Isospectral Property of Hamiltonian Boundary Value Methods (HBVMs) and their blended implementation
For researchers in numerical integration of Hamiltonian systems, this theoretical property simplifies implementation and guarantees stability, though it is an incremental theoretical insight.
The paper proves that Hamiltonian Boundary Value Methods (HBVMs) have the same non-zero spectrum as Gauss-Legendre methods, ensuring A-stability, and enables efficient blended implementation for solving discrete problems.
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.