Numerical integration of Hamiltonian problems by G-symplectic methods

• Published in: Advances in computational mathematics 2014-04, Vol.40 (2), p.553-575
• Main Authors: D’Ambrosio, Raffaele, De Martino, Giuseppe, Paternoster, Beatrice
• Format: Article
• Language: English
• Subjects: Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Conservation; Convergence; Deviation; Joints; Mathematical and Computational Biology; Mathematical Modeling and Industrial Mathematics; Mathematical models; Mathematics; Mathematics and Statistics; Runge-Kutta method; Solvers; Symmetry; Visualization
• ISSN: 1019-7168; 1572-9044
• DOI: 10.1007/s10444-013-9318-z

It is the purpose of this paper to consider the employ of General Linear Methods (GLMs) as geometric numerical solvers for the treatment of Hamiltonian problems. Indeed, even if the numerical flow generated by a GLM cannot be symplectic, we exploit here a concept of near conservation for such methods which, properly combined with other desirable features (such as symmetry and boundedness of parasitic components), allows to achieve an accurate conservation of the Hamiltonian. In this paper we focus our attention on the connection between order of convergence and Hamiltonian deviation by multivalue methods. Moreover, we derive a semi-implicit GLM which results competitive to symplectic Runge-Kutta methods, which are notoriously implicit.