The Convergence Of The Alternating Direction Implicit Waveform Relaxation Method: The Discrete-time Case

To solve large-scale ordinary differential equations is one of the most important researches of computational mathematics.And what we are encountered in the field of modern engineering and science computation would be modeled as a large-scale linear ordinary differential equations. There are a great number of scholars who have done every effort and designed lots of numerical methods of good performance so as to solve large-scale ordinary differential equations.As early as in the the beginning of 1980s, E.Lelarasmee with other people proposed waveform relaxation method to solve differential-algebra equations which were from large-scale integrad circuit,and obtained very good effect.Alike traditional iterative methods to solve linear algebra equations, waveform relaxation method decomposes the large-scale differential-algebra into many subsystems by splitting the coefficient matrices.So it is easy to realize parallel computation and saves much time to compute .Waveform relaxation method has so many good performances that more and more scholars have successfully applied the algorithm to other area besides the problems of electric circuit.In this paper,we first briefly review the background of waveform relaxation method,basic thought and introduce the history and development of research about this approach.In section two,we introduce basic knowledge of waveform relaxation method,and discuss some major theorical results basing on the previous studies.With regards of the realization of the continuous waveform relaxation method on the computer,in the third chapter we use linear multi-step method to discrete the alternating direction implicit waveform relaxation method.By introducing the error vector ,we can get the iterative form of error vector and study its convergence property in detail.The forth section will try to use block SOR to accelerate alternating direction implicit waveform relaxation method.We apply linear multi-step method to discrete it and discuss its convergence character according to the knowledge of block matrices.The results show that this algorithm can improve the speed of convergence by introducing the parameter in some ways .Then in the fifth section ,numerical experiments prove the correctness and effectiveness of our main work.Finally,in the sixth chapter we summarizes the full text and forcasts the development prospect of the waveform relaxation method.