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
It has been then extended by Iavernaro
and Pace ,
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 ,
and then Iavernaro and Trigiante , 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  (see
and, subsequently, developed in .
The isospectral property of HBVMs has
been studied in ,
where the blended
implementation of the methods is also introduced. It has also been used in  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 
Relevant examples have been collected
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.