Approximate preservation of quadratic first integrals by explicit Runge–Kutta methods

The approximate preservation of quadratic first integrals (QFIs) of differential systems in the numerical integration with Runge–Kutta (RK) methods is studied. Conditions on the coefficients of the RK method to preserve all QFIs up to a given order are obtained, showing that the pseudo-symplectic methods studied by Aubry and Chartier (BIT 98(3):439–461, 1998) of algebraic order p preserve QFIs with order q = 2p. An expression of the error of conservation of QFIs by a RK method is given, and a new explicit six-stage formula with classical order four and seventh order of QFI-conservation is obtained by choosing their coefficients so that they minimize both local truncation and conservation errors. Several formulas with algebraic orders 3 and 4 and different orders of conservation have been tested with some problems with quadratic and general first integrals. It is shown that the new fourth-order explicit method preserves much better the qualitative properties of the flow than the standard fourth-order RK method at the price of two extra function evaluations per step and it is a practical and efficient alternative to the fully implicit methods required for a complete preservation of QFIs.     

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     

CHARACTERIZATIONS OF SYMMETRIC MULTISTEP RUNGE-KUTTA METHODS

Some characterizations for symmetric multistep Runge-Kutta(RK) methods are obtained. Symmetric two-step RK methods with one and two-stages are presented. Numerical examples show that symmetry of multistep RK methods alone is not sufficient for long time integration for reversible Hamiltonian systems. This is an important difference between one-step and multistep symmetric RK methods.     

Families of efficient second order Runge–Kutta methods for the weak approximation of Itô stochastic differential equations

Recently, a new class of second order Runge–Kutta methods for Itô stochastic differential equations with a multidimensional Wiener process was introduced by Rößler [A. Rößler, Second order Runge–Kutta methods for Itô stochastic differential equations, Preprint No. 2479, TU Darmstadt, 2006]. In contrast to second order methods earlier proposed by other authors, this class has the advantage that the number of function evaluations depends only linearly on the number of Wiener processes and not quadratically. In this paper, we give a full classification of the coefficients of all explicit methods with minimal stage number. Based on this classification, we calculate the coefficients of an extension with minimized error constant of the well-known RK32 method [J.C. Butcher, Numerical Methods for Ordinary Differential Equations, John Wiley & Sons, West Sussex, 2003] to the stochastic case. For three examples, this method is compared numerically with known order two methods and yields very promising results.     

Parallel iterated methods based on multistep Runge-Kutta methods of Radau type

This paper investigates iterated Multistep Runge-Kutta methods of Radau type as a class of explicit methods suitable for parallel implementation. Using the idea of van der Houwen and Sommeijer [18], the method is designed in such a way that the right-hand side evaluations can be computed in parallel. We use stepsize control and variable order based on iterated approximation of the solution. A code is developed and its performance is compared with codes based on iterated Runge-Kutta methods of Gauss type and various Dormand and Prince pairs [15]. The accuracy of some of our methods are comparable with the PIRK10 methods of van der Houwen and Sommeijer [18], but require fewer processors. In addition at very stringent tolerances these new methods are competitive with RK78 pairs in a sequential implementation.     

具有Gilbert项的Landau-Lifshitz方程的显式平方守恒格式

构造了一种解具有Gilbert项的Landau-Lifshitz方程的显式平方守恒格式．基本思想是离散Landau-Lifshitz方程成常微分方程组,应用李群方法和显式Runge-Kutta方法解常微分方程组．数值试验比较了两方法的保平方守恒特性和精度,得出李群方法(RK-Cayley方法)比相应的Runge-Kutta(RK)方法有更好的精度和保平方守恒特性．     

一种基于直觉模糊集的群决策方法

利用直觉模糊集理论研究专家意见为{满意,不满意,弃权}三种形式的群决策问题.在构建群决策模型的过程中,分如下两种情形分别将专家意见综合为直觉模糊集:一是首先综合单个专家根据各准则给出的方案评估值;二是首先对单个准则综合所有专家给出的方案评估值.由此建立两个群决策模型.对由直觉模糊集表示的群决策结果,利用推广的TOPSIS方法对它们进行排序选优.最后给出一个应用实例.     

