解三类非线性方程的基于分片光滑Lagrange型插值的Galerkin方法
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摘要
本文分为三部分，分别研究正则长波(RLW)方程，Korteweg-de Vries(KdV)方程和非线性Klein-Gordon(NKG)方程的数值解法。
KdV方程和RLW方程是在研究水波、等离子物理等问题时提出的两类非线性发展方程。KdV方程最早由Korteweg和de Vries~[1]提出，由于其良好的性质而得到广泛研究。Peregrine~[2]在研究水波传播问题时，为代替KdV方程提出了RLW方程，在后来的文献中证明，在水波的传播、等离子体波等一些物理现象的研究中，RLW方程可作为KdV方程的一个修正模型，而且较KdV方程，RLW方程具有更好的数学性质，从此得到了广泛的研究。
对于这两类方程，部分文献曾给出其某些特定条件下的解析解，但由于只有小部分方程能够用解析的方法求解，因此研究其数值解法就显得尤为重要。对于RLW方程，目前国内外已有不少的数值解法，早期的算法以差分方法为主，该算法计算简单，计算量小，但是精度不高，于是许多学者将有限元方法应用到正则长波方程的求解中，陆续提出了B-样条有限元法、Petrov—Galerkin方法、Fourier-Galerkin-Center Euler方法等。这些方法提高了数值解的精度，但是计算较为复杂，例如利用B-样条有限元法计算时，由于B-样条不是结点基函数，会给计算造成很多的不便。在本文中，我们构造了一种光滑的分片Lagrange插值型多项式空间，并作为试探函数及检验函数空间，对RLW方程进行求解，由于我们所用的基函数为结点基函数，并且易求导，因此计算简单，实验结果表明此方法对解的精度也有一定提高。
对于KdV方程，目前应用最广泛的数值方法是谱方法，谱方法用于求解KdV方程有几大优点：对光滑函数指数性逼近的谱精度、无相位误差等，但是，用谱方法求解KdV方程计算过于复杂，不易求解。因此，本文利用Galerkin有限元法求解KdV方程。我们注意到，由于KdV方程是一个三阶的偏微分方程，因此在利用有限元求解时，必须在H~2空间中选取基函数，而本文构造的基函数空间恰好满足这一要求，而且通过实验可以证明，Galerkin有限元法用于求解KdV方程可以得到较满意的精度。
正则长波方程和KdV方程都有一个重要的性质，就是具有孤立波解，孤立波的碰撞会产生新的孤立波，并且这些波能在传播中，甚至在碰撞后保持波形不变。另外，孤立波解满足守恒性，已经证明RLW方程有三个守恒量，而KdV方程有无穷多个守恒量，在本文中，我们模拟了单孤立波的传播及多孤立子的碰撞试验。数值结果验证了RLW方程孤波解三个守恒量的数值守恒性。对KdV方程，选取其中比较重要的三个守恒量验证了孤波解的数值守恒性。
Klein—Gordon方程在研究旋转波、非线性光学和其他一些数学物理问题中有着许多重要的应用。已有许多文献对该方程进行了研究，数值方法研究主要有差分方法，有限元法等。本文在所构造的分片光滑Lagrange型插值多项式空间中利用Galerkin方法对NKG方程进行求解，并进行数值实验，将数值解与精确解、本文格式与文献[36]的差分格式所得的数值结果进行了比较，结果表明本文格式保持Klein—Gordon方程的守恒性质，是精确有效的。
本文第一章分析了正则长波方程的Galerkin有限元方法。在本章中，我们在所构造的光滑分片Lagrange型插值多项式空间中，利用Galerkin有限元方法对正则长波方程进行数值求解，给出误差估计并进行数值实验。通过比较误差的L_2及L_∞范数，我们看到，本章所用方法在计算精度上有一定改进，而解的守恒性有很大提高。另外，我们还模拟了孤立波的发展、碰撞及Maxwellian初始条件的发展，取得了满意的数值结果。
第二章分析了Korteweg—de Vries方程的Galerkin有限元法。在本章中，利用第一章中构造的光滑分片多项式空间，利用Galerkin有限元法对Korteweg—de Vries方程进行数值求解，第一节构造了数值格式，第二节分析了格式的线性稳定性，第三节进行了数值实验，数值实验包括小参数KdV方程的小时间区间解的传播和长时间区间的数值解模拟，通过模拟孤立波的传播、几个孤立波的碰撞等大量实验检验了方法的精度、数值解的守恒性，取得了非常满意的结果。
第三章分析了Klein—Gordon方程的Galerkin有限元法。依然在第一章构造的光滑分片多项式空间中利用Galerkin方法对NKG方程进行求解，第一节构造数值格式，第二节通过分析证明所构造格式是能量守恒的，并给出近似解的有界性及收敛性证明，第三节给出了两个数值实验，通过比较数值解与精确解、本文格式与文献[36]的差分格式所得的数值结果说明，对于求解Klein—Gordon方程本文格式是有效的，并且提供了更高的精度。
The paper can be divided into three parts which consider about the numericalsolution for Regularized long wave (RLW) equation, Korteweg-de Vries (KdV)equation and nonlinear Klein-Gordon (NKG) equation, respectively.KdV equation and RLW equation are two kinds of nonlinear equations arisingin the study of a number of physical problems, such as water waves and anharmoniclattices. KdV equation was first derived by Korteweg and de Vries[1], Peregrine[2]introduced RLW equation to describe the behavior of the undular bore and it hasbetter mathematical properties than KdV equation.
Since only limited classes of the two equations are solved by analytical means,the numerical solutions are of practical importance. Many papers have introducedvarious techniques to solve the RLW equation numerically, in which finite differencemethod is the most important one at the earlier period, but they failed to givehigh accuracy. Then many papers consider about the finite element method, andintroduced many methods such as B-spline finite element method, petrov-Galerkin,etc.. These methods give more accurate