On the effectiveness of spectral methods for the numerical solution of multi-frequency highly-oscillatory Hamiltonian problems

Multi-frequency, highly-oscillatory Hamiltonian problems derive from the mathematical modelling of many real life applications. We here propose a variant of Hamiltonian Boundary Value Methods (HBVMs), which is able to efficiently deal with the numerical solution of such problems.Comment: 28 pages, 4 figures (a few typos fixed

Exponentially fitted fifth-order two-step peer explicit methods

The so called peer methods for the numerical solution of Initial Value Problems (IVP) in ordinary differential systems were introduced by R. Weiner et al [6, 7, 11, 12, 13] for solving different types of problems either in sequential or parallel computers. In this work, we study exponentially fitted three-stage peer schemes that are able to fit functional spaces with dimension six. Finally, some numerical experiments are presented to show the behaviour of the new peer schemes for some periodic problems

An eighth-order exponentially fitted two-step hybrid method of explicit type for solving orbital and oscillatory problems

The construction of an eighth-order exponentially fitted (EF) two-step hybrid method for the numerical integration of oscillatory second-order initial value problems (IVPs) is considered. The EF two-step hybrid methods integrate exactly differential systems whose solutions can be expressed as linear combinations of exponential or trigonometric functions and have variable coefficients depending on the frequency of each problem. Based on the order conditions and the EF conditions for this class of methods, we construct an explicit EF two-step hybrid method with symmetric nodes and algebraic order eight which only uses seven function evaluations per step. This new method has the highest algebraic order we know for the case of explicit EF two-step hybrid methods. The numerical experiments carried out with several orbital and oscillatory problems show that the new eighth-order EF scheme is more efficient than other standard and EF two-step hybrid codes recently proposed in the scientific literature

A class of explicit high-order exponentially-fitted two-step methods for solving oscillatory IVPs

The derivation of new exponentially fitted (EF) modified two-step hybrid (MTSH) methods for the numerical integration of oscillatory second-order IVPs is analyzed. These methods are modifications of classical two-step hybrid methods so that they integrate exactly differential systems whose solutions can be expressed as linear combinations of the set of functions {exp(¿t), exp(-¿t)}, ¿¿C, or equivalently {sin(¿t), cos(¿t)} when ¿=i¿, ¿¿R, where ¿ represents an approximation of the main frequency of the problem. The EF conditions and the conditions for this class of EF schemes to have algebraic order p (with p=8) are derived. With the help of these conditions we construct explicit EFMTSH methods with algebraic orders seven and eight which require five and six function evaluation per step, respectively. These new EFMTSH schemes are optimal among the two-step hybrid methods in the sense that they reach a certain order of accuracy with minimal computational cost per step. In order to show the efficiency of the new high order explicit EFMTSH methods in comparison to other EF and standard two-step hybrid codes from the literature some numerical experiments with several orbital and oscillatory problems are presented

On the integral solution of elliptic Kepler’s equation

In a recent paper, Philcox, Goodman and Slepian obtain an explicit solution of the elliptic Kepler’s equation (KE) as a quotient of two contour integrals along a Jordan curve C=C(M,e) that contains the unique real solution of KE but not includes other complex zeros of KE in its interior. The aim of this paper is to study the main issues that arise in the practical implementation of this integral solution. Thus, after a study of the complex zeros of KE, several families of Jordan contours C=C(M,e) that are suitable for this integral solution are proposed. Since contours with minimal length turn out to be the more accurate for numerical purposes, several families that minimize their length are constructed. Secondly, the approximation of the contour integrals by the composite trapezoidal rule is considered. Recall that this rule is employed in the fast Fourier transform and, in spite of its lower order, displays a spectral convergence as a function of the number of nodes, which implies a very fast convergence. Finally, the results of some numerical experiments are presented to show that such a combination of appropriate contours with the composite trapezoidal rule leads to a powerful numerical method to solve KE with any desired accuracy for all values of eccentricity

Fourier methods for oscillatory differential problems with a constant high frequency

The numerical solution of highly oscillatory initial value problems of second order with a unique high frequency is considered. New methods based on Fourier approximations are proposed. These methods can integrate the problems with reasonable stepsizes not dependent on the size of the frequency

Energy-preserving methods for Poisson systems

AbstractWe present and analyze energy-conserving methods for the numerical integration of IVPs of Poisson type that are able to preserve some Casimirs. Their derivation and analysis is done following the ideas of Hamiltonian BVMs (HBVMs) (see Brugnano et al. [10] and references therein). It is seen that the proposed approach allows us to obtain the methods recently derived in Cohen and Hairer (2011) [17], giving an alternative derivation of such methods and a new proof of their order. Sufficient conditions that ensure the existence of a unique solution of the implicit equations defining the formulae are given. A study of the implementation of the methods is provided. In particular, order and preservation properties when the involved integrals are approximated by means of a quadrature formula, are derived

Sixth-order symmetric and symplectic exponentially fitted Runge–Kutta methods of the Gauss type

AbstractThe construction of exponentially fitted Runge–Kutta (EFRK) methods for the numerical integration of Hamiltonian systems with oscillatory solutions is considered. Based on the symplecticness, symmetry, and exponential fitting properties, two new three-stage RK integrators of the Gauss type with fixed or variable nodes, are obtained. The new exponentially fitted RK Gauss type methods integrate exactly differential systems whose solutions can be expressed as linear combinations of the set of functions {exp(λt),exp(−λt)}, λ∈C, and in particular {sin(ωt),cos(ωt)} when λ=iω, ω∈R. The algebraic order of the new integrators is also analyzed, obtaining that they are of sixth-order like the classical three-stage RK Gauss method. Some numerical experiments show that the new methods are more efficient than the symplectic RK Gauss methods (either standard or else exponentially fitted) proposed in the scientific literature

Numerical methods for non conservative perturbations of conservative problems

In this paper the numerical integration of non conservative perturbations of differential systems that possess a first integral, as for example slowly dissipative Hamiltonian systems, is considered. Numerical methods that are able to reproduce appropriately the evolution of the first integral are proposed. These algorithms are based on a combination of standard numerical integration methods and certain projection techniques. Some conditions under which known conservative methods reproduce that desirable evolution in the invariant are analysed. Finally, some numerical experiments in which we compare the behaviour of the new proposed methods, the averaged vector field method AVF proposed by Quispel and McLaren and standard RK methods of orders 3 and 5 are presented. The results confirm the theory and show a good qualitative and quantitative performance of the new projection methods