The Galerkin Alternating-Direction Method For Hyperbolic Equations

A large number of physical problems of significant interest are modeled by hyperbolic equations. For example, the earthquake exploration is an important method to explore petroleum and gas. Because of the complexity of this method, the further research should be given in theory and practical use.The finite difference, which was firstly perfected, is inconvenient in dealing with the second and the third boundary problems. Hence Courant [1], Feng Kang [2,3] began to solve partial differential equations by using finite element method. The important work about the finite element method for hyperbolic problems is [4-9]. Dupont[4], Oden[5] discussed the finite element method for the linear hyperbolic problem. Yuan Yirang and Wang Hong gave further study. [6] gave the stability and convergence of quasi-linear hyperbolic problem. [7] and [8] discussed the semi-discrete, the time-discrete finite element method for nonlinear hyperbolic problem. [9] studied the time-discrete finite element method for nonlinear hyperbolic problems. All the work has important position for later research. However the computation of the finite element method is very complex when dealing high-dimensional problems. So that in order to solve the multi-variables problems, the alternating- direction method appears. It can lower the dimension.In 1971, the alternating-direction finite element method, which was first proposed by Douglas and Dupont [10], is designed to offer a significant reduction in the computing time, storage requirements and to have high accuracy. Dendy, Fairweather [11,12], Bramble, Ewing, Li[13],Wang Shenlin[14],etc, have given further investigation. [10] first gave the alternating-direction finite element method for the simplest hyperbolic equations and gave the H₀¹ -norm error estimates. [11] generalized the work of [10] and obtained the optimal L² -norm error estimates. [12] put forward a new method by transforming the second-order hyperbolic equationinto a first-order system of equations and derived the H₀¹ -norm and the L² -norm error estimates.For this method, although the variable time steps can be employed and the initial conditions can be determined in a more natural way, it investigated a special kind of hyperbolic equation with separable coefficients. All the work [10-14] limited the alternating-direction method to rectangular regions. This is not applicable to many practical problems, see [15, 16] for example.So many researchers generalized this method in computing region. Dendy, Fairweather [17] discussed this method on rectangular polygons. Hayes developed a new method to solve the parabolic problems on curved regions by using patch approximation and an approximation to the Jacobian of the isoparametric map, see [18-20]. Varies questions concerning the practical implementation were addressed by Hayes[21], Hayes, Kennon,Dulikravich[22], Krishnamachari, Hayes[23]. Applications of these methods to problems in fluid flow and heat transfer were described by Hayes, Krishnamachari[24], Lewis,Morgan, Roberts[25]. However only parabolic problems was discussed by this method.In this paper, some alternating-direction methods for some kinds of hyperbolic equations are proposed. The theoretical analyses are given and the numerical examples are also experienced to show that the methods are efficient in practice. My work is divided into four chapters.In chapter 1, Galerkin alternating-direction procedures and an approximation to the Jacobian of the isoparametric map are applied to discuss a three-level and a four-level Galerkin alternating-direction procedures for two kinds of nonlinear hyperbolic equations which are given on a nonrectangular, curved region. The optimal order estimates in L² -norm of the procedure are obtained respectively by using the theory and techniques of priori estimate of differential equations. For the two kind of nonlinear hyperbolic equations, a three-level and a four-level Galerkin alternating-direction procedures are firstly proposed, then the matrix problems are given, and finally the error estimates of the above two schemes are proved. Two papers are written about this chapter. One was accepted by《Applied Mathematics and Computation》and published online, the other was submitted to《Applied Numerical Mathematics》.The new ideas of chapter 1 are: 1 )The work about Galerkin alternating-direction method of nonlinear hyperbolic problems on curved regions has not been found in relative literatures. 