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Hamiltonian Boundary Value Methods (Energy Conserving Discrete Line Integral Methods)
Luigi Brugnano, Felice Iavernaro, Donato Trigiante
TL;DR
The paper addresses how to formulate HBVMs generally and identify their limiting behavior as silent stages grow, while preserving energy for Hamiltonian systems. It develops a line-integral-based framework, relates limits to polynomial bases and nodes, and reports that the limits are characterized independently of abscissae in the stated setting. The paper also identifies finite-precision limitations in non-polynomial examples.
Problem
The paper studies how to generalize HBVM theory and determine optimal limiting formulae with respect to polynomial bases and node distributions.
Method
The authors reformulate HBVMs using discrete line integrals and extended collocation conditions, then analyze the limit as silent stages tend to infinity.
Results
All HBVMs with s fundamental stages tend toward the same limit method, characterized by an operator eigenfunction independent of the abscissae; Lagrange limits coincide with EPCMs.
Takeaways & Limitations
The framework covers polynomial and non-polynomial Hamiltonians, with finite discretizations of limit formulae returning to HBVM(k,s) methods for sufficiently large k.
Takeaways & Limitations
Finite-precision arithmetic can destroy theoretical conservation through cancellation and may make infinite versions inconvenient even when integrals are analytically evaluable.
Abstract
from arXiv · showhide
Recently, a new family of integrators (Hamiltonian Boundary ValueMethods) has been introduced, which is able to precisely conserve the energy function of polynomial Hamiltonian systems and to provide a practical conservation of the energy in the non-polynomial case. We settle the definition and the theory of such methods in a more general framework. Our aim is on the one hand to give account of their good behavior when applied to general Hamiltonian systems and, on the other hand, to find out what are the optimal formulae, in relation to the choice of the polynomial basis and of the distribution of the nodes. Such analysis is based upon the notion of extended collocation conditions and the definition of discrete line integral, and is carried out by looking at the limit of such family of methods as the number of the so called silent stages tends to infinity.
1 Introduction
Hamiltonian Boundary Value Methods (HBVMs) are energy-preserving one-step methods whose silent stages support practical conservation beyond polynomial Hamiltonians. The paper generalizes their framework and studies the limiting methods obtained as silent stages increase.
- Motivation: HBVMs exactly conserve polynomial Hamiltonians, while sufficiently many silent stages provide practical energy conservation for non-polynomial Hamiltonians.In finite precision, the polynomial approximation becomes indistinguishable from H(y) within machine precision.
- Motivation: Silent stages are linear combinations of fundamental stages, so nonlinear-system cost depends essentially on s rather than the number of silent stages.Here s denotes the number of fundamental stages, while r denotes the number of silent stages.
- Research question: The paper asks what limit method emerges when the number of silent stages tends to infinity.This question is motivated by practical energy conservation and the cost structure of silent stages.
- Contributions: Different polynomial bases and abscissae distributions can produce different limit methods; Lagrange-based limits coincide with EPCMs, whereas shifted Legendre bases yield ∞-HBVMs.The paper proves the Lagrange/EPCM coincidence and introduces the shifted-Legendre limit class.
- Contributions: The authors reformulate HBVMs through extended collocation conditions and discrete line integrals, settling their definition and deriving general limit formulae.The framework addresses the construction of HBVM sequences and their infinite-stage limits.
- Contributions: The paper also generalizes the approach beyond polynomial Hamiltonians and determines an optimal node distribution in the polynomial case.It further notes that finite quadrature approximations of the limit methods return to HBVM(k,s) for sufficiently large k.
2 Reformulation of Hamiltonian BVMs
The reformulation represents the approximate solution in a polynomial basis, uses fundamental and silent stages to impose energy-preserving discrete line-integral conditions, and connects finite HBVMs with limit methods.
- Polynomial approximation: HBVMs approximate σ(t) with a polynomial expansion in a suitable basis, with fundamental stages Yi sampled at the chosen abscissae.The coefficients of the expansion are determined through orthogonality conditions involving J∇H.