numerical solutions for the RLW equation,but the computing process are complicated. Take B-spline method for instance, sinceB-spline is not a nodal basis function, it causes lots of inconvenience in computing.In this paper, a kind of smooth piecewise polynomials space based on Lagrangeinterpolation is constructed, we use them as the trial and test function space, solvethe RLW equation by Galerkin approach, since the basis function are nodal basisfunctions, the computing will be very easy.
For the KdV equation, the most extensively used numerical method is spectralmethod, there are several advantage of this method: exponential approximation forsmooth functions, no phase error, etc., but the computing process is complicatedtoo, so we still use Galerkin approach to solve it. Since the finite dimensional spacewe have constructed are in H2, it can be applied to the three-order KdV equation.The test problems show that the Galerkin finite element method can give goodaccuracy in solving the KdV equation.
The RLW equation and KdV equation have been shown to have solitary wavesolutions. The soliton solution of the RLW equation obeys three conservation laws, and the KdV equation has infinite invariants, three of them are taken to verify theconservation property of the numerical solution in this paper.
The solutions of the NKG equation are important in many applications such asspin waves, nonlinear optics and problems in mathematical physics. The numericalscheme include finite difference method and finite element method. In this paper,we solve the NKG equation still on the smooth piecewise polynomials constructed inChapter one, and do the experiment. which shows that the present scheme satisfythe conservation property of the equation and is accurate and effect.
Chapter one discusses the numerical solution of the RLW equation by using aGalerkin method based on the piecewise polynomials space.
Numerical simulations presented in this paper show the motion of the singlesolitary wave with different amplitudes, the interaction of two solitary waves and thenumerical solution of Maxwellian initial condition. Some results are also comparedwith published numerical solutions. The L2 and L∞error norms and conservedquantities are given, which shows that the present method is successful in solvingthe RLW equation.
Chapter two considers about the numerical solution of the Korteweg-de Vries(KdV) equation by using a Galerkin method based on the smooth piecewise polynomialsused in Chapter one.
We construct the numerical algorithm in Section one, then in Section two, alinear stability analysis is given, which shows that our algorithm is unconditionallystable. Section three deals with the test problems including the KdV equation withsmall parameters and equation with bigger parameters on a long time domain. weverify the efficiency of the method and the conservation properties of the solutionsby the simulation including the motion of single solitary wave and the interactionof solitary waves, and get good results.
Chapter three considers about the numerical solution of the NKG equation byusing Galerkin method based on the smooth piecewise polynomials used in Chapterone.