Arbitrarily high-order accurate and energy-stable schemes for solving the conservative Allen–Cahn equation

In this paper, three high-order accurate and unconditionally energy-stable methods are proposed for solving the conservative Allen–Cahn equation with a space–time dependent Lagrange multiplier. One is developed based on an energy linearization Runge–Kutta (EL–RK) method which combines an energy linearization technique with a specific class of RK schemes, the other two are based on the Hamiltonian boundary value method (HBVM) including a Gauss collocation method, which is the particular instance of HBVM, and a general class of cases. The system is first discretized in time by these methods in which the property of unconditional energy stability is proved. Then the Fourier pseudo-spectral method is employed in space along with the proofs of mass conservation. To show the stability and validity of the obtained schemes, a number of 2D and 3D numerical simulations are presented for accurately calculating geometric features of the system. In addition, our numerical results are compared with other known structure-preserving methods in terms of numerical accuracy and conservation properties.     

A Steffensen-like method and its higher-order variants

For solving nonlinear equations, we suggest a second-order parametric Steffensen-like method, which is derivative free and only uses two evaluations of the function in one step. We also suggest a variant of the Steffensen-like method which is still derivative free and uses four evaluations of the function to achieve cubic convergence. Moreover, a fast Steffensen-like method with super quadratic convergence and a fast variant of the Steffensen-like method with super cubic convergence are proposed by using a parameter estimation. The error equations and asymptotic convergence constants are obtained for the discussed methods. The numerical results and the basins of attraction support the proposed methods.     

A criterion forP-stability properties of Runge-Kutta methods

P-stability is an analogous stability property toA-stability with respect to delay differential equations. It is defined by using a scalar test equation similar to the usual test equation ofA-stability. EveryP-stable method isA-stable, but anA-stable method is not necessarilyP-stable. We considerP-stability of Runge-Kutta (RK) methods and its variation which was originally introduced for multistep methods by Bickart, and derive a sufficient condition for an RK method to have the stability properties on the basis of an algebraic characterization ofA-stable RK methods recently obtained by Schere and Müller. By making use of the condition we clarify stability properties of some SIRK and SDIRK methods, which are easier to implement than fully implicit methods, applied to delay differential equations.     

Explicit Runge–Kutta pairs with lower stage-order

Explicit Runge–Kutta pairs of methods of successive orders of accuracy provide effective algorithms for approximating solutions to nonstiff initial value problems. For each explicit RK method of order of accuracy p, there is a minimum number s _p of derivative evaluations required for each step propagating the numerical solution. For p ≤ 8, Butcher has established exact values of s _p , and for p > 8, his work establishes lower bounds; otherwise, upper bounds are established by various published methods. Recently, Khashin has derived some new methods numerically, and shown that the known upper bound on s ₉ for methods of order p = 9 can be reduced from 15 to 13. His results motivate this attempt to identify parametrically exact representations for coefficients of such methods. New pairs of methods of orders 5,6 and 6,7 are characterized in terms of several arbitrary parameters. This approach, modified from an earlier one, increases the known spectrum of types of RK pairs and their derivations, may lead to the derivation of new RK pairs of higher-order, and possibly to other types of explicit algorithms within the class of general linear methods.     

An energy-preserving exponentially-fitted continuous stage Runge–Kutta method for Hamiltonian systems

Recently, the symplectic exponentially-fitted methods for Hamiltonian systems with periodic or oscillatory solutions have been attracting a lot of interest. As an alternative to them, in this paper, we propose a class of energy-preserving exponentially-fitted methods. For this aim, we show sufficient conditions for energy-preservation in terms of the coefficients of continuous stage Runge–Kutta (RK) methods, and extend the theory of exponentially-fitted RK methods in the context of continuous stage RK methods. Then by combining these two theories, we derive second and fourth order energy-preserving exponentially-fitted schemes.     

