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A unifying framework for the derivation and analysis of effective classes of one-step methods for ODEs
Luigi Brugnano, Felice Iavernaro, Donato Trigiante
TL;DR
The paper addresses the derivation and analysis of effective one-step methods, whose order and stability are commonly studied through rooted-tree theory and simplifying assumptions. It discretizes a local Fourier expansion to derive a polynomial-basis subclass of Runge–Kutta methods and applies the framework to Hamiltonian BVMs; numerical tests suggest these methods are less sensitive to a wider class of perturbations than symplectic or symmetric non-energy-conserving methods.
Problem
Order and stability analysis of one-step methods is commonly carried out using rooted-tree theory and, for order, often simplifying assumptions.
Source: §1 · p. 1Method
The paper discretizes a local Fourier expansion of the continuous problem, using a polynomial basis to derive a large subclass of Runge–Kutta methods and analyse Hamiltonian BVMs.
Source: §1 · pp. 1–2, §5 · p. 10Results
A few numerical tests suggest HBVMs are less sensitive to a wider class of perturbations than symplectic or symmetric non-energy-conserving methods.
Source: §5 · p. 10Takeaways & Limitations
For energy-preserving methods, standard mesh-selection strategies can be used without numerical Hamiltonian drift.
Source: §4 · p. 10, §5 · p. 10The quadrature discussion assumes an analytical right-hand side and exactness for polynomials of degree less than q, with q ≥ k ≥ r.
Source: §3 · p. 6
Abstract
from arXiv · showhide
In this paper, we provide a simple framework to derive and analyse several classes of effective one-step methods. The framework consists in the discretization of a local Fourier expansion of the continuous problem. Different choices of the basis lead to different classes of methods, even though we shall here consider only the case of an orthonormal polynomial basis, from which a large subclass of Runge-Kutta methods is derived. The obtained results are then applied to prove, in a simplified way, the order and stability properties of Hamiltonian BVMs (HBVMs), a recently introduced class of energy preserving methods for canonical Hamiltonian systems. A few numerical tests with such methods are also included, in order to confirm the effectiveness of the methods.
1. Introduction.
The paper introduces a framework for deriving and analysing effective one-step methods by discretizing a local Fourier expansion of the continuous problem. With a polynomial basis, it derives a large subclass of Runge–Kutta methods and applies the framework to Hamiltonian BVMs.
1. Introduction.
- Discretizing a local Fourier expansion over each stepsize interval provides a framework for deriving effective one-step methods.Different basis choices yield different method classes; this section focuses on polynomial bases.Source: §1 · p. 1
- For polynomial bases, the framework yields a large subclass of Runge–Kutta methods and simplifies their order and linear stability analysis.The paper contrasts this analysis with rooted-tree order analysis that often uses simplifying assumptions, and describes linear stability through the Lyapunov method.Source: §1 · p. 1
- Applied to canonical Hamiltonian problems, the framework gives an alternative derivation of the energy-preserving Hamiltonian BVMs (HBVMs).The paper also includes numerical examples of these methods.Source: §1 · p. 2
2. Local Fourier expansion of ODEs.
The section constructs a polynomial approximation by truncating a local Fourier expansion of the ODE, then analyzes its accuracy, stability, and Hamiltonian energy behavior.
2. Local Fourier expansion of ODEs.
- In this construction, truncating the series after r terms and imposing the initial condition yields ω, which is a polynomial of degree at most r.The basis is the shifted Legendre polynomials on [0, 1], scaled to be orthonormal.Source: §2 · p. 2
- The endpoint error is O(h^2r+1), and the grid-point approximation over T = Nh is O(h^2r) accurate.The polynomial approximation is extended step by step using its preceding endpoint value.Source: §2 · p. 3
2.1. Linear stability analysis.
- For the linear test problem, the continuous solution and discrete approximation have the same stability properties for every stepsize h > 0.The discrete sequence decays to zero iff Re(λ) < 0 and remains bounded when Re(λ) = 0.Source: §2.1 · p. 5
2.2. The Hamiltonian case.
- For a canonical Hamiltonian problem, the polynomial approximation inherits energy conservation.The continuous Hamiltonian is constant because J is skew-symmetric.Source: §2.2 · p. 5
3. Discretization.
The section discretizes the local polynomial expansion by quadrature, yielding a method whose order depends on both quadrature accuracy and polynomial degree. It then relates the resulting methods to Runge–Kutta collocation and derives order, stability, and energy-conservation properties for HBVMs.
3. Discretization.
3.1. Runge-Kutta formulation.
- The resulting methods form a Runge–Kutta subclass, with the endpoint given by the quadrature-weighted sum of stage derivatives.For Gaussian nodes, the formulation generalizes classical collocation; setting k = r recovers the r-stage Gauss method.Source: §3.1 · p. 7
3.2. Hamiltonian Boundary Value Methods (HBVMs).
- For k ≥ r, HBVM(k, r) has order 2r, is perfectly A-stable, and conserves polynomial Hamiltonians of degree at most 2k/r.Gaussian quadrature applied to canonical Hamiltonian problems yields HBVM(k, r).Source: §3.2 · p. 7, §3.1 · p. 7
4. Numerical Tests.
The numerical tests compare sixth-order methods on the Kepler problem over 1000 periods, using constant steps for moderate eccentricity and variable steps for high eccentricity. The results distinguish methods that preserve energy from those that do not, especially under variable stepping.
4. Numerical Tests.
- The tests compare HBVM(3,3)/GAUSS6, HBVM(4,3), and HBVM(15,3) on the Kepler problem over 1000 periods.The problem becomes harder as eccentricity approaches 1, motivating variable-step integration for e = 0.99.Source: §4 · pp. 8–9
- With e = 0.6 and constant steps, all methods show order 6, while HBVM(4,3) and HBVM(15,3) have smaller error constants than GAUSS6.Hamiltonian conservation is approximate for GAUSS6 and HBVM(4,3), and practically exact for HBVM(15,3).Source: §4 · p. 9, Figure 4.1 · p. 9
- With e = 0.99 and variable steps, HBVM(15,3) has no Hamiltonian drift and linear error growth, unlike the non-energy-preserving HBVM(4,3).The figure reports Hamiltonian and solution errors over 1000 periods at tolerance 10^-10.Source: Figure 4.2 · p. 9, §4 · p. 10
- Energy-preserving methods can use standard mesh-selection strategies without the Hamiltonian drift observed for symplectic methods.The paper notes that refined mesh strategies are used with symplectic methods to avoid drift and quadratic error growth.Source: §4 · p. 10
5. Conclusions.
The paper presents a local-Fourier-expansion framework for effective one-step methods, yielding a large Runge-Kutta subclass when a polynomial basis is used. For canonical Hamiltonian problems, the methods coincide with HBVMs, whose numerical tests suggest reduced sensitivity to perturbations and compatibility with standard mesh selection strategies.
5. Conclusions.
- The framework derives and analyzes effective one-step methods through a local Fourier expansion of the problem.Source: §5 · p. 10
- With a polynomial basis, the framework yields a large subclass of Runge-Kutta methods that generalizes Gauss collocation methods.Source: §5 · p. 10
- Numerical tests suggest HBVMs are less sensitive to a wider class of perturbations than symplectic or symmetric non-energy-conserving methods, and can be used with standard mesh selection strategies.The comparison is limited to the numerical tests reported by the authors.Source: §5 · p. 10
- For canonical Hamiltonian problems, the methods coincide with the energy-preserving HBVM class.Source: §5 · p. 10