A CLASS OF r-POINT (r+1)ST-ORDER A-STABLE ONE-BLOCK METHODS WITH DAMPING AT THE INFINITE POINT

1　ＩｎｔｒｏｄｕｃｔｉｏｎＦｏｒｓｏｌｖｉｎｇｓｔｉｆｆｉｎｉｔｉａｌｖａｌｕｅｐｒｏｂｌｅｍｓｆｏｒｓｙｓｔｅｍｓｏｆＯＤＥｓｙ′=ｆ(ｙ) ,ｙ(ｔ0 ) =ｙ0 ,ｔ0 <ｔ≤Ｔ ,ｙ0 ,ｙ∈Ｒｍ,ｆ :Ω Ｒｍ →Ｒｍ (1 .1 )ｍａｎｙｐａｒｔｉｃｕｌａｒｏｎｅ ｂｌｏｃｋｍｅｔｈｏｄｓｏｆｔｈｅｆｏｒｍＹｎ+1= ＡＹｎ+ｈ( Ｂ0 Ｆ(Ｙｎ) + Ｂ1Ｆ(Ｙｎ+1) ) , Ａ =Ａ Ｉｍ, Ｂｉ=Ｂｉ Ｉｍ,Ａ ,Ｂｉ∈Ｒｒ×ｒ,Ｙｎ =(ＹＴｎｒ,… ,ｙＴ(ｎ+1)ｒ- 1) Ｔ,Ｆ(Ｙｎ) =(ｆＴ(ｙｎｒ) ,… ,ｆＴ(ｙ(ｎ+1)ｒ- 1) ) Ｔ,ｙｊ≈ ｙ(ｔｊ) ,…     

A Class of Multiderivative Hybrid One-step Methods

This paper presents a class of hybrid one-step methods that are obtained by using Cramer's rule and rational approximations to function exp(q). The algorithms fall into the catalogue of implicit formula, which involves sth order derivative and s 1 free parameters. The order of the algorithms satisfies s 1≤p≤2s 2. The stability of the methods is also studied, necessary and sufficient conditions for A-stability and L-stability are given. In addition, some examples are also given to demonstrate the method presented.     

The A-Stability of Methods with Padé and Generalized Padé Stability Functions

This paper surveys some stability results and suggests the use of order arrows as an alternative to order stars in studying questions about the possible A-stability of a numerical method. A discussion of the so-called Butcher–Chipman conjecture includes a proof of a partial result.     

A family of A-stable Runge Kutta collocation methods of higher order for initial-value problems

^{We consider the construction of a special family of Runge–Kutta(RK) collocation methods based on intra-step nodal points ofChebyshev–Gauss–Lobatto type, with A-stability andstiffly accurate characteristics. This feature with its inherentimplicitness makes them suitable for solving stiff initial-valueproblems. In fact, the two simplest cases consist in the well-knowntrapezoidal rule and the fourth-order Runge–Kutta–LobattoIIIA method. We will present here the coefficients up to eighthorder, but we provide the formulas to obtain methods of higherorder. When the number of stages is odd, we have considereda new strategy for changing the step size based on the use ofa pair of methods: the given RK method and a linear multistepone. Some numerical experiments are considered in order to checkthe behaviour of the methods when applied to a variety of initial-valueproblems.}     

一类k步k+2阶解刚性微分方程的混杂法

构造了一类带参数的k步k+2阶混杂方法,讨论了该方法的稳定性质．并给出了与其等价的二阶导数方法．数值实例说明,这类方法更适合求解非线性Stiff问题,对高震荡问题亦会更有效．     

A New Eighth-order A-stable Method for Solving Differential Systems Arising in Chemical Reactions

Implicit Runge–Kutta methods are successful algorithms for the numerical solution of stiff differential equations, as they usually appear in chemical reactions. This article describes the construction of a particular implicit method based on internal stages obtained from certain Chebyshev collocation points. The resulting method has algebraic order 8 and A-stability characteristic. An embedding technique using the Runge–Kutta method and a linear multistep one is provided in order to change the step size. Numerical experiments illustrate the behaviour of the new method, showing that it may reach great accuracy and be competitive with other well-known codes.     

Weak first- or second-order implicit Runge–Kutta methods for stochastic differential equations with a scalar Wiener process

New fully implicit stochastic Runge–Kutta schemes of weak order 1 or 2 are proposed for stochastic differential equations with sufficiently smooth drift and diffusion coefficients and a scalar Wiener process, which are derivative-free and which are A-stable in mean square for a linear test equation in some general settings. They are sought in a transparent way and their convergence order and stability properties are confirmed in numerical experiments.     

Numerical methods for nonlinear stochastic differential equations with jumps

We present and analyse two implicit methods for Ito stochastic differential equations (SDEs) with Poisson-driven jumps. The first method, SSBE, is a split-step extension of the backward Euler method. The second method, CSSBE, arises from the introduction of a compensated, martingale, form of the Poisson process. We show that both methods are amenable to rigorous analysis when a one-sided Lipschitz condition, rather than a more restrictive global Lipschitz condition, holds for the drift. Our analysis covers strong convergence and nonlinear stability. We prove that both methods give strong convergence when the drift coefficient is one-sided Lipschitz and the diffusion and jump coefficients are globally Lipschitz. On the way to proving these results, we show that a compensated form of the Euler–Maruyama method converges strongly when the SDE coefficients satisfy a local Lipschitz condition and the pth moment of the exact and numerical solution are bounded for some p>2. Under our assumptions, both SSBE and CSSBE give well-defined, unique solutions for sufficiently small stepsizes, and SSBE has the advantage that the restriction is independent of the jump intensity. We also study the ability of the methods to reproduce exponential mean-square stability in the case where the drift has a negative one-sided Lipschitz constant. This work extends the deterministic nonlinear stability theory in numerical analysis. We find that SSBE preserves stability under a stepsize constraint that is independent of the initial data. CSSBE satisfies an even stronger condition, and gives a generalization of B-stability. Finally, we specialize to a linear test problem and show that CSSBE has a natural extension of deterministic A-stability. The difference in stability properties of the SSBE and CSSBE methods emphasizes that the addition of a jump term has a significant effect that cannot be deduced directly from the non-jump literature.This work was supported by Engineering and Physical Sciences Research Council grant GR/T19100 and by a Research Fellowship from The Royal Society of Edinburgh/Scottish Executive Education and Lifelong Learning Department.     

求解刚性振荡问题的两类带显式级的三级对角隐式Runge-Kutta方法

针对刚性振荡问题,讨论了两类带显式级的三级对角隐式Runge-Kutta方法的阶、级阶、A-稳定性、相误差和耗散误差,所构造的方法成功应用于一类大气化学反应问题的求解.