Perturbed collocation and symplectic RKN methods

It is known that many Runge-Kutta-Nystr?m methods can be derived by collocation. In this paper we prove that the onlys-stage symplectic Runge-Kutta-Nystr?m method that can be obtained by ordinary collocation is of order 2s and implicit. We also extend the idea of perturbed collocation to Runge-Kutta-Nystr?m methods and derive symplectic Runge-Kutta-Nystr?m methods using this technique. We have obtained symplectic implicits-stage Runge-Kutta-Nystr?m methods of order 2s−1 and 2s−2.     

Perturbed collocation and symplectic RKN methods

It is known that many Runge-Kutta-Nyström methods can be derived by collocation. In this paper we prove that the onlys-stage symplectic Runge-Kutta-Nyström method that can be obtained by ordinary collocation is of order 2s and implicit. We also extend the idea of perturbed collocation to Runge-Kutta-Nyström methods and derive symplectic Runge-Kutta-Nyström methods using this technique. We have obtained symplectic implicits-stage Runge-Kutta-Nyström methods of order 2s?1 and 2s?2.     

On the relations between B2V Ms and Runge-Kutta collocation methods

The principal aim of this paper is a rigorous analysis of the relations between Block Boundary Value Methods (B₂V Ms) with minimal blocksize defined over a suitable nonuniform finer mesh and well-known Runge-Kutta collocation methods. Moreover, a further aspect that will be briefly investigated is the construction of an extended finer mesh for building B₂V Ms with nonminimal blocksize. Some advantages that may arise from the use of the so-obtained methods will be also discussed.     

Higher order methods for a singularly perturbed problem

Summary A finite element method using piecewise polynomials of degree k is used to approximate the problem u+u=f, >0 a small parameter. A very irregular mesh is used. On this mesh error estimates of order0(h ^k+1) are obtained uniformly in ,h the maximum stepsize, fork2. The condition number of the system of linear equations one has to solve in order to get the approximation is estimated. Extension of the results to more complicated problems is briefly indicated. Finally, a numerical example is given.Work performed while visiting the IBM Thomas J. Watson Research Center, Yorktown Heights, N.Y.     

The non-existence of symplectic multi-derivative Runge-Kutta methods

A sufficient condition for the symplecticness ofq-derivative Runge-Kutta methods has been derived by F. M. Lasagni. In the present note we prove that this condition can only be satisfied for methods withq1, i.e., for standard Runge-Kutta methods. We further show that the conditions of Lasagni are also necessary for symplecticness so that no symplectic multi-derivative Runge-Kutta method can exist.This research has been supported by project PB89-0351 (Dirección General de Investigación Científica y Técnica) and by project No. 20-32354.91 of Swiss National Science Foundation.     

On implicit Runge-Kutta methods with high stage order

It is well known that high stage order is a desirable property for implicit Runge-Kutta methods. In this paper it is shown that it is always possible to construct ans-stage IRK method with a given stability function and stage orders−1 if the stability function is an approximation to the exponential function of at least orders. It is further indicated how to construct such methods as well as in which cases the constructed methods will be stiffly accurate.     

A fourth/second order accurate collocation method for singularly perturbed two-point boundary value problems using tension splines

Summary.   The collocation tension spline is considered as a numerical solution of a singularly perturbed two-point boundary value problem: . The collocation points are chosen as a generalization of the classical Gaussian points. Unlike the traditional approach, we employ the B-spline representation in the analysis. This leads to global quadratic convergence of the method for small perturbation parameters, and, for large values, the order of convergence is four. Received October 4, 1996 / Revised version received September 23, 1999 / Published online October 16, 2000     

Family of symplectic implicit Runge-Kutta formulae

A family of formulae for the sympletic IRK method is investigated. Specifically, focus is given to general solutions for formula parameters of IRK under the symplectic and the order conditions. Examples of such formulae are constructed for up to three stages.     

Derivation of continuous explicit two-step Runge–Kutta methods of order three

We describe a construction of continuous extensions to a new representation of two-step Runge–Kutta methods for ordinary differential equations. This representation makes possible the accurate and reliable estimation of local discretization error, facilitates the efficient implementation of these methods in variable stepsize environment, and adapts readily to the numerical solution of a class of delay differential equations. A number of numerical tests carried out on the obtained methods of order 3 with quadratic interpolants show their efficiency and robust performance which allow them to compete with the state-of-the-art dde23 code from Matlab.     

