Beatrice Paternoster

Runge-Kutta(-Nystro¨m) methods for ODEs with periodic solutions based on trigonometric polynomials

We consider the construction of Runge-Kutta(-Nystrom) methods for ordinary differential equations whose solutions are known to be periodic. We assume that the frequency ω can be estimated in advance. The resulting methods depend on the parameter v = ωh, where h is the stepsize. Using the linear stage representation of a Runge-Kutta method given in Albrechts approach, we derive Runge-Kutta and Runge-Kutta-Nystrom methods which integrate trigonometric polynomials exactly.

Two-step hybrid collocation methods for y″=f(x,y)

We consider a new class of two-step collocation methods for the numerical integration of second-order initial value problems having periodic or oscillatory solutions. We describe the constructive technique, discuss the order of the resulting methods and analyse their stability properties.

A Gauss quadrature rule for oscillatory integrands

Abstract We consider the Gauss formula for an integral, ∫ −1 1 y(x) d x≈∑ k=1 N w k y(x k ) , and introduce a procedure for calculating the weights w k and the abscissa points x k , k =1,2,…, N , such that the formula becomes best tuned to oscillatory functions of the form y ( x )= f 1 ( x )sin( ωx )+ f 2 ( x )cos( ωx ) where f 1 ( x ) and f 2 ( x ) are smooth. The weights and the abscissas of the new formula depend on ω and, by the very construction, the formula is exact for any ω provided f 1 ( x ) and f 2 ( x ) are polynomials of class P N−1 . Numerical illustrations are given for N between one and six.

Original article: Trigonometrically fitted two-step hybrid methods for special second order ordinary differential equations

Abstract: The purpose of this paper is to derive two-step hybrid methods for second order ordinary differential equations with oscillatory or periodic solutions. We show the constructive technique of methods based on trigonometric and mixed polynomial fitting and consider the linear stability analysis of such methods. We then carry out some numerical experiments underlining the properties of the derived classes of methods.

Two-step almost collocation methods for Volterra integral equations

In this paper we construct a new class of continuous methods for Volterra integral equations. These methods are obtained by using a collocation technique and by relaxing some of the collocation conditions in order to obtain good stability properties.

A note on the capacitance matrix algorithm, substructuring, and mixed or Neumann boundary conditions

Abstract We develop in this work variants of the capacitance matrix algorithm which can be used to solve discretizations of elliptic partial differential equations when either the original system of equations or one which arises from substructuring has a rank-deficient matrix.

Parameter estimation in exponentially fitted hybrid methods for second order differential problems

In this work we deal with exponentially fitted methods for the numerical solution of second order ordinary differential equations, whose solutions are known to show a prominent exponential behaviour depending on the value of an unknown parameter to be suitably determined. The knowledge of an estimation to the unknown parameter is needed in order to apply the numerical method, since its coefficients depend on the value of the parameter. We present a strategy for the practical estimation of the parameter, which is also tested on some selected problems.

Two-step almost collocation methods for ordinary differential equations

A new class of two-step Runge-Kutta methods for the numerical solution of ordinary differential equations is proposed. These methods are obtained using the collocation approach by relaxing some of the collocation conditions to obtain methods with desirable stability properties. Local error estimation for these methods is also discussed.

Present state-of-the-art in exponential fitting. A contribution dedicated to Liviu Ixaru on his 70th birthday

The standard monograph in this area is the book Exponential fitting by Ixaru and Vanden Berghe (Kluwer, Boston - Dordrecht - London, 2004) but a fresh look on things is necessary because many new contributions have been accumulated in the meantime. With no claim that our investigation is exhaustive we consider various directions of interest, try to integrate the new contributions in a natural, easy to follow way, and also detect some open problems of acute interest.

Exponentially fitted two-step Runge–Kutta methods: Construction and parameter selection

Abstract We derive exponentially fitted two-step Runge–Kutta methods for the numerical solution of y ′ = f ( x , y ) , specially tuned to the behaviour of the solution. Such methods have nonconstant coefficients which depend on a parameter to be suitably estimated. The construction of the methods is shown and a strategy of parameter selection is presented. Some numerical experiments are provided to confirm the theoretical expectations.