Dissipativity of Runge-Kutta methods for dynamical systems with delays

This paper is concerned with the numerical solution of dissipativeinitial value problems with delays by Runge-Kutta methods. Asufficient condition for the dissipativity of the systems isgiven. The concepts of D(l)-dissipativity and GD(l)-dissipativityare introduced. We investigate the dissipativity propertiesof (k,l)-algebraically stable Runge-Kutta methods with piecewiseconstant or linear interpolation procedures for finite-dimensionaland infinite-dimensional dynamical systems with delays.     

积分微分方程Runge-Kutta方法的散逸性

本文针对一类积分微分方程讨论Runge-Kutta方法的散逸性,当积分项用PQ公式逼近时,证明了(k,l)-代数稳定的Runge-Kutta方法是D(l)-散逸的.     

Stability Analysis of Runge-Kutta Methods for Non-Linear Delay Differential Equations

This paper is concerned with the numerical solution of delay differential equations(DDEs). We focus on the stability behaviour of Runge-Kutta methods for nonlinear DDEs. The new concepts of GR(l)-stability, GAR(l)-stability and weak GAR(l)-stability are further introduced. We investigate these stability properties for (k, l)-algebraically stable Runge-Kutta methods with a piecewise constant or linear interpolation procedure.     

Nonlinear Stability of General Linear Methods for Delay Differential Equations

This paper is devoted to a study of nonlinear stability of general linear methods for the numerical solution of delay differential equations in Hilbert spaces. New stability concepts are further introduced. The stability properties of (k,p,q)-algebraically stable general linear methods with piecewise constant or linear interpolation procedure are investigated. We also discuss stability of linear multistep methods viewed as a special subset of the class of general linear methods.     

非线性延迟积分微分方程连续Runge-Kutta方法的稳定性分析

本文主要研究了一般形式的延迟积分微分方程,将连续Runge-Kutt,a方法用于求解该类问题,并讨论了方法的稳定性,证明了(k,l)-代数稳定的Runge-Kutta方法当0k1时对应的连续Runge-Kutta方法是渐近稳定的.最后我们通过数值试验验证了方法的有效性及所获结论的正确性.     

Algebraic stability and the existence of solutions of implicit Runge-Kutta equations

An implicit Runge-Kutta method, applied to an initial value problem, gives systems of algebraic equations. Under natural assumptions concerning the differential system, there are known conditions on the method which guarantee that the algebraic equations have unique solutions. It is shown that these conditions are closely related to the requirement that the method be (k(l),l)-algebraically stable on an interval [0,).     

Nonlinear stability of Runge-Kutta methods applied to infinite-delay-differential equations

In functional differential equations (FDEs), there is a class of infinite delay-differential equations (IDDEs) with proportional delays, which aries in many scientific fields such as electric mechanics, quantum mechanics, and optics. Ones have found that there exist very different mathematical challenges between FDEs with proportional delays and those with constant delays. Some research on the numerical solutions and the corresponding analysis for the linear FDEs with proportional delays have been presented by several authors. However, up to now, the research for nonlinear case still remains to be done. For this, in the present paper, we deal with nonlinear stability of the Runge-Kutta (RK) methods for a class of IDDEs with proportional delays. It is shown under the suitable conditions that a (k, l)-algebraically stable RK method for this kind of nonlinear IDDE is globally and asymptotically stable.     

STABILITY ANALYSIS OF RUNGE-KUTTA METHODS FOR NONLINEAR SYSTEMS OF PANTOGRAPH EQUATIONS

This paper is concerned with numerical stability of nonlinear systems of pantograph equations. Numerical methods based on （k, l）-algebraically stable Runge-Kutta methods are suggested. Global and asymptotic stability conditions for the presented methods are derived.     

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.     

Stability analysis of numerical methods for systems of neutral delay-differential equations

Stability analysis of some representative numerical methods for systems of neutral delay-differential equations (NDDEs) is considered. After the establishment of a sufficient condition of asymptotic stability for linear NDDEs, the stability regions of linear multistep, explicit Runge-Kutta and implicitA-stable Runge-Kutta methods are discussed when they are applied to asymptotically stable linear NDDEs. Some mentioning about the extension of the results for the multiple delay case is given.     

Stabilized multistep methods for index 2 Euler-Lagrange DAEs

We consider multistep discretizations, stabilized by -blocking, for Euler-Lagrange DAEs of index 2. Thus we may use nonstiff multistep methods with an appropriate stabilizing difference correction applied to the Lagrangian multiplier term. We show that orderp =k + 1 can be achieved for the differential variables with orderp =k for the Lagrangian multiplier fork-step difference corrected BDF methods as well as for low orderk-step Adams-Moulton methods. This approach is related to the recently proposed half-explicit Runge-Kutta methods.     

Delay-dependent stability analysis of multistep methods for delay differential equations

This paper deals with the delay-dependent stability of numerical methods for delay differential equations. First, a stability criterion of Runge-Kutta methods is extended to the case of general linear methods. Then, linear multistep methods are considered and a class of r（0）-stable methods are found. Later, some examples of r（0）-stable multistep multistage methods are given. Finally, numerical experiments are presented to confirm the theoretical results.     

