Application of the Modied Local Crank Nicolson Method for Solving the KDV Equation 

Author  GuoRui 
Tutor  ABuDuReXiTi·ABuDuWaiLi 
School  Xinjiang University 
Course  Computational Mathematics 
Keywords  KDV equation CrankNicolson method Modified Local CrankNicolson method Stability Convergence 
CLC  O175.2 
Type  Master's thesis 
Year  2010 
Downloads  35 
Quotes  0 
In recent years, a number of new technical problems and the corresponding theoretical studies, such as the laser, superconductivity, lattice, plasma physics, condensed matter physics research, leads to some nonlinear evolution equations. This requires to construct nonlinear evolution equations and its definite solution of the problem, and to explain the characteristics of the solution, in particular, can explain the solitary wave soliton nature. Therefore, the study of these nonlinear evolution equations become an important topic in mathematics. Nonlinear evolution equations, which KDV equation represents the most typical nonlinear dispersive wave equation, because of its infinite number of conservation laws and richness in different scientific fields such as solid, liquid, gas and plasma Application has been an extremely wide range of studies. Therefore, the study of the the KDV equation of numerical calculation method has important theoretical and practical significance. Corrected Local CrankNicolson method first proposed by Abdul Abdurixit Abdul external force, and with it has been good for solving a numerical solution of the heat conduction equation, which is an efficient and unconditionally stable explicit difference scheme . So there is no need for solving equations to reduce the amount of computation, which is very important in the numerical calculation. I combined the work of their predecessors of the the KDV equation using Modified Local CrankNicolson method. The method is the study of partial differential equations are transformed into ordinary differential equations, reuse the trotter plot of the exponential function formula to approximate the coefficient matrix of ordinary differential equations. Then it splits into a simple matrix, and then you can get a new differential format using the CrankNicolson method, which is a weak nonlinear equations, nonlinear term linearization approximation, that the nonlinear term lagged one time step, the resulting linear equations can be solved iterative method to get the final result. This method is not only solving the KDV equation, but also in solving nonlinear equations, enriched and developed Corrector Local CrankNicolson method for the numerical solution of some partial differential equations provide reference. The full text is divided into four chapters, the first chapter of the preamble describes the background, purpose and significance of the study of the KDV equation described the KDV equation numerically solving the status quo, and finally given the full text of the organizational structure. The second chapter gives the the KDV equation of CrankNicolson method, it is a stable and satisfies the discrete form of primary and secondary conservation laws for the secondorder implicit difference scheme. It is proved that the stability and convergence of the format, the final numerical test, a good description of the physical phenomena of the problem, and the format is good and effective. Chapter KDV equation Modified Local CrankNicolson method, it is an unconditionally stable secondorder explicit difference scheme. The KDV equation Modified Local CrankNicolson format to establish process in detail, and the theoretical analysis, the final numerical experiments, the numerical results is very good, the format is very effective. The fourth chapter is the conclusion, a summary of the full text of the two methods were compared.