法不仅具有无条件稳定和能在计算机上实现并行计算的特点,而且它们在空间上都具有4阶精度。通过数值算例,我们也容易看到,数值结果和理论分析是一致的。数值算例说明,新算法们在精度和稳定性上都优于算法AGE([11])和ASEI([12])。论文的部分内容已在国际国内刊物上公开发表([68—72])。全文共分为三章:第一章介绍色散方程的高精度并行迭代法。本章导出了一种数值求解色散方程的高精度交替分组迭代格式,此格式收敛速度快并可以在并行计算机上直接应用。本章内容公开发表在([71])。第二章介绍色散方程基于6点差分格式的高精度并行算法。在第一节中,我们介绍了色散方程的高精度交替6点分组算法。本节,我们将给出一类Saul’yev型非对称差分格式来求解色散方程。基于这些Saul’yev型格式,我们给出了求解带周期边界条件的色散方程的新的交替六点分组算法。这个新算法不仅具有无条件稳定和能在计算机上实现并行计算的特点,而且它在空间上具有4阶精度。通过数值算例,我们也容易看到,数值结果和理论分析是一致的。数值算例说明,新算法在精度和稳定性上都优于算法AGE([11])。在第二节中,介绍了色散方程的一类新的高精度交替分组显隐算法。本节针对色散方程提出的nAGEI新方法不但绝对稳定、本质并行,而且误差分析和数值试验表明,其数值解关于空间步长的收敛速度几乎是4阶的。通过与AGE([11])和ASEI([12])等方法的数值比较,我们容易看到本文方法确实具有更高的精度。本节内容已在《应用数学和力学》发表,请见[69]。第三章介绍了色散方程基于12点差分格式的高精度并行算法。在第一节中,我们介绍了色散方程的高精度交替12点分组算法。近年来,随着并行计算机的发展,并行数值计算也越来越多的受到人们的关注和重视。像区域分裂算法一样([38—49,63,64,66]),交替分组方法也因其绝对稳定、本质并行的特点而日渐成为行之有效的并行数值方法之一。1983年,Evans首先提出了交替分组显方法(AGE)([15—16]),历经近20年的发展,交替分组算法的思想已经被成功运用到求解扩散方程([15—17,19—21])、色散方程([1—13])以及Kdv([22,50])方程等方程中去。但是,在已有交替分组方法材料中,它们的数值解在空间上都是有接近2阶的收敛速度。我们在本节给出的新算法不仅仍然具有绝对稳定、本质并行的优良特性,而且我们随后的截断误差分析和数值算例将表明新算法的数值解在空间上具有接近4阶的收敛速度。我们在数值算例中给出了本节算法与已有算法AGE([11])的数值比较。在第二节中,我们介绍了色散方程的一类4阶交替分段Crank-Nicolson算法。在本节,我们将给出一个新的4阶nASCN算法来求解色散方程,这个算法不仅绝对稳定,而且可以直接应用到并行计算中去。事实上,交替分组方法是随着并行计算机的发展而蓬勃发展起来的。目前,两类主要的并行算法就是:交替分组方法([2,4,9—24])和区域分裂算法([38—49,63,64,66])。前者是无条件稳定的,所以我们通常可以采用比较大的时间步长,而后者是条件稳定的,因此,在使用过程中,我们通常需要选取比较小的时间步长来进行计算。1983年,Evans率先提出了交替分组显式计算方法(AGE),此后,又有人提出了交替分组显隐算法(ASEI)以及交替分段Crank-Nicolson([19,20])(ASCN)算法。近年来,我们也开始看到交替分组方法被应用到求解色散方程、Kdv方程等方程中。不过,在已看到的交替分组算法文献中,几乎所有算法的数值解在空间上都只能接近2阶。新算法nASCN不仅格式无条件稳定,而且还具有本质并行的特点。此外,我们随后的截断误差分析和数值试验表明新算法可在空间上达到4阶收敛,这比已知的AGE([11])和ASEI([12])都精确。本节内容已发表在《Computers and Mathematics with Applications》请见[68]。考虑到一维算法为高维算法的基础和依托,我们也正在将本文的高精度并行算法推广应用到Burgers方程、Kdv方程以及相关的二维问题的求解过程中去。更多还原

【Abstract】 The dispersive equation is one of the important equations of mathematical physics and its numerical solving methods are widely studied.The dispersive equation also occupies a concernful position in the physical problems of the nonlinear wave and the soliton theory.Considering the applied value of the dispersive equation in the physical area,more and more experts begin to study the numerical solving methods for the dispersive equation([1-13]).In[4],many difference schemes and the relevant nature of stability for the dispersive equation are divided.There are two kinds of schemes in the difference schemes of[4].They are the explicit scheme and the implicit scheme. We know that the explicit difference scheme,which is simple and able to be used on parallel computers straightly,often needs some strict stable conditions.Though some experts tries to improve the strict stable conditions([7,8]),the improvement is not very obvious.In[5],the author discusses a class of unconditional stable explicit difference schemes with two parameters.While in these schemes,we have much difficulty to find the appropriate parameters.While the stable implicit method can’t be used for parallel computation directly.With the coming of the information age and the development of the computers,parallel computation attracts more and more concern([10-37]) for its character of solving the large and complicated computing problems rapidly.So,how to find a stable numerical solving method,which can be used on parallel computers directly,becomes an important problem needed to be solved as quickly as possible.So far,the two main parallel algorithms are:alternating group methods([2,4,9-24]) and domain decomposition methods([38-49,63,64,66]).The former methods are unconditionally stable,so we can choose a bigger time step.To the latter methods, which are conditionally stable,we usually need to choose a smaller time step in our computation.The alternating group method has become one of