2010/02/08 by Luigi Brugnano, Brugnano, Luigi, Felice Iavernaro +3
Computer Science · Engineering · Mathematics · #65H10 #65L05 #65L06 #65L80 #65P10 #Advanced Numerical Methods in Computational Mathematics #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.1002.1387
openalex publication_date 2010/02/08 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/28
One main issue, when numerically integrating autonomous Hamiltonian systems, is the long-term conservation of some of its invariants, among which the Hamiltonian function itself. Recently, a new class of methods, named "Hamiltonian Boundary Value Methods (HBVMs)" has been introduced and analysed, which are able to exactly preserve polynomial Hamiltonians of arbitrarily high degree. We here study a further property of such methods, namely that of having, when cast as Runge-Kutta methods, a matrix of the Butcher tableau with the same spectrum (apart the zero eigenvalues) as that of the corresponding Gauss-Legendre method, independently of the considered abscissae. Consequently, HBVMs are always perfectly A-stable methods. Moreover, this allows their efficient "blended" implementation, for solving the generated discrete problems.