Joseba Makazaga

Joseba Makazaga
Universidad del País Vasco / Euskal Herriko Unibertsitatea | UPV/EHU · Computer Sciences and Artificial Intelligence

PhD

About

17
Publications
1,548
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214
Citations

Publications

Publications (17)
Article
Full-text available
We present FCIRK16, a 16th-order implicit symplectic integrator for long-term high-precision Solar System simulations. Our integrator takes advantage of the near-Keplerian motion of the planets around the Sun by alternating Keplerian motions with corrections accounting for the planetary interactions. Compared to other symplectic integrators (the Wi...
Preprint
Full-text available
Compared to other symplectic integrators (the Wisdom and Holman map and its higher order generalizations) that also take advantage of the hierarchical nature of the motion of the planets around the central star, our methods require solving implicit equations at each time-step. We claim that, despite this disadvantage, FCIRK16 is more efficient than...
Article
Full-text available
This work considers the gravitational N-body problem and introduces global time-renormalization functions that allow the efficient numerical integration with fixed time-steps. First, a lower bound of the radius of convergence of the solution to the original equations is derived, which suggests an appropriate time-renormalization. In the new fictiti...
Preprint
This work considers the gravitational $N$-body problem and introduces time-reparametrization functions that allow to define globally solutions of the $N$-body equations. First, a lower bound of the radius of convergence of the solution to the original equations is derived, which suggests an appropriate time-reparametrization. In the new fictitious...
Preprint
Full-text available
We present an algorithm based on continuation techniques that can be applied to solve numerically minimization problems with equality constraints. We focus on problems with a great number of local minima which are hard to obtain by local minimization algorithms with random starting guesses. We are particularly interested in the computation of minim...
Article
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We propose a family of integrators, Flow-Composed Implicit Runge-Kutta (FCIRK) methods, for perturbations of nonlinear ordinary differential equations, consisting of the composition of flows of the unperturbed part alternated with one step of an implicit Runge-Kutta (IRK) method applied to a transformed system. The resulting integration schemes are...
Article
Full-text available
Konposizio metodoek, Ekuazio Diferentzial Arruntak (EDAak) ebazteko oinarrizko zenbakizko integrazio-metodo bat modu egokian konposatuz emaitzak hobetzeko aukera ematen dute. Bigarren ordenako zehaztasuna duen oinarrizko integratzaile simetriko bat erabiliz lortzen den konposizio metodo simetrikoei erreparatuko diegu lan honetan. Asko dira honelako...
Article
Full-text available
We are concerned with the efficient implementation of symplectic implicit Runge-Kutta (IRK) methods applied to systems of (non-necessarily Hamiltonian) ordinary differential equations by means of Newton-like iterations. We pay particular attention to symmetric symplectic IRK schemes (such as collocation methods with Gaussian nodes). For a $s$-stage...
Article
Full-text available
We propose an implementation of symplectic implicit Runge-Kutta schemes for highly accurate numerical integration of non-stiff Hamiltonian systems based on fixed point iteration. Provided that the computations are done in a given floating point arithmetic, the precision of the results is limited by round-off error propagation. We claim that our imp...
Article
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We introduce a new class of multi-revolution composition methods for the approximation of the Nth-iterate of a given near-identity map. When applied to the numerical integration of highly oscillatory systems of differential equations, the technique benefits from the properties of standard composition methods: it is intrinsically geometric and well-...
Article
We present new splitting methods designed for the numerical integration of near-integrable Hamiltonian systems, and in particular for planetary N-body problems, when one is interested in very accurate results over a large time span. We derive in a systematic way an independent set of necessary and sufficient conditions to be satisfied by the coeffi...
Article
Using a Newtonian model of the Solar System with all 8 planets, we perform extensive tests on various symplectic integrators of high orders, searching for the best splitting scheme for long term studies in the Solar System. These comparisons are made in Jacobi and Heliocentric coordinates and the implementation of the algorithms is fully detailed f...
Article
Full-text available
We present a new family of one-step symplectic integration schemes for Hamiltonian systems of the general form [(y)\dot]=J-1ÑH(y)T{\dot y=J^{-1}\nabla H(y)^T} . Such a class of methods contains as particular cases the methods of Miesbach and Pesch (Numer Math 61:501–521, 1992), and also the family of symplectic Runge-Kutta methods. As in the case o...
Thesis
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En este trabajo se estudia el error cometido por los mé́todos explícitos de Runge-Kutta en la resolució́n de problemas de valor inicial de sistemas de ecuaciones diferenciales ordinarias. El trabajo se basa en el aná́lisis de las condiciones de orden de los méttodos así como en el estudio de las cotas de los errores, tanto los errores locales (es d...
Article
We present a particular 5th order one-step integrator for ODEs that provides an estimation of the global error. It's based on the class of one-step integrator for ODEs of Murua and Makazaga considered as a generalization of the globally embedded RK methods of Dormand, Gilmore and Prince. The scheme we present cheaply gives useful information on the...
Article
Full-text available
We propose a class of one-step integrators for ODEs that provide an estimation of the global error along with the numerical solution. The schemes that we propose can be considered as a generalization of the globally embedded RK methods of Dormand, Gilmore and Prince. We present preliminary numerical experiments testing a particular 5th order scheme...

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