解微分代数方程的波形松弛方法
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摘要
本文首先综述了波形松弛方法的起源,发展和现状,其中包括方法的基本思想和应用领域。然后我们主要研究解线性微分代数方程的波形松弛方法,包括连续时间情形和离散时间情形的波形松弛方法。波形松弛方法是一种具有良好并行性的方法,它被广泛应用于常微分方程系统,微分代数系统,延迟微分动力系统,泛函微分动力系统,热传导方程,双曲型微分方程,对流-扩散方程,随机微分动力系统等的数值求解中。
波形松弛方法是一种动态迭代方法,在应用其求解各种问题时,首先要考虑的是它的收敛性,包括连续情形和离散情形的收敛性。与连续波形松弛方法收敛性紧密相关的是它的分裂函数,而离散波形松弛方法的收敛性不仅依赖于连续情形的收敛性,而且还与离散方法的稳定性有关。本文我们主要研究方法的收敛性,即对反映方法收敛性态的迭代算子谱半径进行研究和分析。
文中,我们首先总结前人求解1指标的线性微分代数方程组的连续波形松弛方法的主要结果,并且用不同方法给出连续时间波形松弛算子谱半径和文献相同的表达式。此外,我们还给出一类连续波形松弛算子在无限时间区间上的谱半径表达式和收敛性结果。随后,我们主要研究离散波形松弛方法的收敛性。包括求解微分代数系统的线性多步方法,Runge-Kutta方法,边值方法和分块边值方法的波形松弛方法。由于微分代数系统的刚性性质,我们主要研究具有良好稳定性质的隐式方法,其中边值方法是具有良好稳定性质和阶精度的新方法。对于线性多步方法的波形松弛迭代,我们结合离散方法的稳定性性质,分析算子的谱半径,给出方法的收敛性结果。对于解1指标的线性微分代数系统的Runge-Kutta方法,边值方法和分块边值方法的波形松弛迭代,我们首次给出了算子谱半径的表达式,并且结合离散方法的稳定性质给出了方法的离散波形松弛迭代的收敛性结果。最后,我们进行数值实验,数值结果很好地验证前面给出的结论。
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.
波形松弛方法是一种动态迭代方法,在应用其求解各种问题时,首先要考虑的是它的收敛性,包括连续情形和离散情形的收敛性。与连续波形松弛方法收敛性紧密相关的是它的分裂函数,而离散波形松弛方法的收敛性不仅依赖于连续情形的收敛性,而且还与离散方法的稳定性有关。本文我们主要研究方法的收敛性,即对反映方法收敛性态的迭代算子谱半径进行研究和分析。
文中,我们首先总结前人求解1指标的线性微分代数方程组的连续波形松弛方法的主要结果,并且用不同方法给出连续时间波形松弛算子谱半径和文献相同的表达式。此外,我们还给出一类连续波形松弛算子在无限时间区间上的谱半径表达式和收敛性结果。随后,我们主要研究离散波形松弛方法的收敛性。包括求解微分代数系统的线性多步方法,Runge-Kutta方法,边值方法和分块边值方法的波形松弛方法。由于微分代数系统的刚性性质,我们主要研究具有良好稳定性质的隐式方法,其中边值方法是具有良好稳定性质和阶精度的新方法。对于线性多步方法的波形松弛迭代,我们结合离散方法的稳定性性质,分析算子的谱半径,给出方法的收敛性结果。对于解1指标的线性微分代数系统的Runge-Kutta方法,边值方法和分块边值方法的波形松弛迭代,我们首次给出了算子谱半径的表达式,并且结合离散方法的稳定性质给出了方法的离散波形松弛迭代的收敛性结果。最后,我们进行数值实验,数值结果很好地验证前面给出的结论。
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.
引文
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[3] White J, Sangiovanni-Vincentelli A. Relaxation techniques for the simulation of VLSI cir-cuits. Kluwer Academic Publishers Norwell, MA, USA, 1987
[4] Riechelt M, White J, Allen J, et al. Waveform relaxation applied to transient device simula-tion. Circuits and Systems, 1988., IEEE International Symposium on, 1988, pages 1647–1650
[5] Erdman D, Rose D. A Newton waveform relaxation algorithm for circuit simulation.Computer-Aided Design, 1989. ICCAD-89. Digest of Technical Papers., 1989 IEEE Inter-national Conference on, 1989, pages 404–407
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[7] La Scala M, Bose A, Tylavsky D, et al. A highly parallel method for transient stability anal-ysis. IEEE Transactions on Power Systems, 1990, 5(4):1439–1446
[8] Saleh R, White J. Accelerating relaxation algorithms for circuit simulation using waveform-Newton and step-size refinement. Computer-Aided Design of Integrated Circuits and Sys-tems, IEEE Transactions on, 1990, 9(9):951–958
[9] Crow M, Ilic M. The parallel implementation of the waveform relaxation method fortransientstability simulations. Power Systems, IEEE Transactions on, 1990, 5(3):922–932
[10] Erdman D, Rose D. Newton waveform relaxation techniques for tightly coupled systems.Computer-Aided Design of Integrated Circuits and Systems, IEEE Transactions on, 1992,11(5):598–606
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