求解比例延迟微分方程的Rosenbrock方法的渐近稳定性(英文)

本文讨论了一类Rosenbrock方法求解比例延迟微分方程,y′(t)=λy(t) μy(qt),λ,μ∈C,0     

Maximal order for second derivative general linear methods with Runge-Kutta stability

An extension of general linear methods (GLMs), so-called SGLMs (GLMs with second derivative), was introduced to the case in which second derivatives, as well as first derivatives, can be calculated. SGLMs are divided into four types, depending on the nature of the differential system to be solved and the computer architecture that is used to implement these methods. In this paper, we obtain maximal order for two types of SGLMs with Runge-Kutta stability (RKS) property. Also, we construct methods of these types which possess RKS property and A-stability. Efficiency of the constructed methods is shown by numerical experiments.     

Some linear stability results for iterative schemes for implicit Runge-Kutta methods

This article examines stability properties of some linear iterative schemes that have been proposed for the solution of the nonlinear algebraic equations arising in the use of implicit Runge-Kutta methods to solve a differential systemx =f(x). Each iteration step requires the solution of a set of linear equations, with constant matrixI –hJ, whereJ is the Jacobian off evaluated at some fixed point. It is shown that the stability properties of a Runge-Kutta method can be preserved only if is an eigenvalue of the coefficient matrixA. SupposeA has minimal polynomial (x – ) ^m p(x),p() 0. Then stability can be preserved only if the order of the method is at mostm + 2 (at mostm + 1 except for one case).This work was partially supported by a grant from the Science and Engineering Research Council.     

Asymptotic Stability of Runge-Kutta Methods for the Pantograph Equations

This paper considers the asymptotic stability analysis of both exact and numerical solutions of the following neutral delay differential equation with pantograph delay.where $B,C,D\in C^{d\times d},q\in (0,1)$,and $B$ is regular. After transforming the above equation to non-automatic neutral equation with constant delay, we determine sufficient conditions for the asymptotic stability of the zero solution. Furthermore, we focus on the asymptotic stability behavior of Runge-Kutta method with variable stepsize. It is proved that a L-stable Runge-Kutta method can preserve the above-mentioned stability properties.     

Delay-dependent stability of Runge-Kutta methods for neutral systems with distributed delays

The aim of this paper is to analyze the asymptotic stability of Runge-Kutta (RK) methods for neutral systems with distributed delays. With an adaptation of the argument principle, some sufficient criteria for weak delay-dependent stability of numerical solutions are proposed. Several numerical examples are performed to confirm the effectiveness of our theoretical results.     

Stability analysis of block boundary value methods for neutral pantograph equation

This paper deals with the convergence and stability properties of block boundary value methods (BBVMs) for the neutral pantograph equation. Due to its unbounded time lags and limited computer memory, a change in the independent variable is used to transform a pantograph equation into a non-autonomous differential equation with a constant delay but variable coefficients. It is shown under the classical Lipschitz condition that a BBVM is convergent of order p if the underlying boundary value method is consistent with order p. Furthermore, it is proved under a certain condition that BBVMs can preserve the asymptotic stability of exact solutions for the neutral pantograph equation. Meanwhile, some numerical experiments are given to confirm the main conclusions.     

The th moment stability for the stochastic pantograph equation

In this paper, we investigate the αth moment asymptotical stability of the analytic solution and the numerical methods for the stochastic pantograph equation by using the Razumikhin technique. Especially the linear stochastic pantograph equations and the semi-implicit Euler method applying them are considered. The convergence result of the semi-implicit Euler method is obtained. The stability conditions of the analytic solution of those equations and the numerical method are given. Finally, some experiments are given.     

Stability of Runge-Kutta methods for the generalized pantograph equation

Summary. This paper deals with stability properties of Runge-Kutta (RK) methods applied to a non-autonomous delay differential equation (DDE) with a constant delay which is obtained from the so-called generalized pantograph equation, an autonomous DDE with a variable delay by a change of the independent variable. It is shown that in the case where the RK matrix is regular stability properties of the RK method for the DDE are derived from those for a difference equation, which are examined by similar techniques to those in the case of autonomous DDEs with a constant delay. As a result, it is shown that some RK methods based on classical quadrature have a superior stability property with respect to the generalized pantograph equation. Stability of algebraically stable natural RK methods is also considered. Received May 5, 1998 / Revised version received November 17, 1998 / Published online September 24, 1999     

Parallel linear system solvers for Runge-Kutta methods

If the nonlinear systems arising in implicit Runge-Kutta methods like the Radau IIA methods are iterated by (modified) Newton, then we have to solve linear systems whose matrix of coefficients is of the form I-A hJ with A the Runge-Kutta matrix and J an approximation to the Jacobian of the righthand side function of the system of differential equations. For larger systems of differential equations, the solution of these linear systems by a direct linear solver is very costly, mainly because of the LU-decompositions. We try to reduce these costs by solving the linear systems by a second (inner) iteration process. This inner iteration process is such that each inner iteration again requires the solution of a linear system. However, the matrix of coefficients in these new linear systems is of the form I - B hJ where B is similar to a diagonal matrix with positive diagonal entries. Hence, after performing a similarity transformation, the linear systems are decoupled into s subsystems, so that the costs of the LU-decomposition are reduced to the costs of s LU-decompositions of dimension d. Since these LU-decompositions can be computed in parallel, the effective LU-costs on a parallel computer system are reduced by a factor s ³ . It will be shown that matrices B can be constructed such that the inner iterations converge whenever A and J have their eigenvalues in the positive and nonpositive halfplane, respectively. The theoretical results will be illustrated by a few numerical examples. A parallel implementation on the four-processor Cray-C98/4256 shows a speed-up ranging from at least 2.4 until at least 3.1 with respect to RADAU5 applied in one-processor mode.     

