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.     

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．     

QUADRATIC INVARIANTS AND SYMPLECTIC STRUCTURE OF GENERAL LINEAR METHODS

1. IntroductionInvestigating whether a numerical method inherits some dynamical properties possessed bythe differential equation systems being integrated is an important field of numerical analysisand has received much attention in recent years See the review articlesof Sanz-Serna[9] and Section 11.16 of Hairer et. al.[2] for more detail concerning the symplectic methods. Most of the work on canonical iotegrators has dealt with one-step formulaesuch as Runge-Kutta methods and Runge'methods …     

Geometric Properties of Runge-Kutta Discretizations for Nonautonomous Index 2 Differential Algebraic Systems

We analyze Runge-Kutta discretizations applied to nonautonomous index 2 differential algebraic equations (DAEs) in semi-explicit form. It is shown that for half-explicit and projected Runge-Kutta methods there is an attractive invariant manifold for the discrete system which is close to the invariant manifold of the DAE. The proof combines reduction techniques to autonomou index 2 differential algebraic equations with some invariant manifold results of Schropp [9]. The results support the favourable behavior of these Runge-Kutta methods applied to index 2 DAEs for t = 0.     

Runge-Kutta schemes that maintain invariant solutions for differential algebraic systems

Implicit Runge-Kutta (IRK) methods and projected IRK methods for the solution of semiexplicit index-2 systems of differential algebraic systems (DAEs) have been proposed by several authors. In this paper we prove that if a method satisfiesBA+A ^t B–bb ^t =0, it conserves quadratic invariants of DAEs.     

Efficient Runge-Kutta integrators for index-2 differential algebraic equations

In seeking suitable Runge-Kutta methods for differential algebraic equations, we consider singly-implicit methods to which are appended diagonally-implicit stages. Methods of this type are either similar to those of Butcher and Cash or else allow for the importation of a final derivative from a previous step. For these two classes, with up to three additional diagonally-implicit stages, we derive methods that satisfy appropriate order conditions for index-2 DAEs.

     

Implicit‐explicit multistep finite element methods for nonlinear convection‐diffusion problems

Implicit‐explicit multistep finite element methods for nonlinear convection‐diffusion equations are presented and analyzed. In space we discretize by finite element methods. The discretization in time is based on linear multistep schemes. The linear part of the equation is discretized implicitly and the nonlinear part of the equation explicitly. The schemes are stable and very efficient. We derive optimal order error estimates. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:93–104, 2001     

Non-linear stability of a general class of differential equation methods

For a class of methods sufficiently general as to include linear multistep and Runge-Kutta methods as special cases, a concept known as algebraic stability is defined. This property is based on a non-linear test problem and extends existing results on Runge-Kutta methods and on linear multistep and one-leg methods. The algebraic stability properties of a number of particular methods in these families are studied and a generalization is made which enables estimates of error growth to be provided for certain classes of methods.     

Numerical integration of positive linear differential-algebraic systems

In the simulation of dynamical processes in economy, social sciences, biology or chemistry, the analyzed values often represent non-negative quantities like the amount of goods or individuals or the density of a chemical or biological species. Such systems are typically described by positive ordinary differential equations (ODEs) that have a non-negative solution for every non-negative initial value. Besides positivity, these processes often are subject to algebraic constraints that result from conservation laws, limitation of resources, or balance conditions and thus the models are differential-algebraic equations (DAEs). In this work, we present conditions under which both these properties, the positivity as well as the algebraic constraints, are preserved in the numerical simulation by Runge–Kutta or multistep discretization methods. Using a decomposition approach, we separate the dynamic and the algebraic equations of a given linear, positive DAE to give positivity preserving conditions for each part separately. For the dynamic part, we generalize the results for positive ODEs to DAEs using the solution representation via Drazin inverses. For the algebraic part, we use the consistency conditions of the discretization method to derive conditions under which this part of the approximation overestimates the exact solution and thus is non-negative. We analyze these conditions for some common Runge–Kutta and multistep methods and observe that for index-1 systems and stiffly accurate Runge–Kutta methods, positivity is conditionally preserved under similar conditions as for ODEs. For higher index problems, however, none of the common methods is suitable.     

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.     

一般线性方法的正则性

１．引言数值方法的动力特征近年引起了人们的广泛关注。其中之一就是系统的平衡态和数值方法的平衡态相一致的问题,即用一个数值方法沿定步长求解系统时,是否会出现伪平衡。不可能出现伪平衡的方法称为是正则的。ＲＫ方法和线性多步法的正则性已被众多的文献研究［２,３,４］,其它方法的正则性显然是一亟待研究的问题。本文讨论较ＲＫ方法和线性多步法远为广泛的一般线性方法的正则性。设ｆ：Ｒ~（Ｎ）→Ｒ~（Ｎ）是一充分光滑的映射,考虑求解初值问题：的一般线性方法［１］：其中步长逼近于逼近于关于微分方程真解ｙ（ｔ）在第ｎ层…     