- Discrete line integral: For polynomial H(y), additional abscissae are required because the line-integral integrand can have too high a degree for quadrature based only on the fundamental nodes.The associated quadrature uses weights for both the fundamental and silent abscissae.
- Relation to existing methods: The formulation applied directly to the original line integral yields the same HBVM(k,s) method as the discrete-line-integral construction.In the non-polynomial case, EPCMs arise as the limit of HBVMs with a Lagrange basis as silent stages tend to infinity.
- Stage structure: Silent stages are evaluated from σ at additional abscissae and are linear combinations of the fundamental stages rather than independent unknowns.They increase the quadrature precision while the nonlinear system depends essentially only on s.
- Functional formulation: The orthogonality conditions produce extended collocation conditions and a master functional equation in which σ is an eigenfunction of the associated operator for eigenvalue 1.The resulting formulation defines the HBVM class for the selected polynomial basis.
- Basis choice: The polynomial basis affects the resulting method: monomial choices can force σ to be a line, whereas shifted Legendre choices yield order 2s.The monomial example gives η1 = 1 and ηj = 0 for j > 1; the Legendre choice gives ηj = 2j −1.
- Non-polynomial Hamiltonians: With sufficiently many silent stages, HBVMs are indistinguishable in finite precision from their limit formulae and provide practical energy conservation for non-polynomial Hamiltonians.This practical behavior means the method cannot distinguish H(y) from a polynomial approximation within machine precision ε in many general situations.
3 Infinity Hamiltonian Boundary Value Methods
The infinity formulation selects the shifted Legendre basis to obtain basis-independent limit methods with order 2s and perfect A-stability, while preserving energy-conserving behavior through line-integral conditions.
- Basis selection: The choice of polynomial basis is central because different bases can produce different limit formulae, motivating the search for an optimal basis.The paper identifies the shifted Legendre basis as the preferred choice for the subsequent HBVM formulation.
- Conservation conditions: For an orthogonal basis, the orthogonality conditions are not only sufficient but necessary for the discrete line-integral conservation condition, assuming suitable differentiability.Theorem 1 states that each term in the conservation sum must vanish.
- Legendre basis: The shifted Legendre basis yields the scaling ηj = 2j −1 and supports the order and stability properties sought for the limit methods.Its orthogonality and independence from the abscissae distinguish it from Newton and Lagrange bases.
- ∞-HBVM definition: The limit formula based on the Legendre basis is called HBVM(∞, s), or an Infinity Hamiltonian Boundary Value Method.For polynomial Hamiltonians, its integral is exactly computed by a sufficiently accurate quadrature; for non-polynomial Hamiltonians, it is the limit of HBVM(k,s).
- Order and stability: HBVM(k,s) has order p = 2s when the quadrature order is at least 2s, independently of the fundamental abscissae.The same condition gives perfect A-stability, and these properties transfer to HBVM(∞,s).
- Limit-method properties: HBVM(∞, s) has order 2s and is perfectly A-stable for any choice of the fundamental abscissae.It produces the same polynomial σ as optimal EPCMs, while EPCMs reach order 2s only when their abscissae define a quadrature of order at least 2s −1.
- Node distributions: For the Lobatto-node construction, deg(σ) = s and HBVM(k,s) has order 2s with a quadrature satisfying B(2k).Symmetric distributions of all abscissae also make HBVM(k,s) symmetric.
4 Generalization of Hamiltonian BVMs
The paper generalizes energy-preserving Hamiltonian Boundary Value Methods from polynomial settings to broader function spaces, using discrete line integrals and extended collocation conditions. It also develops concrete second- and fourth-order examples while identifying integrability and finite-precision boundaries.
- General framework: The construction uses a curve in a finite-dimensional function space W and extended collocation conditions to zero the discrete line integral.The basis functions may be any linearly independent functions, provided the relevant integrals are tractable.
