The approach of using
discrete line integrals has been used, at first, by Iavernaro and Trigiante, in
connection with the study of the properties of the trapezoidal rule
[1,2,3].
It has been then extended by Iavernaro
and Pace [4],
thus providing the first example of
conservative Runge-Kutta methods, basically an extension of the trapezoidal rule,
named s-stage trapezoidal
methods: this is a family of energy-preserving methods of order 2, able to preserve
polynomial Hamiltonian functions of arbitrarily high degree.
Later generalizations allowed
Iavernaro and Pace [5],
and then Iavernaro and Trigiante [6], to
derive energy preserving methods of higher order.
The general approach, involving the
shifted Legendre polynomial basis, which has allowed a full
complete analysis of HBVMs, has been introduced in [9] (see
also [8])
and, subsequently, developed in [10].
The isospectral property of HBVMs has
been studied in [11],
where the blended
implementation of the methods is also introduced. It has also been used in [13] to study the existing connections between HBVMs and Runge-Kutta collocation methods.
Computational aspects, concerning both
the computational cost and the efficient numerical implementation
of HBVMs, have been studied in [7]
and [11].
Relevant examples have been collected
in [12],
where the potentialities of HBVMs are clearly
outlined, also demonstrating their effectiveness with respect to
standard symmetric and symplectic methods. More recent developments can be foundhere.