the most popular parallel numerical methods for its nature of unconditional stability and parallelism.To see the development of the parallel methods,we can take some literature for example.In([15,23,25,63-67]),we can see the parallel methods for the parabolic equation.In([15-17,19-21]),we can see the parallel methods being used to the diffusion equation and the convection-diffusion equation.In recent years,we begin to see the alternating group methods being used to the third order dispersive equation, the Korteweg-de Vries equation etc([1-13,22,50]).But for the third order dispersive equation,the similar application is not much.In fact,as early as 1983,Evans and Abdullah first proposed the Alternating Group Explicit(AGE) in[15,16].Later,Bao-lin Zhang developed the Alternating Segment Explicit-Implicit(ASE-I) methods in[19].Since 2000,Shao-hong Zhu extended the AGE methods to the third-order dispersive equation in[11,12].All these methods are capable of parallel implementation and are unconditionally stable,but all their accuracies in space are nearly the second order.On the other hand,we are all trying to improve the accuracy of our numerical algorithms in our numerical computation([50-62]).So,how to improve the accuracy of the method becomes another important problem to be solved quickly.According to the above discussion and under the guidance of my super advisor, the author provides a class of high accuracy,unconditionally stable and parallelizable algorithms for the third order dispersive equation with period boundary condition.In Chapter 1,the author introduces a high accuracy parallelizable iterative algorithm for the third order dispersive equation.In Chapters 2,3,the author provides four kinds of Sauryev asymmetrical difference schemes to solve the dispersive equation.Basing on the above four Saul’yev asymmetrical difference schemes,the author gives out the relevant numerical solving algorithms. They are the alternating 6-point group algorithm,the new alternating group explicitimplicit algorithm,the alternating 12-point group algorithm,and a 4-order alternating segment crank-nicolson algorithm.These four algorithms are not only unconditionally stable but also have the parallel nature.Besides,our truncation error analysis and numerical experiment show that the numerical solution from the four algorithms all have a four-order rate of convergence in space,which is higher than the accuracy of AGE([11]) and ASEI([12]).Some results of this dissertation have been published in[68-72].The dissertation is divided into three chapters:In Chapter 1,we give out a high accuracy parallelizable iterative algorithm for the third order dispersive equation.This algorithm has a rapid convergence rate and can be used on parallel computers directly.You can see the results of this chapter in([71]).In Chapter 2,we introduce a class of high accuracy parallelizable algorithms based on the relevant 6-point difference schemes.In Section 2.1,we introduce the high accuracy alternating 6-point group algorithm for the dispersive equation.In this section,we give out a group of Saul’yev type asymmetric difference formulas to approach the dispersive equation.Basing on these formulae we derive a new alternating 6-point group algorithm to solve the dispersive equation with the periodic boundary condition.The parallel algorithm has the fourth-order accuracy in space and the unconditional stability.The theoretical results are conformed to the numerical simulation.Numerical examples show that the AG-6p method is better in both the accuracy and the stability than the known method in AGE([11]).In Section 2.2,we introduce the new high accuracy alternating group explicitimplicit algorithm for the dispersive