An improvement on the positivity results for 2-stage explicit Runge-Kutta methods

In this paper, we investigate the positivity property for a class of 2-stage explicit Runge-Kutta (RK2) methods of order two when applied to the numerical solution of special nonlinear initial value problems (IVPs) for ordinary differential equations (ODEs). We also pay particular attention to monotonicity property. We obtain new results for positivity which are important in practical applications. We provide some numerical examples to illustrate our results.     

Stein implicit Runge-Kutta methods with high stage order for large-scale ordinary differential equations

The Runge-Kutta method is one of the most popular implicit methods for the solution of stiff ordinary differential equations. For large problems, the main drawback of such methods is the cost required at each integration step for computing the solution of a nonlinear system of equations. In this paper, we propose to reduce the cost of the computation by transforming the linear systems arising in the application of Newton's method to Stein matrix equations. We propose an iterative projection method onto block Krylov subspaces for solving numerically such Stein matrix equations. Numerical examples are given to illustrate the performance of our proposed method.     

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.     

Dissipativity of Runge-Kutta methods for Volterra functional differential equations

This paper is concerned with the numerical dissipativity of nonlinear Volterra functional differential equations (VFDEs). We give some dissipativity results of Runge-Kutta methods when they are applied to VFDEs. These results provide unified theoretical foundation for the numerical dissipativity analysis of systems in ordinary differential equations (ODEs), delay differential equations (DDEs), integro-differential equations (IDEs), Volterra delay integro-differential equations (VDIDEs) and VFDEs of other type which appear in practice. Numerical examples are given to confirm our theoretical results.     

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     

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.     

求解多延迟微分方程的Runge-Kutta方法的收缩性

该文涉及多延迟微分方程MDDEs系统的理论解与数值解的收缩性．为此,一些新的稳定性概念诸如：BN_f^(m)-稳定性及GRN_m-稳定性稳定性被引入．该探讨得出：Ｒｕｎｇｅ Ｋｕｔｔａ（ＲＫ）方法及相应的连续插值的BN^(m)-稳定性导致求解MDDEs的方法的收缩性（GRN_m-稳定性）．     

Two-Stage Stochastic Runge-Kutta Methods for Stochastic Differential Equations

In this paper we discuss two-stage diagonally implicit stochastic Runge-Kutta methods with strong order 1.0 for strong solutions of Stratonovich stochastic differential equations. Five stochastic Runge-Kutta methods are presented in this paper. They are an explicit method with a large MS-stability region, a semi-implicit method with minimum principal error coefficients, a semi-implicit method with a large MS-stability region, an implicit method with minimum principal error coefficients and another implicit method. We also consider composite stochastic Runge-Kutta methods which are the combination of semi-implicit Runge-Kutta methods and implicit Runge-Kutta methods. Two composite methods are presented in this paper. Numerical results are reported to compare the convergence properties and stability properties of these stochastic Runge-Kutta methods.     

An efficient scheme for the implementation of implicit Runge-Kutta methods

A scheme is proposed for solving nonlinear algebraic equations arising in the implementation of the implicit Runge-Kutta methods. In contrast to the available schemes, not only the starting values of the variables but also those of the derivatives are predicted. This makes it possible to reduce the number of evaluations of the function (the right-hand side) at each implicit stage without significantly reducing the accuracy of integration.     

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.     

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.     

求解延迟微分代数方程的多步Runge-Kutta方法的渐近稳定性

延迟微分代数方程(DDAEs)广泛出现于科学与工程应用领域．本文将多步Runge-Kutta方法应用于求解线性常系数延迟微分代数方程，讨论了该方法的渐近稳定性．数值试验表明该方法对求解DDAEs是有效的．     

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.     

Runge-Kutta Discretizations of Singularly Perturbed Gradient Equations

We analyze Runge-Kutta discretizations applied to singularly perturbed gradient systems. It is shown in which sense the discrete dynamics preserve the geometric properties and the longtime behavior of the underlying ordinary differential equation. If the continuous system has an attractive invariant manifold then numerical trajectories started in some neighbourhood (the size of which is independent of the step-size and the stiffness parameter) approach an equilibrium in a nearby manifold. The proof combines invariant manifold techniques developed by Nipp and Stoffer for singularly perturbed systems with some recent results of the second author on the global behavior of discretized gradient systems. The results support the favorable behavior of ODE methods for stiff minimization problems.     

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.     

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.     

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.     

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.     

一类多步并行Runge—Kutta公式

本文构造了一类适合在多处理机系统上实现的并行Ｒｕｎｇｅ－Ｋｕｔｔａ公式，对于其中的具体公式证明了收敛性，给出它的稳定区域，数值例子表明，该公式可以有效地求解常微分方程初值问题。     

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.