[Paper Review] Chebyshev symplectic methods based on continuous-stage Runge-Kutta methods

Wensheng Tang|arXiv (Cornell University)|May 29, 2018

Numerical methods for differential equations21 references3 citations

TL;DR

This paper introduces a novel family of symplectic Runge-Kutta methods based on Chebyshev orthogonal polynomials of the first and second kind, constructed via a new framework of weighted continuous-stage Runge-Kutta methods. The proposed methods achieve high-order accuracy and preserve the symplectic structure exactly, demonstrating superior long-term stability and energy conservation in numerical experiments on Hamiltonian systems, outperforming standard Gauss-Legendre methods in qualitative behavior despite slightly higher initial error growth.

ABSTRACT

We develop Chebyshev symplectic methods based on Chebyshev orthogonal polynomials of the first and second kind separately in this paper. Such type of symplectic methods can be conveniently constructed with the newly-built theory of weighted continuous-stage Runge-Kutta methods. A few numerical experiments are well performed to verify the efficiency of our new methods.

Motivation & Objective

• To develop new symplectic integrators for Hamiltonian systems that preserve geometric structure over long time intervals.
• To extend the theory of continuous-stage Runge-Kutta methods to construct symplectic methods using Chebyshev polynomials of the first and second kind.
• To provide an alternative to the classical W-transformation technique, which is limited to Legendre polynomials, by enabling construction with arbitrary weighted orthogonal polynomials.
• To validate the efficiency and geometric accuracy of the proposed methods through numerical experiments on perturbed Kepler problems.

Proposed method

• Leverages the newly developed theory of weighted continuous-stage Runge-Kutta (csRK) methods to construct symplectic integrators.
• Uses Chebyshev polynomials of the first and second kind as basis functions to define the coefficients of the csRK method.
• Applies simplifying assumptions (B̌(ξ), Č(η), Ď(ζ)) to ensure high-order accuracy and symplecticity.
• Employs interpolatory quadrature rules with Chebyshev abscissae to ensure exact symplecticity in the discrete flow.
• Derives specific Butcher-type tables for 3-stage 4th-order and 5-stage 6th-order methods based on orthogonal polynomial roots.
• Implements the methods using continuous-stage formulations with functions Aτ,σ, Bτ, and Cτ satisfying integral conditions for order and symplecticity.

Experimental results

Research questions

• RQ1 Can symplectic integrators be systematically constructed using Chebyshev polynomials within the continuous-stage Runge-Kutta framework?
• RQ2 How does the performance of Chebyshev-based symplectic methods compare to classical Gauss-Legendre methods in terms of energy conservation and error growth?
• RQ3 Can the framework be generalized beyond Legendre polynomials, such as to Chebyshev polynomials, to generate new symplectic methods?
• RQ4 What is the order and stability behavior of the resulting symplectic methods when applied to stiff or perturbed Hamiltonian systems?

Key findings

• The proposed Chebyshev symplectic methods of order 4 and 6 exhibit linear error growth and bounded energy error over long integration intervals, confirming their geometric accuracy.
• Numerical results show that the Chebyshev I and II order 4 methods perform comparably to the 4th-order Gauss-Legendre method, with slightly higher initial solution error but similar energy conservation.
• The 6th-order Chebyshev symplectic methods (both I and II) demonstrate nearly identical performance to the 6th-order Gauss-Legendre method in both solution error and energy conservation.
• The methods are exactly symplectic, as confirmed by the preservation of the symplectic structure in the discrete flow, unlike spectral collocation methods that only approximate it.
• The construction framework is general and can be extended to other weighted orthogonal polynomials beyond Chebyshev, such as Jacobi or Legendre polynomials.