Two Kinds Of Parallel Difference Computing Methods For MKdV Equation

Two kinds of parallel difference computing methods are given for solving the three-order nonlinear modified Korteweg-de Vries (mKdV) equation with periodic boundary value problem. One is alternating segment difference method, and the other is alternating group iterative method. The first one derive a general alternating difference scheme on the base of difference schemes with intrinsic parallelism for dispersive equation by zhu shaohong, which plus a nonlinear item u~2u_x. And then, the alternating segment explicit-implicit method (ASE-I) and alternating segment Crank-Nicolson method (ASC-N) are obtained by the proper valuation of the discrete variable. The second one is also on the base of dispersive equation. First, the alternating group iterative method is given for dispersive equation. Its theoretical analysis and numerical results are also given. And then, the method is applied to the nonliear mKdV equation. The alternating segment scheme and iterative process are unconditionally stable and convergence by analysis of linearization procedure. Finally, numerical examples for a single soliton solution and the collision of two solitons by the two methods are given, which also compare the accuracy and convergence rate between the numercal solution and the exact solution for a single soliton solution. Numerical results show that two methods have good practicability and accuracy, and are very suitable to parallel computing.The paper is composed of three parts as shown below:In first part, we mainly introduce the background and some effective numerical methods for mKdV equation, and the development and some results of the two methods we use.In the second part, we give general alternating difference scheme for the mKdV equation, and then, obtain two kinds of nonsymmetrical difference scheme by the proper valuation of the discrete variable. We design three difference methods using the nonsymmetrical difference scheme and explicit scheme and implicit scheme: one is an alternating segment difference method (AGE), the other is the ASE-I method in general case, another is the ASE-I method in special case. We get ASC-N method using the other kind of nonsymmetrial difference scheme and Crank-Nicolson scheme.In order to analysis the linear stability of the difference scheme, we give the forms of the concrete matrix of the ASE-I method in general case and the ASC-N method. Numerical examples proof the stability and accuracy of those methods.In the third part, we first introduce the alternating group iterative method for dispersive equation, theoretical analysis and numerical results proof the practicability and accuracy, and then, an explicit finite difference scheme for three-order nonlinear mKdV equation with periodic boundary values is given. Then an alternating group iterative method is suggested which can be used to solve mKdV equation parallel. The linear convergence of the iterative process is proved. Finally, numerical examples for one solitary wave and two solitary wave are proposed.