2) For hyperbolic problems, a four-level Galerkin alternating-direction procedures are firstly proposed.Chapter 2 mainly used the method proposed in [12] and discussed the Galerkin alternating-direction procedure for two- and three-dimensional hyperbolic equations. The second-order quasi-linear hyperbolic equation is firstly transformed into a first-order system of equations, then the Galerkin alternating-direction procedure for the system is derived. The optimal orderestimates in H¹ norm and L² norm of the procedure are obtained respectively by using the theory and techniques of priori estimate of differential equations. The numerical experiment is also given to support the theoretical analysis. Comparing the results of numerical example with the theoretical analysis, they are uniform. The work of§2.1 was accepted by《Applied Mathematics and Computation》and published online. The work of§2.2 was submitted to《Journal of Computational Mathematics》.The new ideas about chapter 2 are: 1) Originally, the method is given for a kind of hyperbolic equation that has separable coefficients. No literature gave discussion for the general form of hyperbolic equations. I do it. 2) No numerical analysis has been given from the method was proposed, however the numerical examples are given to support the theoretical analysis in my paper.Chapter 3 also discussed the Galerkin alternating-direction method for a kind of nonlinear hyperbolic equation. The structure is as the same as that of chapter 2. The work of this chapter has not been found in past literatures. The work about this chapter was submitted to《J. of Computational and Applied Mathematics》.In chapter 4, some kinds of generalized KdV equations are discussed. As a typical nonlinear phenomena, soliton has been widely investigated in physics and engineering. Many theoretical researches was given in [26-29]. A quite lot of difference schemes about KdV equations have also been used in numerical studies (see [30-36] for instance). Although most of the schemes showed their efficiency in numerical analysis, there is no faithful comparison of the numerical solutions with the exact ones. It is well known that the analytic solution of GKdV equations decays to zero as (?)±∞. In this case, even the absolute error may be small, the numerical solution is several times larger than the analytic one, which means that the relative error is large. Thus the relative error should be considered when choosing a scheme. However, few analysis for relative error of difference schemes for GKdV equations can be found in the relative literatures. Although other methods for regularized long wave equations and the generalized forms, such as pseudo-spectral method are found to be accurate and efficient, see [30,35], the error comparison is only taken under the L², L^∞norms or the computing time, which is deficient in dealing with the case of the analytic solution tends to zero.In§4.1, two difference schemes, I and IV, are developed for GKdV equations. The stability of the schemes I and IV are obtained by employing linearized stability method. Both the numerical dissipation and the numerical dispersion for scheme I , II and III are analyzed by Fourier method. The results are in a good agreement with that given in [31,32]. The comparison of the numerical solution with exact ones among these four schemes is carried out. Both the absolute and the relative errors are considered in this study. The results given in this paper enable readers to consider the relative error when dealing with the case of the analytic solution tends to zero. The work about§4.1 was published on《Communications in Numerical Methods in Engineering》.In§4.2, Cell mapping method was used to study the global domain of attraction of GKdV equations. Cell mapping is developed by discretization of the point mapping which is a useful tool in solving nonlinear dynamical systems. The general method of point mapping can be traced back to the time of Poincare and Birkhoff in 1960's and 1970's, see [37-38]. The simple cell mapping method (SCM) was proposed and developed in 1980's, see [39] for instance. The SCM method was found to be potentially a very powerful tool for global analysis, which has been used successfully for global analysis of nonlinear dynamical systems with dimension of 2,3 and 4, see also in [40-42]. Numerical results show that the domain of attraction can also be obtained by SCM method. Different initial values were chosen to require the phase portraits. But when SCM method was used to get the domain of attraction, it is unnecessary to choose the proper different initial values and the domain of attraction can easily be obtained. Numerical results computed by using Forth order Runge-Kutta method and the center point method. Comparing these results with their corresponding phase portraits, they are uniform. It means that soliton can be discussed from the point of view of cell mapping. This has not been found in relative lectures. The work about§4.2 was published in《Chaos, Solitons & Fractals》.The new ideas of chapter 4 are:1) Analysis for relative error of difference schemes for GKdV equations are discussed. 2) It is the first time to discuss the global domain of attraction of GKdV equations by applying the cell mapping method. This has not been found in relative lectures.There are four papers in English dissertation. The first paper has been published on《Applied Mathematics and Computation》, the second on《Applied Mathematics andComputation》, the third was submitted to《Journal of Computational and AppliedMathematics》and the fourth has been published on《Communications in Numerical Methods inEngineering》.