Dissertation > Mathematical sciences and chemical > Mathematics > Mathematical Analysis > Differential equations, integral equations > Partial Differential Equations

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 Crank-Nicolson method Modified Local Crank-Nicolson method Stability Convergence
CLC O175.2
Type Master's thesis
Year 2010
Downloads 35
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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 Crank-Nicolson 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 Crank-Nicolson 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 Crank-Nicolson 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 Crank-Nicolson 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 Crank-Nicolson method, it is a stable and satisfies the discrete form of primary and secondary conservation laws for the second-order 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 Crank-Nicolson method, it is an unconditionally stable second-order explicit difference scheme. The KDV equation Modified Local Crank-Nicolson 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.

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