Asymptotic behavior of multistep runge-kutta methods for systems of delay differential equations

1. IntroductionIn order to assess the asymptotic behavior of numerical methods for DDEs, much attention has been given in the literature to the scalar case (cL [1-6]). UP to now) only partialresults (of. [7-10]) have dealt with the delay systemswhere y(t) = (yi(t), so(t),' ) yp(t))" E Cd, which is unknown for t > 0, L and M areconstat complex p x Hmatrices, T > 0 is a constat delay and W(t) 6 CP is a specifiedinitial function.In [111, C.J. Zhang and S.Z. Zhou made an investigation on…     

The equivalence ofB-stability andA-stability

It is a well-known result of Dahlquist that the linear (A-stability) and non-linear (G-stability) stability concepts are equivalent for multistep methods in their one-leg formulation. We show to what extent this result also holds for Runge-Kutta methods. Dedicated to Germund Dahlquist on the occasion of his 60th birthday.     

Total Variation Diminishing Implicit Runge-Kutta Methods for Dissipative Advection-Diffusion Problems in Astrophysics

We investigate the properties of dissipative full discretizations for the equations of motion associated with models of flow and radiative transport inside stars. We derive dissipative space discretizations and demonstrate that together with specially adapted total-variation-diminishing (TVD) or strongly stable Runge-Kutta time discretizations with adaptive step-size control this yields reliable and efficient integrators for the underlying high-dimensional nonlinear evolution equations. For the most general problem class, fully implicit SDIRK methods are demonstrated to be competitive when compared to popular explicit Runge-Kutta schemes as the additional effort for the solution of the associated nonlinear equations is compensated by the larger step-sizes admissible for strong stability and dissipativity. For the parameter regime associated with semiconvection we can use partitioned IMEX Runge-Kutta schemes, where the solution of the implicit part can be reduced to the solution of an elliptic problem. This yields a significant gain in performance as compared to either fully implicit or explicit time integrators. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Stability and -convergence properties of multistep Runge-Kutta methods

This paper continues earlier work by the same author concerning the stability and -convergence properties of multistep Runge-Kutta methods for the numerical solution of nonlinear stiff initial-value problems in a Hilbert space. A series of sufficient conditions and necessary conditions for a multistep Runge-Kutta method to be algebraically stable, diagonally stable, - or optimally -convergent are established, by means of which six classes of high order algebraically stable and -convergent multistep Runge-Kutta methods are constructed in a unified pattern. These methods include the class constructed by Burrage in 1987 as special case, and most of them can be regarded as extension of the Gauss, RadauIA, RadauIIA and LobattoIIIC Runge-Kutta methods. We find that the classes of multistep Runge-Kutta methods constructed in the present paper are superior in many respects to the corresponding existing one-step Runge-Kutta schemes.

     

一类线性隐式A稳定的单步法

1.引言 对于Stiff方程组初值问题的数值解法,Dahlquist在[1]中引进了 A稳定的概念,并且证明了显式的线性多步法(包括显式的Runge-Kutta方法)不可能是A稳定的.现在已经有许许多多隐式A稳定或Stiff稳定的方法,但绝大多数在数值解的过程中必须解由于隐式方法所产生的非线性方程组,而非线性方程组的求解过程往往又要采用Newton-Raphson迭代方法,因此需要计算方程y’=f(x,y)的右函数f(x,y)的Jacobi矩阵以及与此有关的逆矩阵.本文的主要思想是:既然在数值解过程中要计算f(x,y)的Jacobi矩阵,那么不妨在数值公式中明显的出现f(x,y)的一阶偏导数.我们将A稳定公式     

Convergence Aspects of Step-Parallel Iteration of Runge-Kutta Methods for Delay Differential Equations

Implicit Runge-Kutta methods are known as highly accurate and stable methods for solving differential equations. However, the iteration technique used to solve implicit Runge-Kutta methods requires a lot of computational efforts. To lessen the computational effort, one can iterate simultaneously at a number of points along the t-axis. In this paper, we extend the PDIRK (Parallel Diagonal Iterated Runge-Kutta) methods to delay differential equations (DDEs). We give the region of convergence and analyze the speed of convergence in three parts for the P-stability region of the Runge-Kutta corrector. It is proved that PDIRK methods to DDEs are efficient, and the diagonal matrix D of the PDIRK methods for DDES can be selected in the same way as for ordinary differential equations (ODEs).     

Linearly implicit methods for nonlinear parabolic equations

We construct and analyze combinations of rational implicit and explicit multistep methods for nonlinear parabolic equations. The resulting schemes are linearly implicit and include as particular cases implicit-explicit multistep schemes as well as the combination of implicit Runge-Kutta schemes and extrapolation. An optimal condition for the stability constant is derived under which the schemes are locally stable. We establish optimal order error estimates.

     

Dissipativity of Runge-Kutta methods in Hilbert spaces

This paper concerns the discretization by Runge-Kutta methods of the initial value problemu _t =f(u), under the dissipative structural condition that there exist α≥0, β>0, such thatf:W→H, ℜe, ∀w∈W, for complex Hilbert spacesW⊆H. It is shown that strong A-stability is necessary to ensure the dissipativity of the method, whilst algebraic stability plus |R(∞)|<1 is a sufficient condition in the case of DJ-irreducible methods.