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.}     

Characterization and Construction of Linear Symplectic RK-Methods

A characterization of linear symplectic Runge-Kutta methods, which is based on the W-transformation of Hairer and Wanner, is presented. Using this charac-terization three classes of high order linear symplectic Runge-Kutta methods are constructed. They include and extend known classes of high order linear symplectic Runge-Kutta methods.     

一类满足FSAL技术的RKNd方法

陈丙振和游雄给出了内级阶比传统 RK 方法高一阶的 RKNd 方法.FSAL技术是一种常用的节省函数计算量的手段. 其主要思想是,方法的更新与内级的最后一步相同. 本文正是给出满足FSAL技术的RKNd 方法. 数值试验表明, 本文的 RKNdF 方法比RKNd 方法在计算效率上具有一定的优越性.     

Convergence of a class of runge-kutta methods for differential-algebraic systems of index 2

This paper deals with convergence results for a special class of Runge-Kutta (RK) methods as applied to differential-algebraic equations (DAE's) of index 2 in Hessenberg form. The considered methods are stiffly accurate, with a singular RK matrix whose first row vanishes, but which possesses a nonsingular submatrix. Under certain hypotheses, global superconvergence for the differential components is shown, so that a conjecture related to the Lobatto IIIA schemes is proved. Extensions of the presented results to projected RK methods are discussed. Some numerical examples in line with the theoretical results are included.     

Efficient fourth orderP-stable formulae

In this paper, a family of fourth orderP-stable methods for solving second order initial value problems is considered. When applied to a nonlinear differential system, all the methods in the family give rise to a nonlinear system which may be solved using a modified Newton method. The classical methods of this type involve at least three (new) function evaluations per iteration (that is, they are 3-stage methods) and most involve using complex arithmetic in factorising their iteration matrix. We derive methods which require only two (new) function evaluations per iteration and for which the iteration matrix is a true real perfect square. This implies that real arithmetic will be used and that at most one real matrix must be factorised at each step. Also we consider various computational aspects such as local error estimation and a strategy for changing the step size.     

Algorithm for forming derivative-free optimal methods

We develop a simple yet effective and applicable scheme for constructing derivative free optimal iterative methods, consisting of one parameter, for solving nonlinear equations. According to the, still unproved, Kung-Traub conjecture an optimal iterative method based on k+1 evaluations could achieve a maximum convergence order of $2^{k}$ . Through the scheme, we construct derivative free optimal iterative methods of orders two, four and eight which request evaluations of two, three and four functions, respectively. The scheme can be further applied to develop iterative methods of even higher orders. An optimal value of the free-parameter is obtained through optimization and this optimal value is applied adaptively to enhance the convergence order without increasing the functional evaluations. Computational results demonstrate that the developed methods are efficient and robust as compared with many well known methods.     

A class of symplectic partitioned Runge–Kutta methods

This paper deals with some relevant properties of Runge–Kutta (RK) methods and symplectic partitioned Runge–Kutta (PRK) methods. First, it is shown that the arithmetic mean of a RK method and its adjoint counterpart is symmetric. Second, the symplectic adjoint method is introduced and a simple way to construct symplectic PRK methods via the symplectic adjoint method is provided. Some relevant properties of the adjoint method and the symplectic adjoint method are discussed. Third, a class of symplectic PRK methods are proposed based on Radau IA, Radau IIA and their adjoint methods. The structure of the PRK methods is similar to that of Lobatto IIIA–IIIB pairs and is of block forms. Finally, some examples of symplectic partitioned Runge–Kutta methods are presented.     

An embedded pair of exponentially fitted explicit Runge–Kutta methods

An embedded pair of exponentially fitted explicit Runge–Kutta (RK) methods for the numerical integration of IVPs with oscillatory solutions is derived. This pair is based on the exponentially fitted explicit RK method constructed in Vanden Berghe et al., and we confirm that the methods which constitute the pair have algebraic order 4 and 3. Some numerical experiments show the efficiency of our pair when it is compared with the variable step code proposed by Vanden Berghe et al. (J. Comput. Appl. Math. 125 (2000) 107).