equation.The new method of this section is not only unconditionally stable but also has the parallel nature.Besides,our truncation error analysis and numerical experiment show that the numerical solution from the nAGEI has the fourth-order rate of convergence in space,which is much higher than the accuracy of AGE([11]) and ASEI([12]).The results of this section are published in "Applied Mathematics and Mechanics" ([69]).In Chapter 3,we introduce a class of high accuracy parallelizable algorithms based on the relevant 12-point difference schemes.In Section 3.1,we introduce the high accuracy alternating 12-point group algorithm for the dispersive equation.In recent years parallel computers and the numerical parallel computation are more and more popular for their efficiency.As the domain decomposition method([38-49,63,64,66]),the alternating group method which is unconditionally stable and has the parallelizable nature has also become one of the efficient parallel numerical methods. In 1983,Evans first proposed the Alternating Group Explicit(AGE) strategy in[15-16]. After near twenty years’ development,the study of the alternating group method has been introduced into solving the diffusion equation([15-17,19-21]),the dispersive equation([1-13]) and the KdV equation etc.But in the known alternating group literatures,nearly all of their numerical solutions’s rate of convergence was only near two-order in space.The new method of this section is not only unconditionally stable but also has the parallel nature.Besides,our truncation error analysis and numerical experiment show that the numerical solution from the AG-12p has nearly four-order rate of convergence in space,which is higher than the accuracy of the AGE([11]).In Section 3.2,we introduce the four-order alternating segment Crank-Nicolson algorithm for the dispersive equation.The dispersive equation is popular as one of the applied equations and its numerical solving methods was widely studied([1-13]).We know that the explicit difference scheme is simple and can be used on parallel computers straightly.But it often needs some strict stable conditions.While the stable implicit method can’t be used for parallel computation directly.In this paper,we will give out a new four-order method (nASCN) to solve the dispersive equation.The nASCN is not only unconditionally stable but also can be used for parallel computation directly.In fact,the study of alternating segment algorithms develops with the development of parallel computers and the parallel numerical computation.Currently,there are two major types of parallel schemes:the alternating schemes([2,4,9-24]) and the domain decomposition schemes([38-49,63,64,66]).The former which allow large time steps is unconditionally stable.But the latter is usually conditionally stable and for this we often have to choose very small time steps.In 1983,Evans first proposed the Alternating Group Explicit (AGE).Afterward the Alternating Segment Explicit-Implicit(ASEI) scheme and the Alternating Segment Crank-Nicolson(ASCN) scheme were introduced([19,20]).In recent years,we see the use of alternating segment methods in the dispersive equation and the KdV equation etc.But in the known alternating segment literatures,almost all of their numerical solutions’s rates of convergence were near two-order in space.The nASCN is not only unconditionally stable but also has the parallel nature.Besides,our truncation error analysis and numerical experiment show that the numerical solution from the nASCN has a four-order rate of convergence in space,which is higher than the accuracy of AGE([11]) and ASEI([2]).The results of this section are published in "Computers and Mathematics with Applications"([68]).更多还原