Reversible methods of Runge-Kutta type for Index-2 DAEs

Summary. A new interpretation of Runge-Kutta methods for differential algebraic equations (DAEs) of index 2 is presented, where a step of the method is described in terms of a smooth map (smooth also with respect to the stepsize). This leads to a better understanding of the convergence behavior of Runge-Kutta methods that are not stiffly accurate. In particular, our new framework allows for the unified study of two order-improving techniques for symmetric Runge-Kutta methods (namely post-projection and symmetric projection) specially suited for solving reversible index-2 DAEs.Mathematics Subject Classification (1991): 65L05, 65L06     

Runge-Kutta stability on a Floquet problem

This work examines the stability of explicit Runge-Kutta methods applied to a certain linear ordinary differential equation with periodic coefficients. On this problem naïve use of the eigenvalues of the Jacobian results in misleading conclusions about stable behaviour. It is shown, however, that a valid analogue of the classical absolute stability theory can be developed. Further, using a suitable generalisation of the equilibrium theory of Hall [ACM Trans. on Math. Soft. 11 (1985), pp. 289–301], accurate predictions are made about the performance of modern, adaptive algorithms.Supported by the University of Dundee Research Initiatives Fund.     

Stability and Dynamics of Numerical Methods for Nonlinear Ordinary Differential Equations

Stability of numerical methods for nonlinear autonomous ordinarydifferential equations is approached from the point of viewof dynamical systems. It is proved that multistep methods (withnonlinear algebraic equations exactly solved) with bounded trajectoriesalways produce correct asymptotic behaviour, but this is notthe case with Runge-Kutta. Examples are given of Runge-Kuttaschemes converging to wrong solutions in a deceptively ‘smooth’manner and a characterization of such two-stage methods is presented.PE(CE)^m schemes are examined as well, and it is demonstratedthat they, like Runge-Kutta, may lead to false asymptotics.     

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…     

One- and multistep discretizations of index 2 differential algebraic systems and their use in optimization

An approach to solve constrained minimization problems is to integrate a corresponding index 2 differential algebraic equation (DAE). Here, corresponding means that the ω-limit sets of the DAE dynamics are local solutions of the minimization problem. In order to obtain an efficient optimization code, we analyze the behavior of certain Runge–Kutta and linear multistep discretizations applied to these DAEs. It is shown that the discrete dynamics reproduces the geometric properties and the long-time behavior of the continuous system correctly. Finally, we compare the DAE approach with a classical SQP-method.     

Analytical and numerical stability of partial differential equations with piecewise constant arguments

This article is concerned with the stability analysis of the analytic and numerical solutions of a partial differential equation with piecewise constant arguments of mixed type. First, by means of the similar technique in Wiener and Debnath [Int J Math Math Sci 15 (1992), 781–788], the sufficient conditions under which the analytic solutions asymptotically stable are obtained. Then, the θ‐methods are used to solve the above‐mentioned equation, the sufficient conditions for the asymptotic stability of numerical methods are derived. Finally, some numerical experiments are given to demonstrate the conclusions.Copyright © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1‐16, 2014     

Stochastic Runge-Kutta methods with deterministic high order for ordinary differential equations

We consider embedding deterministic Runge-Kutta methods with high order into weak order stochastic Runge-Kutta (SRK) methods for non-commutative stochastic differential equations (SDEs). As a result, we have obtained weak second order SRK methods which have good properties with respect to not only practical errors but also mean square stability. In our stability analysis, as well as a scalar test equation with complex-valued parameters, we have used a multi-dimensional non-commutative test SDE. The performance of our new schemes will be shown through comparisons with an efficient and optimal weak second order scheme proposed by Debrabant and Rößler (Appl. Numer. Math. 59:582–594, 2009).     

A parametric transformation at numerical integration of ODE

The initial value problem for ordinary differential equation (ODE) is investigated when the linear parametric transformation (rotation of coordinate axes) is applied. It is shown that for transformed equation the principal term of asymptotic error expansion of numerical method can be minimized by an angle of rotation. The dependence of the optimal angle φ_opt(λ)on λis plotted for the model equation solved by linear multistep methods and Runge-Kutta methods.     

Four-stage symplectic and P-stable SDIRKN methods with dispersion of high order

New SDIRKN methods specially adapted to the numerical integration of second-order stiff ODE systems with periodic solutions are obtained. Our interest is focused on the dispersion (phase errors) of the dominant components in the numerical oscillations when these methods are applied to the homogeneous linear test model. Based on this homogeneous test model we derive the dispersion and P-stability conditions for SDIRKN methods which are assumed to be zero dissipative. Two four-stage symplectic and P-stable methods with algebraic order 4 and high order of dispersion are obtained. One of the methods is symmetric and sixth-order dispersive whereas the other method is nonsymmetric and eighth-order dispersive. These methods have been applied to a number of test problems (linear as well as nonlinear) and some numerical results are presented to show their efficiency when they are compared with other methods derived by Sharp et al. [IMA J. Numer. Anal. 10 (1990) 489–504].