Hamiltonian
Boundary
Value
Methods
(HBVMs)
## Energy Preserving Discrete Line Integral Methods |

NOTES (pdf full)

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 found here.

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 found here.

REFERENCES
(downloadable)

[14 ff.] RECENT DEVELOPMENTS.

[13] L.Brugnano, F.Iavernaro, D.Trigiante. Isospectral Property of HBVMs and their connections with Runge-Kutta collocation methods. Preprint, 2010 (arXiv:1002.4394).

[12] L.Brugnano, F.Iavernaro, T.Susca. Numerical comparisons between Gauss-Legendre methods and Hamiltonian BVMs defined over Gauss points. Monografias de la Real Academia de Ciencias de Zaragoza, Special Issue devoted to the 65th birthday of Manuel Calvo, 33 (2010) 95-112. (arXiv:1002.2727).

[11] L.Brugnano, F.Iavernaro, D.Trigiante. Isospectral Property of HBVMs and their Blended Implementation. Preprint (2010) (arXiv:1002.1387).

[10] L.Brugnano, F.Iavernaro, D.Trigiante. Hamiltonian Boundary Value Methods (Energy Conserving Discrete Line Integral Methods). Jour. Numer. Anal., Industrial and Appl. Math., 5, 1-2 (2010) 17-37 (arXiv:0910.3621).

[9] L.Brugnano, F.Iavernaro, D.Trigiante. Analisys of Hamiltonian Boundary Value Methods (HBVMs): a class of energy-preserving methods for the numerical solution of polynomial Hamiltonian dynamical systems. Preprint (2009) (arXiv:0909.5659).

[8] L.Brugnano, F.Iavernaro, D.Trigiante. Hamiltonian BVMs (HBVMs): a family of "drift-free" methods for integrating polynomial Hamiltonian systems. "Proceedings of ICNAAM 2009", AIP Conf. Proc. 1168 (2009) 715-718. <Permalink>

[7] L.Brugnano, F.Iavernaro, T.Susca. Hamiltonian BVMs (HBVMs): implementation details and applications. "Proceedings of ICNAAM 2009", AIP Conf. Proc. 1168 (2009) 723-726. <Permalink>

[6] F.Iavernaro, D.Trigiante. High-order symmetric schemes for the energy conservation of polynomial Hamiltonian problems. J. Numer. Anal. Ind. Appl. Math. 4, 1-2 (2009) 87-101.

[5] F.Iavernaro, B.Pace. Conservative Block-Boundary Value Methods for the Solution of Polynomial Hamiltonian Systems. AIP Conf. Proc. 1048 (2008) 888-891. <Permalink>

[4] F.Iavernaro, B.Pace. s-Stage Trapezoidal Methods for the Conservation of Hamiltonian Functions of Polynomial Type. AIP Conf. Proc. 936 (2007) 603-606. <Permalink>

[3] F.Iavernaro, D.Trigiante. State-dependent symplecticity and area preserving numerical methods. J. Comput. Appl. Math. 205 (2007) 814-825.

[2] F.Iavernaro, D.Trigiante. Discrete conservative vector fields induced by the trapezoidal method. J. Numer. Anal. Ind. Appl. Math. 1, 1 (2006) 113-130.

[1] F.Iavernaro, D.Trigiante. On some conservation properties of the Trapezoidal Method applied to Hamiltonian systems. ICNAAM 2005 proceedings, T.E.Simos, G.Psihoyios, Ch.Tsitouras (Eds.). Wiley-VCH, Weinheim, 2005, pp. 254-257 (ISBN: 3527406522).

[14 ff.] RECENT DEVELOPMENTS.

[13] L.Brugnano, F.Iavernaro, D.Trigiante. Isospectral Property of HBVMs and their connections with Runge-Kutta collocation methods. Preprint, 2010 (arXiv:1002.4394).

[12] L.Brugnano, F.Iavernaro, T.Susca. Numerical comparisons between Gauss-Legendre methods and Hamiltonian BVMs defined over Gauss points. Monografias de la Real Academia de Ciencias de Zaragoza, Special Issue devoted to the 65th birthday of Manuel Calvo, 33 (2010) 95-112. (arXiv:1002.2727).

[11] L.Brugnano, F.Iavernaro, D.Trigiante. Isospectral Property of HBVMs and their Blended Implementation. Preprint (2010) (arXiv:1002.1387).

[10] L.Brugnano, F.Iavernaro, D.Trigiante. Hamiltonian Boundary Value Methods (Energy Conserving Discrete Line Integral Methods). Jour. Numer. Anal., Industrial and Appl. Math., 5, 1-2 (2010) 17-37 (arXiv:0910.3621).

[9] L.Brugnano, F.Iavernaro, D.Trigiante. Analisys of Hamiltonian Boundary Value Methods (HBVMs): a class of energy-preserving methods for the numerical solution of polynomial Hamiltonian dynamical systems. Preprint (2009) (arXiv:0909.5659).

[8] L.Brugnano, F.Iavernaro, D.Trigiante. Hamiltonian BVMs (HBVMs): a family of "drift-free" methods for integrating polynomial Hamiltonian systems. "Proceedings of ICNAAM 2009", AIP Conf. Proc. 1168 (2009) 715-718. <Permalink>

[7] L.Brugnano, F.Iavernaro, T.Susca. Hamiltonian BVMs (HBVMs): implementation details and applications. "Proceedings of ICNAAM 2009", AIP Conf. Proc. 1168 (2009) 723-726. <Permalink>

[6] F.Iavernaro, D.Trigiante. High-order symmetric schemes for the energy conservation of polynomial Hamiltonian problems. J. Numer. Anal. Ind. Appl. Math. 4, 1-2 (2009) 87-101.

[5] F.Iavernaro, B.Pace. Conservative Block-Boundary Value Methods for the Solution of Polynomial Hamiltonian Systems. AIP Conf. Proc. 1048 (2008) 888-891. <Permalink>

[4] F.Iavernaro, B.Pace. s-Stage Trapezoidal Methods for the Conservation of Hamiltonian Functions of Polynomial Type. AIP Conf. Proc. 936 (2007) 603-606. <Permalink>

[3] F.Iavernaro, D.Trigiante. State-dependent symplecticity and area preserving numerical methods. J. Comput. Appl. Math. 205 (2007) 814-825.

[2] F.Iavernaro, D.Trigiante. Discrete conservative vector fields induced by the trapezoidal method. J. Numer. Anal. Ind. Appl. Math. 1, 1 (2006) 113-130.

[1] F.Iavernaro, D.Trigiante. On some conservation properties of the Trapezoidal Method applied to Hamiltonian systems. ICNAAM 2005 proceedings, T.E.Simos, G.Psihoyios, Ch.Tsitouras (Eds.). Wiley-VCH, Weinheim, 2005, pp. 254-257 (ISBN: 3527406522).