Juan I. Montijano

Iterative schemes for three-stage implicit Runge-Kutta methods

Abstract In this paper we consider a class of iterative schemes for implicit Runge-Kutta methods. Taking into account convergence and linear stability properties, we propose a technique to develop efficient schemes for three-stage methods and we construct particular schemes for the case of Gauss and RadauII A methods. Finally, some numerical experiments are included in order to show the efficiency of the algorithms.

A fifth-order interpolant for the Dornand and Prince Runge-Kutta method

Abstract A family of fifth-order interpolants for the fifth-order solution provided by the Dormand and Prince Runge-Kutta pair RK5(4)7M which requires two additional function evaluations per step is presented. An optimal interpolant in this family has been determined by choosing the parameters to minimize the leading coefficients of the local truncation error of the continuous solution. Some numerical experiments with the nonstiff DETEST problems show that the proposed optimal method has a good interpolatory behavior.

Energy-preserving methods for Poisson systems

We 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.

Explicit Runge-Kutta methods for initial value problems with oscillating solutions

New pairs of embedded Runge-Kutta methods specially adapted to the numerical solution of first order systems of differential equations which are assumed to possess oscillating solutions are obtained. These pairs have been derived taking into account not only the usual properties of accuracy, stability and reliability of the local error estimator to adjust the stepsize of the underlying formulas but also the dispersion and dissipation orders of the advancing formula as defined by Van der Houwen and Sommeijer (1989). Three nine-stage embedded pairs of Runge-Kutta methods with algebraic orders 7 and 5 and higher orders of dispersion and/or dissipation are selected among the members of a family of pairs depending on several free parameters. Some numerical results are presented to show the efficiency of the new methods.

Iterative schemes for Gauss methods

Abstract In this paper, we consider a class of iterative schemes for implicit Runge-Kutta methods and we study the convergence of these schemes for a family of nonlinear stiff problems. A particular convergent scheme for the two stage Gauss method is proposed and the order and linear stability properties are analyzed. Finally, some numerical experiments are included in order to show the efficiency of the method.

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

Abstract The construction of symmetric and symplectic exponentially fitted modified Runge–Kutta (RK) methods for the numerical integration of Hamiltonian systems with oscillatory solutions is considered. In a previous paper [H. Van de Vyver, A fourth order symplectic exponentially fitted integrator, Comput. Phys. Comm. 176 (2006) 255–262] a two-stage fourth-order symplectic exponentially fitted modified RK method has been proposed. Here, two three-stage symmetric and symplectic exponentially fitted integrators of Gauss type, either with fixed nodes or variable nodes, are derived. The algebraic order of the new integrators is also analyzed, obtaining that they possess sixth-order as the classical three-stage RK Gauss method. Numerical experiments with some oscillatory problems are presented to show that the new methods are more efficient than other symplectic RK Gauss codes proposed in the scientific literature.

Stepsize selection for tolerance proportionality in explicit Runge-Kutta codes

The potential for adaptive explicit Runge–Kutta (ERK) codes to produce global errors that decrease linearly as a function of the error tolerance is studied. It is shown that this desirable property may not hold, in general, if the leading term of the locally computed error estimate passes through zero. However, it is also shown that certain methods are insensitive to a vanishing leading term. Moreover, a new stepchanging policy is introduced that, at negligible extra cost, ensures a robust global error behaviour. The results are supported by theoretical and numerical analysis on widely used formulas and test problems. Overall, the modified stepchanging strategy allows a strong guarantee to be attached to the complete numerical process.

A new embedded pair of Runge-Kutta formulas of orders 5 and 6

A new pair of embedded Runge-Kutta (RK) formulas of orders 5 and 6 is presented. It is derived from a family of RK methods depending on eight parameters by using certain measures of accuracy and stability. Numerical tests comparing its efficiency to other formulas of the same order in current use are presented. With an extra function evaluation per step, a C^1-continuous interpolant of order 5 can be obtained.

On the Preservation of Invariants by Explicit Runge--Kutta Methods

A new strategy to preserve invariants in the numerical integration of initial value problems with explicit Runge--Kutta methods is presented. It is proved that this technique retains the order of the original method, has an easy and cheap implementation, and can be used in adaptive Runge--Kutta codes. Some numerical experiments with the classical code of Dormand and Prince, DoPri5(4), based on a pair of embedded methods with orders 5 and 4, are presented to show the behavior of the new method for several problems which possess invariants.

Chebyshev Polynomials in Distributed Consensus Applications

In this paper we analyze the use of Chebyshev polynomials in distributed consensus applications. It is well known that the use of polynomials speeds up the convergence to the consensus in a significant way. However, existing solutions only work for low degree polynomials and require the topology of the network to be fixed and known. We propose a distributed algorithm based on the second order difference equation that describes the Chebyshev polynomials of first kind. The contributions of our algorithm are three: (i) Since the evaluation of Chebyshev polynomials is stable, there is no limitation in the degree of the polynomial. (ii) Instead of the knowledge of the whole network topology, it only requires a partial knowledge or an approximation to it. (iii) It can be applied to time varying topologies. In the paper we characterize the main properties of the algorithm for both fixed and time-varying communication topologies. Theoretical results, as well as experiments with synthetic data, show the benefits of using our algorithm compared to existing methods.