- General framework: The approach extends beyond polynomial Hamiltonians when the products Pj(τ)∇H(σ(t0 + τh)) admit elementary primitives.The line-integral procedure can then be repeated using the primitive rather than a quadrature sum.
- 4.1 A method of order two: In separable Hamiltonian systems, the simplest resulting discrete-gradient method is second order and symmetric, although its general form is first order and nonsymmetric.It replaces partial derivatives with increments along the q and p axes.
- 4.2 A method of order four: A fourth-order method is constructed from a degree-two curve using abscissae c0 = 0, c1 = 1/2, and c2 = 1.The method introduces midpoint and endpoint unknowns before applying the generalized formula.
- 4.2 A method of order four: The general fourth-order construction lacks a universal formula when elementary integrability is not guaranteed, though explicit primitives exist in several cases.The paper uses one such case for later numerical tests and relates it to a Lotka-Volterra-type system.
- 4.2 A method of order four: Even when the integrals are analytically evaluable, finite-precision arithmetic may make the infinite version of the methods inconvenient.The paper identifies this boundary in an example reserved for numerical testing.
5 HBVMs based upon Gauss quadrature
Gauss-based HBVMs retain order four, symmetry, and A-stability while achieving exact polynomial-energy conservation under a degree-dependent condition. Numerical tests show HBVMs outperform classical fourth-order methods in energy preservation, including practical conservation for a non-polynomial Hamiltonian.
- HBVM(k,s) uses k Gauss-Legendre abscissae, a degree-s polynomial, and has order 2s, symmetry, and perfect A-stability.For k=s, the method reduces to Gauss-Legendre collocation.
- For sufficiently large k, Gauss- and Lobatto-based HBVM(k,s) methods define the same polynomial σ and converge to the same HBVM(∞,s) in the non-polynomial case.The numerical experiments reported the same results for the two node choices on polynomial test problems.
- Test problem 1: For the degree-6 polynomial test, fourth-order HBVM(6,2) preserves the Hamiltonian to approximately 10^-16, whereas fourth-order Lobatto IIIA exhibits drift and Gauss-Legendre preserves H only to approximately 10^-6.All methods use h = 0.16 for problem (43).
- Test problem 4 (non-polynomial Hamiltonian): In the non-polynomial charged-particle problem, fourth-order HBVM(6,2) reduces the Hamiltonian error to approximately 10^-15, while Gauss-Legendre gives approximately 10^-3 and Lobatto IIIA drifts.The methods use h = 0.1; the Hamiltonian has a Biot-Savart potential.
- Test problem 4 (non-polynomial Hamiltonian): Finite arithmetic can destroy theoretical conservation when formulas become ill-conditioned, but HBVM behavior suggests practical energy conservation in the reported non-polynomial experiment.Cancellation can cause trajectory jumps for the Itoh-Abe and fourth-order formulas, whereas the HBVM result appears conserved in finite arithmetic.
6 Conclusions
The paper presents HBVMs in a unified discrete-line-integral framework that exactly preserves polynomial Hamiltonians and converges to an abscissa-independent limit as silent stages increase. Numerical tests also show practical energy conservation and identify ill-conditioning issues in competing formulae.
- 6 Conclusions: HBVMs are re-derived as energy-preserving methods based on discrete line integrals, which are exact for polynomial Hamiltonians.The framework unifies the methods and explains exact preservation for polynomial Hamiltonians.
- 6 Conclusions: The fourth-order HBVM with k = 10 Gaussian abscissae achieved |H − H0| approximately 10^-12, while the compared Itoh-Abe formulae exhibited jumps from ill-conditioning.The jumps occur for certain solution values.
- 6 Conclusions: In the Itoh-Abe trajectory, closely spaced consecutive q values caused loss of significant digits in the subsequent branch.The reported points occur at t = 2830.5 and t = 2831.
- 6 Conclusions: As the number of silent stages tends to infinity, HBVMs with s fundamental stages converge to the same limit method, independent of the abscissae.The limit method is characterized by the unit-eigenvalue eigenfunction of an operator independent of the node choices.