M. P. Laburta

Starting algorithms for IRK methods

In this paper some classes of starting algorithms for the iterations of IRK methods are studied. They are of three types, according to their additional cost. By means of B-series, the order conditions for them are obtained. The maximum order attained by these algorithms and their construction are derived too.

Construction of starting algorithms for the RK-Gauss methods

In this paper starting algorithms for the numerical solution of stage equations in Runge-Kutta-Gauss formulae with 2, 3 and 4 stages are constructed. For each of these formulae, three types of starting algorithms are given according to their requirement of none, one or two additional function evaluations per step. Numerical experiments with Hamiltonian systems are presented to show the superior performance of the new starting algorithms of high order.

Starting algorithms for Gauss Runge-Kutta methods for Hamiltonian systems

Abstract Among the symplectic integrators for the numerical solution of general Hamiltonian systems, implicit Runge-Kutta methods of Gauss type (RKG) play an important role. To improve the efficiency of the algorithms to be used in the solution of the nonlinear equations of stages, accurate starting values for the iterative process are required. In this paper, a class of starting algorithms, which are based on numerical information computed in two previous steps, is studied. For two- and three-stages RKG methods, explicit starting algorithms for the stage equations with orders three and four are derived. Finally, some numerical experiments comparing the behaviour of the new starting algorithms with the standard first iterant based on Lagrange interpolation of stages in the previous step are presented.

Two-Step High Order Starting Values for Implicit Runge–Kutta Methods

In this article a simple form of expressing and studying the order conditions to be satisfied by starting algorithms for Runge–Kutta methods, which use information from the two previous steps is presented. In particular, starting algorithms of highest order for Runge–Kutta–Gauss methods up to seven stages are derived. Some numerical experiments with Hamiltonian systems to compare the behaviour of the new starting algorithms with other existing ones are presented.

Runge---Kutta projection methods with low dispersion and dissipation errors

In this paper new one-step methods that combine Runge–Kutta (RK) formulae with a suitable projection after the step are proposed for the numerical solution of Initial Value Problems. The aim of this projection is to preserve some first integral in the numerical integration. In contrast with standard orthogonal projection, the direction of the projection at each step is obtained from another suitable embedded formula so that the overall method is affine invariant. A study of the local errors of these projection methods is carried out, showing that by choosing proper embedded formulae the order can be increased for the harmonic oscillator. Particular embedded formulae for the third order method by Bogacki and Shampine (BS3) are provided. Some criteria to get appropriate dynamical directions for general problems as well as sufficient conditions that ensure the existence of RK methods embedded in BS3 according to them are given. Finally, some numerical experiments to test the behaviour of the new projection methods are presented.

Original article: Error growth in the numerical integration of periodic orbits

This paper is concerned with the long term behaviour of the error generated by one step methods in the numerical integration of periodic flows. Assuming numerical methods where the global error possesses an asymptotic expansion and a periodic flow with the period depending smoothly on the starting point, some conditions that ensure an asymptotically linear growth of the error with the number of periods are given. A study of the error growth of first integrals is also carried out. The error behaviour of Runge-Kutta methods implemented with fixed or variable step size with a smooth step size function, with a projection technique on the invariants of the problem is considered.

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 qu2009=u20092p. 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.

Numerical methods for non conservative perturbations of conservative problems

Abstract 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.

On the numerical integration of orthogonal flows with Runge-Kutta methods

Abstract This paper deals with the numerical integration of matrix differential equations of type Y ′( t )= F ( t , Y ( t )) Y ( t ) where F maps, for all t , orthogonal to skew-symmetric matrices. It has been shown (Dieci et al., SIAM J. Numer. Anal. 31 (1994) 261–281; Iserles and Zanna, Technical Report NA5, Univ. of Cambridge, 1995) that Gauss–Legendre Runge–Kutta (GLRK) methods preserve the orthogonality of the flow generated by Y ′= F ( t , Y ) Y whenever F ( t , Y ) is a skew-symmetric matrix, but the implicit nature of the methods is a serious drawback in practical applications. Recently, Higham (Appl. Numer. Math. 22 (1996) 217–223) has shown that there exist linearly implicit methods based on the GLRK methods with orders ⩽2 which preserve the orthogonality of the flow. The aim of this paper is to study the order and stability properties of a class of linearly implicit orthogonal methods of GLRK type obtained by extending Highams approach. Also two particular linearly implicit schemes with orders 3 and 4 based on the two-stage GLRK method that minimize the local truncation error are proposed. In addition, the results of several numerical experiments are presented to test the behaviour of the new methods.

On the Preservation of Lyapunov Functions by Runge—Kutta Methods

In this paper we study the preservation of Lyapunov functions in the numerical integration of ordinary differential equations. By means of a continuous extension and a projection technique we extend the technique proposed by Grimm and Quispel (BIT 45, 2005), so that it can be applied to other families Runge—Kutta methods such as the well known Dormand and Prince 5(4) pair.