Stability analysis of Runge-Kutta methods for nonlinear neutral delay integro-differential equations

The sufficient conditions for the stability and asymptotic stability of Runge-Kutta methods for nonlinear neutral delay integro-differential equations are derived. A numerical test that confirms the theoretical results is given in the end.     

On fifth order Runge-Kutta methods

A 6 stage Runge-Kutta method is derived with the property that its order is 5 when used to solve a scalar differential equation but only 4 when used to solve a general system of differential equations. The existence of such a method underlines the necessity of carrying out theoretical analyses in a vector valued setting rather than in a one-dimensional setting as in the work of Kutta and some more recent authors.This research was supported by the New Zealand Foundation for Research, Science and Technology     

Numerical stability of higher-order derivative methods for the pantograph equation

Dealing with numerical stability of higher-order derivative methods with variable stepsize is the purpose of this paper for pantograph equations. A new way to compute this kind of equation is provided, and a sufficient condition for the numerical stability of high order derivative forms is given. Some numerical examples are presented to confirm our theoretical analysis.     

A stability property of A-stable collocation-based Runge-Kutta methods for neutral delay differential equations

We consider a linear homogeneous system of neutral delay differential equations with a constant delay whose zero solution is asymptotically stable independent of the value of the delay, and discuss the stability of collocation-based Runge-Kutta methods for the system. We show that anA-stable method preserves the asymptotic stability of the analytical solutions of the system whenever a constant step-size of a special form is used.     

The multi-symplecticity of partitioned Runge-Kutta methods for Hamiltonian PDEs

In this article we consider partitioned Runge-Kutta (PRK) methods for Hamiltonian partial differential equations (PDEs) and present some sufficient conditions for multi-symplecticity of PRK methods of Hamiltonian PDEs.

     

非线性中立型泛函微分方程Runge-Kutta 法的稳定性和收敛性

本文涉及Runge-Kutta 法变步长求解非线性中立型泛函微分方程(NFDEs) 的稳定性和收敛性.为此, 基于Volterra 泛函微分方程Runge-Kutta 方法的B- 理论, 引入了中立型泛函微分方程Runge-Kutta 方法的EB (expanded B-theory)-稳定性和EB-收敛性概念. 之后获得了Runge-Kutta 方法变步长求解此类方程的EB - 稳定性和EB- 收敛性. 这些结果对中立型延迟微分方程和中立型延迟积分微分方程也是新的.     

Order conditions for partitioned Runge-Kutta methods

We illustrate the use of the recent approach by P. Albrecht to the derivation of order conditions for partitioned Runge-Kutta methods for ordinary differential equations.     

A modified Runge-Kutta method for the numerical solution of ODE''s with oscillation solutions

A modified Runge-Kutta method with minimal phase-lag is developed for the numerical solution of Ordinary Differential Equations with oscillating solutions. The method is based on the accurate Runge-Kutta method of Sharp and Smart RK4SS(5) (see [1]) of order five. Numerical and theoretical results show that this new approach is more efficient, compared with the fifth order Runge-Kutta Sharp and Smart method.     

B-Theory of Runge-Kutta methods for stiff Volterra functional differential equations

B-stability and B-convergence theories of Runge-Kutta methods for nonlinear stiff Volterra func-tional differential equations(VFDEs)are established which provide unified theoretical foundation for the studyof Runge-Kutta methods when applied to nonlinear stiff initial value problems(IVPs)in ordinary differentialequations(ODEs),delay differential equations(DDEs),integro-differential equatioons(IDEs)and VFDEs of     

Construction of two-step Runge-Kutta methods of high order for ordinary differential equations

The construction of two-step Runge-Kutta methods of order p and stage order q=p with stability polynomial given in advance is described. This polynomial is chosen to have a large interval of absolute stability for explicit methods and to be A-stable and L-stable for implicit methods. After satisfying the order and stage order conditions the remaining free parameters are computed by minimizing the sum of squares of the difference between the stability function of the method and a given polynomial at a sufficiently large number of points in the complex plane. This revised version was published online in August 2006 with corrections to the Cover Date.     

Runge-Kutta Methods for the Numerical Solution of Stiff Semilinear Systems

This paper studies the stability and convergence properties of general Runge-Kutta methods when they are applied to stiff semilinear systems y(t) = J(t)y(t) + g(t, y(t)) with the stiffness contained in the variable coefficient linear part.We consider two assumptions on the relative variation of the matrix J(t) and show that for each of them there is a family of implicit Runge-Kutta methods that is suitable for the numerical integration of the corresponding stiff semilinear systems, i.e. the methods of the family are stable, convergent and the stage equations possess a unique solution. The conditions on the coefficients of a method to belong to these families turn out to be essentially weaker than the usual algebraic stability condition which appears in connection with the B-stability and convergence for stiff nonlinear systems. Thus there are important RK methods which are not algebraically stable but, according to our theory, they are suitable for the numerical integration of semilinear problems.This paper also extends previous results of Burrage, Hundsdorfer and Verwer on the optimal convergence of implicit Runge-Kutta methods for stiff semilinear systems with a constant coefficients linear part.