Waveform Relaxation Method For Solving Differential-algebraic Equations

In this document, we first summarize the appearance, development and actuality of thewaveform relaxation (WR) method, including the basic idea of the method and its appli-cation. Then we mainly study the WR method for solving differential-algebraic equations(DAEs) at the continuous-time case and discrete-time case. WR method has excellent paral-lel property and has been applied broadly to solving numerical solution of ordinary differen-tial equations, differential-algebraic equations, delay differential equations, functional dif-ferential equations, heat equations, hyperbolic differential equations, convection-diffusionequations, stochastic differential equations and so on.WR method is a dynamic method. When using it for solving problems, we should con-sider the convergence of the method. Closely related to the convergence of the continuous-time WR method is the splitting function. Unlike the continuous-time case, the convergenceof discrete-time WR method not only relates to continuous-time WR method, but also de-pends on the stability of the discrete methods. Here we mainly study the convergence of themethod, i.e. studying and analyzing the spectral radius of the WR operator which responsethe convergence of the method.In the second chapter, we summarize the main results of continuous-time WRmethod for solving linear index 1 DAEs and get the same expression of spectral radiusof continuous-time WR operator as in the literature in a different way. Besides, we getthe expression of spectral radius of a class of WR operators and convergent result at theinfinite interval case. After that, we study the discrete-time WR method in the third chapterand fourth chapter. Combining the WR method with discrete methods, we get a seriesof discrete-time WR methods. Here we consider implicit discrete methods which haveexcellent stable properties, because differential-algebraic system is stiff. Among thesediscrete methods, boundary value method (BVM) is a new method with well stabilities andaccuracy order. We analyze the spectral radius of the linear multi-step formulae (LMF)WR operator and get its expression by using the stable domain. This result is better thanthat in the exist literature. Meanwhile, we get the spectral radius of Runge-Kutta (RK) WRoperator, BVM WR operator and BBVM WR operator for linear index 1 DAEs for the firsttime.