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Explicit invariant-preserving integration of differential equations using homogeneous projection

Benjamin Kwanen Tapley

TL;DR

The paper introduces a general projection framework to preserve invariants in numerical integration by exploiting homogeneous symmetries of the invariants. When the invariant is homogeneous with respect to a known flow, the base time-stepping method can be post-processed with a closed-form projection hatΦ_h = ψ_s ∘ Φ_h that preserves the invariant exactly without nonlinear solves; for nonhomogeneous cases, a pseudo-invariant-preserving extension applies a small, cheap auxiliary step to achieve high-order invariant accuracy. The authors extend the theory to multiple invariants and nonlinear symmetries, including alternating projections and extended phase-space formulations, and provide an open-source implementation compatible with adaptive solvers like Dormand-Prince 8(5,3). Numerical experiments on ODEs (double pendulum, Kepler) and semidiscretised PDEs (KdV, Camassa-Holm) show substantial gains in accuracy and efficiency over standard approaches, with robust behavior under adaptivity. The framework offers a practical, explicit, high-order route to structure-preserving time integration applicable to a wide range of systems, including those with multiple invariants and PDE discretizations."

Abstract

We develop a general framework for numerically solving differential equations while preserving invariants. As in standard projection methods, we project an arbitrary base integrator onto an invariant-preserving manifold, however, our method exploits homogeneous symmetries to evaluate the projection exactly and in closed form. This yields explicit invariant-preserving integrators for a broad class of nonlinear systems, as well as pseudo-invariant-preserving schemes capable of preserving multiple invariants to arbitrarily high precision. The resulting methods are high-order and introduce negligible computational overhead relative to the base solver. When incorporated into adaptive solvers such as Dormand-Prince 8(5,3), they provide error-controlled, invariant-preserving, high-order time-stepping schemes. Numerical experiments on double-pendulum and Kepler ODEs as well as semidiscretised KdV and Camassa-Holm PDEs demonstrate substantial improvements in both accuracy and efficiency over standard approaches.

Explicit invariant-preserving integration of differential equations using homogeneous projection

TL;DR

The paper introduces a general projection framework to preserve invariants in numerical integration by exploiting homogeneous symmetries of the invariants. When the invariant is homogeneous with respect to a known flow, the base time-stepping method can be post-processed with a closed-form projection hatΦ_h = ψ_s ∘ Φ_h that preserves the invariant exactly without nonlinear solves; for nonhomogeneous cases, a pseudo-invariant-preserving extension applies a small, cheap auxiliary step to achieve high-order invariant accuracy. The authors extend the theory to multiple invariants and nonlinear symmetries, including alternating projections and extended phase-space formulations, and provide an open-source implementation compatible with adaptive solvers like Dormand-Prince 8(5,3). Numerical experiments on ODEs (double pendulum, Kepler) and semidiscretised PDEs (KdV, Camassa-Holm) show substantial gains in accuracy and efficiency over standard approaches, with robust behavior under adaptivity. The framework offers a practical, explicit, high-order route to structure-preserving time integration applicable to a wide range of systems, including those with multiple invariants and PDE discretizations."

Abstract

We develop a general framework for numerically solving differential equations while preserving invariants. As in standard projection methods, we project an arbitrary base integrator onto an invariant-preserving manifold, however, our method exploits homogeneous symmetries to evaluate the projection exactly and in closed form. This yields explicit invariant-preserving integrators for a broad class of nonlinear systems, as well as pseudo-invariant-preserving schemes capable of preserving multiple invariants to arbitrarily high precision. The resulting methods are high-order and introduce negligible computational overhead relative to the base solver. When incorporated into adaptive solvers such as Dormand-Prince 8(5,3), they provide error-controlled, invariant-preserving, high-order time-stepping schemes. Numerical experiments on double-pendulum and Kepler ODEs as well as semidiscretised KdV and Camassa-Holm PDEs demonstrate substantial improvements in both accuracy and efficiency over standard approaches.

Paper Structure

This paper contains 42 sections, 8 theorems, 59 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Key contributions.
  3. Related work
  4. Symplectic vs. energy-preserving methods
  5. Adaptive step size geometric methods
  6. Extended phase space and auxiliary variable methods
  7. Projection methods and discrete gradients
  8. Homogeneous projection
  9. Motivating example: a planar separable polynomial Hamiltonian
  10. General framework
  11. Multiple homogeneous invariants
  12. Linear homogeneous symmetries
  13. Isotropic dilations.
  14. Weighted dilations.
  15. Matrix actions.
  16. ...and 27 more sections

Key Result

Theorem 3.3

The homogeneous projection method $\widehat{\Phi}_h$ satisfies the following:

Figures (9)

  • Figure 1: The error, energy preservation and runtime of the double pendulum systems averaged over 10 random initial conditions. The error bars and shaded regions indicate one standard deviation.
  • Figure 2: The error versus runtime of adaptive time-stepping methods compared to fixed step size symplectic methods for the torsioned joint double pendulum.
  • Figure 3: The error as a function of function evaluations (dashed lines) and runtime (solid lines) for the Kepler problem over a time interval of $[0, 1000]$ with varying eccentricities.
  • Figure 4: Results for the Kepler problem with eccentricity $e=0.95$ over $10000$ periods with tolerance $10^{-6}$ and step size of $h=0.01$ for the Suzuki8 method.
  • Figure 5: KdV equation with increasingly challenging initial conditions.
  • ...and 4 more figures

Theorems & Definitions (22)

  • Definition 2.1: Projection method
  • Definition 3.1: Homogeneous function
  • Definition 3.2: Homogeneous projection method
  • Theorem 3.3: Invariant-preservation and order-retention
  • proof
  • proof
  • Remark 3.4: Pseudo-invariant-preserving methods
  • Corollary 3.5: Invariant-dissipating methods
  • proof
  • Proposition 3.6: Equivalence with standard projection methods
  • ...and 12 more