On the Existence of Solutions for Algebraically Stable Runge--Kutta Methods

An implicit Runge—Kutta method, applied to an initial-valueproblem, gives systems of algebraic equations. It is shown that,under natural assumptions concerning the differential system,these equations have unique solutions if the method satisfiesa condition related to algebraic stability. In particular, thiscondition is satisfied if the method is irreducible and (k,l)-algebraically stable for some l 0.     

Global Error Estimation with Runge--Kutta Methods

An analysis of global error estimation for Runge—Kuttasolutions of ordinary differential equations is presented. Thebasic technique is that of Zadunaisky in which the global erroris computed from a numerical solution of a neighbouring problemrelated to the main problem by some method of interpolation.It is shown that Runge—Kutta formulae which permit validglobal error estimation using low-degree interpolation can bedeveloped, thus leading to more accurate and computationallyconvenient algorithms than was hitherto expected. Some specialRunge—Kutta processes up to order 4 are presented togetherwith numerical results.     

Global Error Estimation with Runge--Kutta Methods II

An analysis of global error estimation using the Zadunaiskytechnique with Runge—Kutta methods is presented. Threeforms of interpolant which can lead to valid asymptotic estimationare considered. Test results indicate that the Hermite formcoupled with special Runge—Kutta formulae is to be preferred,particularly when two-term global error estimation can be obtained.Very reliable estimation can be achieved and it is suggestedthat the technique could form the basis of a production code.     

An Iteration Scheme for Implicit Runge--Kutta Methods

In the implementation of implicit Runge-Kutta methods the computationaleffort is typically dominated by the cost of solving large setsof non-linear equations. As an alternative to the usual Newtonmethod, an iteration scheme is considered that sacrifices superlinearconvergence for reduced linear algebra costs. By modifying thisnew scheme, using the successive over-relaxation technique,an improvement is achieved in the rate of convergence.     

A Generalization of Algebraic Stability for Runge--Kutta Methods

The theory of algebraic stability does not give detailed informationabout the stability properties of explicit methods. This theoryis generalized by considering a monotonic and bounded test problem.The results give additional information about the stabilityproperties of explicit and implicit methods and show how particularexplicit methods may be selected.     

Solution of Differential--Algebraic Systems Using Diagonally Implicit Runge--Kutta Methods

Diagonally Implicit Runge—Kutta (DIRK) methods are developedand applied to differential—algebraic systems arisingfrom dynamic process simulation. In particular, an embeddedfamily of DIRK methods is developed for implementation as avariable-step variable-order algorithm. The methods developedallow easy assessment of local solution error as well as theability to change the order of approximation. The stabilityproperties of the methods are chosen to make them suitable foruse on stiff systems. Some important aspects of implementation of DIRK methods arediscussed within the context of the solution of differential—algebraicsystems. The performance of this algorithm is compared withan alternative variable-order approach based on "triples" whichallows the patching together of several fixed-order formulae.The results indicate that the fully embedded DIRK algorithmis generally more efficient than the algorithm based on "triples".Areas of further investigation in the context of differential—algebraicsystems are outlined.     

Convergence and Stability Analysis for Modified Runge--Kutta Methods in the Numerical Treatment of Second-Kind Volterra Integral Equations

In this paper modified and conventional Runge—Kutta methodsfor second kind Volterra integral equations are discussed ina uniform way. The modification presented takes into accountthe residual of the previous step with the aim of improvingthe stability behaviour. A general convergence theorem is givenwhich establishes that the modified methods may lose one orderof accuracy. Furthermore, the stability behaviour of the methodsis analysed and explicit stability results are derived. It transpiresthat every A-stable Runge—Kutta method for ordinary differentialequations generates mixed methods which can be made A-stableby a suitable modification.     

Parallel Implicit-Explicit General Linear Methods

High-order discretizations of partial differential equations(PDEs)necessitate high-order time integration schemes capable of handling both stiff and nonstiff op...     

On the Utility of Graphics Cards to Perform Massively Parallel Simulation of Advanced Monte Carlo Methods

We present a case study on the utility of graphics cards to perform massively parallel simulation of advanced Monte Carlo methods. Graphics cards, containing multiple Graphics Processing Units (GPUs), are self-contained parallel computational devices that can be housed in conventional desktop and laptop computers and can be thought of as prototypes of the next generation of many-core processors. For certain classes of population-based Monte Carlo algorithms they offer massively parallel simulation, with the added advantage over conventional distributed multicore processors that they are cheap, easily accessible, easy to maintain, easy to code, dedicated local devices with low power consumption. On a canonical set of stochastic simulation examples including population-based Markov chain Monte Carlo methods and Sequential Monte Carlo methods, we find speedups from 35- to 500-fold over conventional single-threaded computer code. Our findings suggest that GPUs have the potential to facilitate the growth of statistical modeling into complex data-rich domains through the availability of cheap and accessible many-core computation. We believe the speedup we observe should motivate wider use of parallelizable simulation methods and greater methodological attention to their design. This article has supplementary material online.     

广义并行矩阵多分裂松弛算法

求解大型线性代数方程组的并行矩阵多分裂算法讨论的大多为系数矩阵是非奇日矩阵的情况，[2]提出了当系数矩阵是非奇H矩阵时的广义矩阵多分裂松弛算法．对系数矩阵是奇异日矩阵的情况研究较少，本文给出了当系数矩阵G是不可约奇异H矩阵时的齐次线性方程组Gx=0的广义矩阵多分裂松弛算法并讨论其收敛性。     

一类块隐式混合单步并行计算方法

本文讨论了一类解常微分方程初值问题的块隐式混合单步并行算法，这种算法的块数为K，精度阶为2d＋2，可在S台处理机上进行并行计算，其中K＝S·d．本文讨论了方法的一般性质，给出了方法的稳定性定理，最后给出了一个数值例子．     

求解广义鞍点问题的混合并行迭代法(英文)

为了在高性能计算机上求解广义鞍点问题,对于合适的系数矩阵,本文提出混合并行迭代法及其加速形式.并详细讨论了新方法的收敛性.     

On the United Theory of the Family of Euler-Halley Type Methods with Cubical Convergence in Banach Spaces

The convergence problem of the family of Euler-Halley methods is considered under the Lipschitz condition with the L-average, and a united convergence theory with its applications is presented.     

On the United Theory of the Family of Euler—Haller Type Methods with Cubical Convergence in Banach Spaces

The convergence problem of the family of Euler-Halley methods is considered under the Lipschitz condition with the L-average,and a united convergence theory with its applications is presented.     

Algebraic Theory of Two-Grid Methods

About thirty years ago, Achi Brandt wrote a seminal paper providing a convergence theory for algebraic multigrid methods [Appl. Math. Comput., 19 (1986), pp. 23–56]. Since then, this theory has been improved and extended in a number of ways, and these results have been used in many works to analyze algebraic multigrid methods and guide their developments. This paper makes a concise exposition of the state of the art. Results for symmetric and nonsymmetric matrices are presented in a unified way, highlighting the influence of the smoothing scheme on the convergence estimates. Attention is also paid to sharp eigenvalue bounds for the case where one uses a single smoothing step, allowing straightforward application to deflation-based preconditioners and two-level domain decomposition methods. Some new results are introduced whenever needed to complete the picture, and the material is self-contained thanks to a collection of new proofs, often shorter than the original ones.     

关于奇异摄动问题一致有效渐近解

In this paper, the defect of [     

并行多分裂迭代收敛速率的估计

本文给出了并多分裂迭代（PMI）收敛速度的一个估计式,利用此估计式可以简化和统一PMI方法的收敛性证明。