Optimized Waveform Relaxation Method For Pentadiagonal Linear Differential Equations

Waveform relaxation method is a parallel algorithm based on system decoupling; ithas been widely used to solve ordinary and partial differential systems. Optimizedwaveform relaxation method can improve the convergence rate by introducing newtransmission conditions, which have a tremendous influence on the convergence of thewaveform relaxation algorithm, the more information these conditions transmit, thefaster the convergence rate is. In this thesis, we mainly consider solving pentadiagonallinear differential system by using waveform relaxation method.In chapter one, we describe the background of waveform relaxation method,overview the current development of this method and briefly describe the structure of thisthesis.In chapter two, we discuss a small pentadiagonal differential system. Newtransmission conditions with a parameter are proposed to construct the optimizedwaveform relaxation method. After decoupling the system, we calculate the convergencefactor by doing Laplace transform and some algebra operations. And then by solving amaximum-minimum problem about the parameter via numerical method to find out theexact value of the parameter, we can get the optimized waveform relaxation method.Finally, we compare the convergence rate with the classical waveform relaxation methodthrough their spectral radius.In chapter three, we study a large pentadiagonal differential system. After decouplingthe system, we construct a new waveform relaxation method. We present theconvergence rate by the solutions of difference equations and judge the convergence byspectral radiuses. And then, compare convergence rate with classical waveform relaxationmethod. Numerical result shows that no matter which system it is applied to, small systemor large system, optimized waveform relaxation method can remarkably improve theconvergence rate.