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Symmetric integrators based on continuous-stage Runge-Kutta-Nystrom methods for reversible systems

Wensheng Tang, Jingjing Zhang

TL;DR

The paper develops symmetric integrators for reversible second-order ODEs by leveraging continuous-stage Runge-Kutta-Nyström (csRKN) methods and Legendre polynomial expansions. Through symmetry constraints and Gauss/Lobatto-type quadratures, it constructs new symmetric (and often symplectic) RKN-type schemes and analyzes their order via csRKN theory. Numerical experiments on a reversible perturbed pendulum illustrate how symmetry interacts with energy preservation, highlighting that symplecticity typically yields superior long-time behavior for Hamiltonian-like systems. The work provides a flexible framework for designing high-order, reversibility-preserving integrators suitable for reversible dynamics and Hamiltonian-type problems.

Abstract

In this paper, we study symmetric integrators for solving second-order ordinary differential equations on the basis of the notion of continuous-stage Runge-Kutta-Nystrom methods. The construction of such methods heavily relies on the Legendre expansion technique in conjunction with the symmetric conditions and simplifying assumptions for order conditions. New families of symmetric integrators as illustrative examples are presented. For comparing the numerical behaviors of the presented methods, some numerical experiments are also reported.

Symmetric integrators based on continuous-stage Runge-Kutta-Nystrom methods for reversible systems

TL;DR

The paper develops symmetric integrators for reversible second-order ODEs by leveraging continuous-stage Runge-Kutta-Nyström (csRKN) methods and Legendre polynomial expansions. Through symmetry constraints and Gauss/Lobatto-type quadratures, it constructs new symmetric (and often symplectic) RKN-type schemes and analyzes their order via csRKN theory. Numerical experiments on a reversible perturbed pendulum illustrate how symmetry interacts with energy preservation, highlighting that symplecticity typically yields superior long-time behavior for Hamiltonian-like systems. The work provides a flexible framework for designing high-order, reversibility-preserving integrators suitable for reversible dynamics and Hamiltonian-type problems.

Abstract

In this paper, we study symmetric integrators for solving second-order ordinary differential equations on the basis of the notion of continuous-stage Runge-Kutta-Nystrom methods. The construction of such methods heavily relies on the Legendre expansion technique in conjunction with the symmetric conditions and simplifying assumptions for order conditions. New families of symmetric integrators as illustrative examples are presented. For comparing the numerical behaviors of the presented methods, some numerical experiments are also reported.

Paper Structure

This paper contains 8 sections, 10 theorems, 43 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Continuous-stage RKN method and its order theory
  3. Continuous-stage RKN method
  4. Order theory for RKN-type method
  5. Conditions for the symmetry of csRKN methods
  6. Symmetric RKN method
  7. Numerical experiments
  8. Concluding remarks

Key Result

Theorem 1.1

Stofferz95vsf A Runge-Kutta method or a Runge-Kutta-Nyström (RKN) method is reversible iff it is symmetric.

Figures (3)

  • Figure 5.1: Global errors of the numerical solutions by RKN-IIIB method(black line), RKN-Diagsymp method (blue line), RKN-A method (red line) and RKN-B method (green line) for the perturbed pendululm equation \ref{['eq:second1']}. The reference line has slope 4 in every subplots.
  • Figure 5.2: Energy errors of the numerical solutions by RKN-IIIB method(black line), RKN-Diagsymp method (blue line), RKN-A method (red line) and RKN-B method (green line) for the perturbed pendululm equation \ref{['eq:second1']}: step size $h=0.16,$ integration interval $[0, 1600].$
  • Figure 5.3: Energy errors of the numerical solutions by RKN-Diagsymp method (blue line), and RKN-A method (red line) for the perturbed pendululm equation \ref{['eq:second1']}: step size $h=0.16,$ integration interval $[0, 1.6 \times 10^5].$

Theorems & Definitions (20)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Definition 3.1
  • Theorem 3.2
  • proof
  • ...and 10 more