A Class Of High Accuracy Parallel Algorithms For The Dispersive Equation

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]).