We construct the numerical algorithm in Section one, and proved that it isenergy conserved in Section two, we also proved the numerical solution is boundedand convergent. Section three gives two experiments to verify the accuracy of ourmethod and get good results.
KdV方程和RLW方程是在研究水波、等离子物理等问题时提出的两类非线性发展方程。KdV方程最早由Korteweg和de Vries~[1]提出，由于其良好的性质而得到广泛研究。Peregrine~[2]在研究水波传播问题时，为代替KdV方程提出了RLW方程，在后来的文献中证明，在水波的传播、等离子体波等一些物理现象的研究中，RLW方程可作为KdV方程的一个修正模型，而且较KdV方程，RLW方程具有更好的数学性质，从此得到了广泛的研究。
对于这两类方程，部分文献曾给出其某些特定条件下的解析解，但由于只有小部分方程能够用解析的方法求解，因此研究其数值解法就显得尤为重要。对于RLW方程，目前国内外已有不少的数值解法，早期的算法以差分方法为主，该算法计算简单，计算量小，但是精度不高，于是许多学者将有限元方法应用到正则长波方程的求解中，陆续提出了B-样条有限元法、Petrov—Galerkin方法、Fourier-Galerkin-Center Euler方法等。这些方法提高了数值解的精度，但是计算较为复杂，例如利用B-样条有限元法计算时，由于B-样条不是结点基函数，会给计算造成很多的不便。在本文中，我们构造了一种光滑的分片Lagrange插值型多项式空间，并作为试探函数及检验函数空间，对RLW方程进行求解，由于我们所用的基函数为结点基函数，并且易求导，因此计算简单，实验结果表明此方法对解的精度也有一定提高。
对于KdV方程，目前应用最广泛的数值方法是谱方法，谱方法用于求解KdV方程有几大优点：对光滑函数指数性逼近的谱精度、无相位误差等，但是，用谱方法求解KdV方程计算过于复杂，不易求解。因此，本文利用Galerkin有限元法求解KdV方程。我们注意到，由于KdV方程是一个三阶的偏微分方程，因此在利用有限元求解时，必须在H~2空间中选取基函数，而本文构造的基函数空间恰好满足这一要求，而且通过实验可以证明，Galerkin有限元法用于求解KdV方程可以得到较满意的精度。
正则长波方程和KdV方程都有一个重要的性质，就是具有孤立波解，孤立波的碰撞会产生新的孤立波，并且这些波能在传播中，甚至在碰撞后保持波形不变。另外，孤立波解满足守恒性，已经证明RLW方程有三个守恒量，而KdV方程有无穷多个守恒量，在本文中，我们模拟了单孤立波的传播及多孤立子的碰撞试验。数值结果验证了RLW方程孤波解三个守恒量的数值守恒性。对KdV方程，选取其中比较重要的三个守恒量验证了孤波解的数值守恒性。
Klein—Gordon方程在研究旋转波、非线性光学和其他一些数学物理问题中有着许多重要的应用。已有许多文献对该方程进行了研究，数值方法研究主要有差分方法，有限元法等。本文在所构造的分片光滑Lagrange型插值多项式空间中利用Galerkin方法对NKG方程进行求解，并进行数值实验，将数值解与精确解、本文格式与文献[36]的差分格式所得的数值结果进行了比较，结果表明本文格式保持Klein—Gordon方程的守恒性质，是精确有效的。
本文第一章分析了正则长波方程的Galerkin有限元方法。在本章中，我们在所构造的光滑分片Lagrange型插值多项式空间中，利用Galerkin有限元方法对正则长波方程进行数值求解，给出误差估计并进行数值实验。通过比较误差的L_2及L_∞范数，我们看到，本章所用方法在计算精度上有一定改进，而解的守恒性有很大提高。另外，我们还模拟了孤立波的发展、碰撞及Maxwellian初始条件的发展，取得了满意的数值结果。
第二章分析了Korteweg—de Vries方程的Galerkin有限元法。在本章中，利用第一章中构造的光滑分片多项式空间，利用Galerkin有限元法对Korteweg—de Vries方程进行数值求解，第一节构造了数值格式，第二节分析了格式的线性稳定性，第三节进行了数值实验，数值实验包括小参数KdV方程的小时间区间解的传播和长时间区间的数值解模拟，通过模拟孤立波的传播、几个孤立波的碰撞等大量实验检验了方法的精度、数值解的守恒性，取得了非常满意的结果。
第三章分析了Klein—Gordon方程的Galerkin有限元法。依然在第一章构造的光滑分片多项式空间中利用Galerkin方法对NKG方程进行求解，第一节构造数值格式，第二节通过分析证明所构造格式是能量守恒的，并给出近似解的有界性及收敛性证明，第三节给出了两个数值实验，通过比较数值解与精确解、本文格式与文献[36]的差分格式所得的数值结果说明，对于求解Klein—Gordon方程本文格式是有效的，并且提供了更高的精度。
The paper can be divided into three parts which consider about the numericalsolution for Regularized long wave (RLW) equation, Korteweg-de Vries (KdV)equation and nonlinear Klein-Gordon (NKG) equation, respectively.KdV equation and RLW equation are two kinds of nonlinear equations arisingin the study of a number of physical problems, such as water waves and anharmoniclattices. KdV equation was first derived by Korteweg and de Vries[1], Peregrine[2]introduced RLW equation to describe the behavior of the undular bore and it hasbetter mathematical properties than KdV equation.
