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Pseudo-Energy-Preserving Explicit Runge-Kutta Methods

Gabriel A. Barrios de León, David I. Ketcheson, Hendrik Ranocha

TL;DR

The paper addresses finite-dimensional Hamiltonian system integration by introducing maximal pseudo-energy-preserving (PEP) explicit Runge-Kutta methods, which optimize the energy-conservation error to order $q$ while maintaining classical order $p$. It develops a B-series framework and explicit algebraic conditions on RK coefficients that guarantee PEP up to a chosen order, translating flow-based constraints into Butcher coefficient relations. Through numerical optimization and a curated set of canonical and non-canonical tests, the authors construct and demonstrate RK schemes with PEP orders up to $6$, showing improved long-time energy behavior and lower energy-related errors compared to standard methods of the same classical order. The results indicate that PEP methods offer a practical balance between efficiency and energy preservation, delivering better qualitative and quantitative performance for moderately long-time simulations in Hamiltonian systems and related PDE discretizations. These methods extend the toolbox for structure-preserving computation by enabling explicit, high-PEP-order schemes without fully implicit CSRK formulations or projection steps.

Abstract

Using a recent characterization of energy-preserving B-series, we derive the explicit conditions on the coefficients of a Runge-Kutta method that ensure energy preservation (for Hamiltonian systems) up to a given order in the step size, which we refer to as the pseudo-energy-preserving (PEP) order. We study explicit Runge-Kutta methods with PEP order higher than their classical order. We provide examples of such methods up to PEP order six, and test them on Hamiltonian ODE and PDE systems. We find that these methods behave similarly to exactly energy-conservative methods over moderate time intervals and exhibit significantly smaller errors, relative to other Runge-Kutta methods of the same order, for moderately long-time simulations.

Pseudo-Energy-Preserving Explicit Runge-Kutta Methods

TL;DR

The paper addresses finite-dimensional Hamiltonian system integration by introducing maximal pseudo-energy-preserving (PEP) explicit Runge-Kutta methods, which optimize the energy-conservation error to order while maintaining classical order . It develops a B-series framework and explicit algebraic conditions on RK coefficients that guarantee PEP up to a chosen order, translating flow-based constraints into Butcher coefficient relations. Through numerical optimization and a curated set of canonical and non-canonical tests, the authors construct and demonstrate RK schemes with PEP orders up to , showing improved long-time energy behavior and lower energy-related errors compared to standard methods of the same classical order. The results indicate that PEP methods offer a practical balance between efficiency and energy preservation, delivering better qualitative and quantitative performance for moderately long-time simulations in Hamiltonian systems and related PDE discretizations. These methods extend the toolbox for structure-preserving computation by enabling explicit, high-PEP-order schemes without fully implicit CSRK formulations or projection steps.

Abstract

Using a recent characterization of energy-preserving B-series, we derive the explicit conditions on the coefficients of a Runge-Kutta method that ensure energy preservation (for Hamiltonian systems) up to a given order in the step size, which we refer to as the pseudo-energy-preserving (PEP) order. We study explicit Runge-Kutta methods with PEP order higher than their classical order. We provide examples of such methods up to PEP order six, and test them on Hamiltonian ODE and PDE systems. We find that these methods behave similarly to exactly energy-conservative methods over moderate time intervals and exhibit significantly smaller errors, relative to other Runge-Kutta methods of the same order, for moderately long-time simulations.
Paper Structure (19 sections, 1 theorem, 42 equations, 8 figures, 12 tables)

This paper contains 19 sections, 1 theorem, 42 equations, 8 figures, 12 tables.

Table of Contents

  1. Introduction
  2. Related ideas
  3. B-series and Runge-Kutta methods
  4. Pseudo-energy-preservation conditions for Runge-Kutta methods applied to Hamiltonian systems
  5. Energy-preserving B-series flows
  6. Conditions on the flow coefficients
  7. Relation between elementary weights of the map and flow
  8. Energy-preserving conditions on the RK coefficients
  9. Methods with maximal pseudo-energy-preserving order
  10. Numerical tests
  11. Canonical Hamiltonian systems
  12. Exponential entropy system
  13. Undamped Duffing Oscillator
  14. Hénon-Heiles System
  15. Benjamin-Bona-Mahony Equation
  16. ...and 4 more sections

Key Result

Theorem 1

A B-series flow Bflow for the canonical Hamiltonian system hamiltonian-system is pseudo-energy-preserving of order $q$ if and only if there exist coefficients $\mu_{j,\tau}$ such that Here the sequence $\ell_j$ is an ordering of the leaves of $\tau$ and $m_j(\tau)$ is the distance from the root to leaf $\ell_j$.

Figures (8)

  • Figure 1: Simulation of the undamped duffing oscillator comparing RK methods for which $p = q$ with RK methods with higher PEP.
  • Figure 2: Numerical energy evolution for the Hénon-Heiles system, comparing PS and PEP methods.
  • Figure 3: Error growth for the BBM equation comparing the midpoint method with with PEP(4,2,5) with $h_0 = 0.05$.
  • Figure 4: Error growth for the nonlinear oscillator problem \ref{['eq:nonlinearoscillator']}, comparing RK methods for which $p = q$ with RK methods with higher PEP order.
  • Figure 5: Error growth for the nonlinear oscillator comparing PS with PEP methods.
  • ...and 3 more figures

Theorems & Definitions (7)

  • Definition 1
  • Definition 2
  • Theorem 1: celledoni2010energy
  • Example 1: Bushy trees
  • Example 2: Tall trees
  • Example 3: Self-conjugate trees
  • Remark 1