The Modified Local Crank-Nicolson Method For Burgers Equation

It is well-known that Burgers equation is one of the most simple model ofnonlinear convection di?usion problem. It is widely used in turbulence, heat andmass transfer, air and water pollution, continuous stochastic process and so on.This problem can form shock, which makes it di?culty to solve Burgers equation.Therefore, the study of the numerical method for Burgers equation is importantin theoretical and practical.The Modified Local Crank-Nicolson method is given by Abduwali first, andhe used this method to solve the heat equation very well. This method is an ex-plicit di?erence scheme with unconditional stability. Moreover, it avoids solvingthe linear equations. It is very important in numerical computation.In this paper, we give the Modified Local Crank-Nicolson method for Burgersequation based on the work of the predecessor. This method transforms thepartial di?erential equation into the ordinary di?erential equations, and uses theTrotter Product formula of exponential function to approximate the coe?cientmatrix of these ordinary di?erential equations. Then separates it into some small-block matrices, and employs Crank-Nicolson method to obtain a new di?erentscheme. It is a weak nonlinear system, and linearization approach is applied; i.e.,it is linearized by allowing the nonlinearities to lag one time step behind, and theobtained system of linear equations is solved by using iterative algorithms. Ourwork in this paper, is not only used to solve the Burgers equation, but also candevelop 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 researchbackground, purpose and significance, and describe the research situation ofnumerical solution for Burgers equation . Finally, the organizational structureof this work is given.In section 2, we give the Crank-Nicolson scheme for Burgers equation. Itis an implicit two order di?erence scheme with unconditional stability. In thispaper, we proof the stability and convergence, and finally make a numericalexperiment and the numerical results are in line with the physical situation ofthis problem. It is showed that the scheme is e?ective.In section 3, we give the Modified Local Crank-Nicolson scheme for Burg-ers equation. It is an explicit two order di?erence scheme with unconditionalstability. In this paper, the Modified Local Crank-Nicolson schemes for one-andtwo-dimensional Burgers equations are discussed in detail. Then, we also supportthe theoretical analysis, and finally several numerical experiments are given toverify the numerical results, and it is seen that they are in excellent agreement.Section 4 is conclusion, we make a conclusion on the whole work. By thecomparison of the two numerical methods, we find that the absolute error be-tween the exact solution and the numerical solution using the Modified LocalCrank-Nicolson scheme is less than that of the Crank-Nicolson scheme, and theModified Local Crank-Nicolson scheme uses less CPU time. It is shown thatthe Modified Local Crank-Nicolson method is an e?ective numerical method forsolving partial di?erential equation.