Since only limited classes of the two equations are solved by analytical means,the numerical solutions are of practical importance. Many papers have introducedvarious techniques to solve the RLW equation numerically, in which finite differencemethod is the most important one at the earlier period, but they failed to givehigh accuracy. Then many papers consider about the finite element method, andintroduced many methods such as B-spline finite element method, petrov-Galerkin,etc.. These methods give more accurate numerical solutions for the RLW equation,but the computing process are complicated. Take B-spline method for instance, sinceB-spline is not a nodal basis function, it causes lots of inconvenience in computing.In this paper, a kind of smooth piecewise polynomials space based on Lagrangeinterpolation is constructed, we use them as the trial and test function space, solvethe RLW equation by Galerkin approach, since the basis function are nodal basisfunctions, the computing will be very easy.
For the KdV equation, the most extensively used numerical method is spectralmethod, there are several advantage of this method: exponential approximation forsmooth functions, no phase error, etc., but the computing process is complicatedtoo, so we still use Galerkin approach to solve it. Since the finite dimensional spacewe have constructed are in H2, it can be applied to the three-order KdV equation.The test problems show that the Galerkin finite element method can give goodaccuracy in solving the KdV equation.
The RLW equation and KdV equation have been shown to have solitary wavesolutions. The soliton solution of the RLW equation obeys three conservation laws, and the KdV equation has infinite invariants, three of them are taken to verify theconservation property of the numerical solution in this paper.
The solutions of the NKG equation are important in many applications such asspin waves, nonlinear optics and problems in mathematical physics. The numericalscheme include finite difference method and finite element method. In this paper,we solve the NKG equation still on the smooth piecewise polynomials constructed inChapter one, and do the experiment. which shows that the present scheme satisfythe conservation property of the equation and is accurate and effect.
Chapter one discusses the numerical solution of the RLW equation by using aGalerkin method based on the piecewise polynomials space.
Numerical simulations presented in this paper show the motion of the singlesolitary wave with different amplitudes, the interaction of two solitary waves and thenumerical solution of Maxwellian initial condition. Some results are also comparedwith published numerical solutions. The L2 and L∞error norms and conservedquantities are given, which shows that the present method is successful in solvingthe RLW equation.
Chapter two considers about the numerical solution of the Korteweg-de Vries(KdV) equation by using a Galerkin method based on the smooth piecewise polynomialsused in Chapter one.
We construct the numerical algorithm in Section one, then in Section two, alinear stability analysis is given, which shows that our algorithm is unconditionallystable. Section three deals with the test problems including the KdV equation withsmall parameters and equation with bigger parameters on a long time domain. weverify the efficiency of the method and the conservation properties of the solutionsby the simulation including the motion of single solitary wave and the interactionof solitary waves, and get good results.
Chapter three considers about the numerical solution of the NKG equation byusing Galerkin method based on the smooth piecewise polynomials used in Chapterone.
We construct the numerical algorithm in Section one, and proved that it isenergy conserved in Section two, we also proved the numerical solution is boundedand convergent. Section three gives two experiments to verify the accuracy of ourmethod and get good results.
引文
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