Application Of The Modied Local Crank -Nicolson Method For Solving The KDV Equation

In recent year , It appear many nonlinear evolution equations in order to sovle the newtechnique problem and corresponding theory research,such as laser,superconduct,crystallattice,plasma physics,agglomerate physics and so on.It's need to construct these equationsand their's solution of steady problem ,to explain solution's character,especially clarifyisolator's character.hence,investigating these nonlinear evolution equations has been animportant task of maths.In numerous nonlinear evolution equations,the most typical rep-resentation of nonlinear dispersive wave equations is KDV equation.It is widely studiedfor having boundless conservation laws and being applied to di?erent science fields includ-ing solid,liquid,gas,plasma etc. Therefore, the study of the numerical method for KDVequation is important in theoretical and practical.The Modified Local Crank-Nicolson method is given by Abduwali first, and he usedthis method to solve the heat equation very well. This method is an explicit di?erencescheme with unconditional stability. Moreover, it avoids solving the linear equations. Itis very important in numerical computation.In this paper, we give the Modified Local Crank-Nicolson method for KDV equationbased on the work of the predecessor. This method transforms the partial di?erentialequation into the ordinary di?erential equations, and uses the Trotter Product formula ofexponential function to approximate the coe?cient matrix of these ordinary di?erentialequations. Then separates it into some small block matrices, and employs Crank-Nicolsonmethod to obtain a new di?erent scheme. It is a weak nonlinear system, and linearizationapproach is applied; i.e., it is linearized by allowing the nonlinearities to lag one timestep behind, and the obtained system of linear equations is solved by using iterativealgorithms. Our work in this paper, is not only used to solve the KDV equation, butalso can develop the Modified Local Crank-Nicolson method in dealing with nonlinearequation, and provide a reference for some other partial di?erential equations.This work consists four sections. Section 1 is preface. we introduce research back-ground, purpose and significance, and describe the research situation of numerical solutionfor KDV equation . Finally, the organizational structure of this work is given.In section 2, we give the Crank-Nicolson scheme for KDV equation. It is an implicittwo order di?erence scheme with ,which is stability and satisfies the first and two discrete conservation laws. In this paper, we proof the stability and convergence, and finally makea numerical experiment and the numerical results are in line with the physical phenomenaof this problem. It is showed that the scheme is effective.In section 3, we give the Modified Local Crank-Nicolson scheme for KDV equation. Itis an explicit two order difference scheme with unconditional stability. In this paper, theModified Local Crank-Nicolson schemes for one-dimensional KDV equations is discussedin detail. Then, we also support the theoretical analysis, and finally several numericalexperiments are given to verify the numerical results, and it is seen that they are inexcellent agreement.Section 4 is conclusion, we make a conclusion on the whole work. By the comparisonof the two numerical methods, we find that the Modified Local Crank-Nicolson scheme isalso an effective numerical method for solving partial differential equation as the Crank-Nicolson scheme do.