A bound on the maximum strong order of stochastic Runge-Kutta methods for stochastic ordinary differential equations

In Burrage and Burrage [1] it was shown that by introducing a very general formulation for stochastic Runge-Kutta methods, the previous strong order barrier of order one could be broken without having to use higher derivative terms. In particular, methods of strong order 1.5 were developed in which a Stratonovich integral of order one and one of order two were present in the formulation. In this present paper, general order results are proven about the maximum attainable strong order of these stochastic Runge-Kutta methods (SRKs) in terms of the order of the Stratonovich integrals appearing in the Runge-Kutta formulation. In particular, it will be shown that if ans-stage SRK contains Stratonovich integrals up to orderp then the strong order of the SRK cannot exceed min{(p+1)/2, (s−1)/2},p≥2,s≥3 or 1 ifp=1.     

Stability properties of collocation methods

Lower bounds for are given for which equidistant s-point collocation methods areA()-stable for arbitrarys.     

Weak second order S-ROCK methods for Stratonovich stochastic differential equations

It is well known that the numerical solution of stiff stochastic ordinary differential equations leads to a step size reduction when explicit methods are used. This has led to a plethora of implicit or semi-implicit methods with a wide variety of stability properties. However, for stiff stochastic problems in which the eigenvalues of a drift term lie near the negative real axis, such as those arising from stochastic partial differential equations, explicit methods with extended stability regions can be very effective. In the present paper our aim is to derive explicit Runge–Kutta schemes for non-commutative Stratonovich stochastic differential equations, which are of weak order two and which have large stability regions. This will be achieved by the use of a technique in Chebyshev methods for ordinary differential equations.     

A robust trigonometrically fitted embedded pair for perturbed oscillators

A new kind of trigonometrically fitted embedded pair of explicit ARKN methods for the numerical integration of perturbed oscillators is presented in this paper. This new pair is based on the trigonometrically fitted ARKN method of order five derived by Yang and Wu in [H.L. Yang, X.Y. Wu, Trigonometrically-fitted ARKN methods for perturbed oscillators, Appl. Numer. Math. 9 (2008) 1375–1395]. We analyze the stability properties, phase-lag (dispersion) and dissipation of the higher-order method of the new pair. Numerical experiments carried out show that our new embedded pair is very competitive in comparison with the embedded pairs proposed in the scientific literature.     

General linear methods: connection to one step methods and invariant curves

Summary We generalize a result of Kirchgraber (1986) on multistep methods. We show that every strictly stable general linear method is essentially conjugate to a one step method of the same order. This result may be used to show that general properties of one step methods carry over to general linear methods. As examples we treat the existence of invariant curves and the construction of attracting sets.     

Computation of order conditions for symplectic partitioned Runge-Kutta schemes with application to optimal control

We derive order conditions for the discretization of (unconstrained) optimal control problems, when the scheme for the state equation is of Runge-Kutta type. This problem appears to be essentially the one of checking order conditions for symplectic partitioned Runge-Kutta schemes. We show that the computations using bi-coloured trees are naturally expressed in this case in terms of oriented free tree. This gives a way to compute them by an appropriate computer program. This study is supported by CNES and ONERA, in the framework of the CNES fellowship of the second author.     

Supplement: Efficient weak second order stochastic Runge-Kutta methods for non-commutative Stratonovich stochastic differential equations

This paper gives a modification of a class of stochastic Runge-Kutta methods proposed in a paper by Komori (2007). The slight modification can reduce the computational costs of the methods significantly.     

B-spline collocation for solution of two-point boundary value problems

A numerical method based on B-spline is developed to solve the general nonlinear two-point boundary value problems up to order 6. The standard formulation of sextic spline for the solution of boundary value problems leads to non-optimal approximations. In order to derive higher orders of accuracy, high order perturbations of the problem are generated and applied to construct the numerical algorithm. The error analysis and convergence properties of the method are studied via Green’s function approach. O(h⁶) global error estimates are obtained for numerical solution of these classes of problems. Numerical results are given to illustrate the efficiency of the proposed method. Results of numerical experiments verify the theoretical behavior of the orders of convergence.     

Efficient computation of characteristic roots of delay differential equations using LMS methods

We aim at the efficient computation of the rightmost, stability-determining characteristic roots of a system of delay differential equations. The approach we use is based on the discretization of the time integration operator by a linear multistep (LMS) method. The size of the resulting algebraic eigenvalue problem is inversely proportional to the steplength. We summarize theoretical results on the location and numerical preservation of roots. Furthermore, we select nonstandard LMS methods, which are better suited for our purpose. We present a new procedure that aims at computing efficiently and accurately all roots in any right half-plane. The performance of the new procedure is demonstrated for small- and large-scale systems of delay differential equations.     

Superconvergent explicit two-step peer methods

We consider explicit two-step peer methods for the solution of nonstiff differential systems. By an additional condition a subclass of optimally zero-stable methods is identified that is superconvergent of order p=s+1$p=s+1$, where s$s$ is the number of stages. The new condition allows us to reduce the number of coefficients in a numerical search for good methods. We present methods with 4–7 stages which are tested in FORTRAN90 and compared with DOPRI5 and DOP853. The results confirm the high potential of the new class of methods.