Galerkin Method For Three Nonlinear Equations Based On Smooth Piecewise Lagrange Interpolation

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.