[Paper Review] Continuous-stage Runge-Kutta-NystrÖm methods

Wensheng Tang|arXiv (Cornell University)|Jul 9, 2018

Numerical methods for differential equations40 references4 citations

TL;DR

This paper introduces a novel framework for continuous-stage Runge-Kutta-Nyström (csRKN) methods by incorporating a weight function and generalizing the integration interval to any finite or infinite domain. By leveraging orthogonal polynomial expansions and simplifying assumptions, the method enables systematic construction of high-order symplectic and symmetric integrators for second-order differential equations, with numerical results confirming near-conservation of energy and invariants in Hamiltonian systems.

ABSTRACT

We develop continuous-stage Runge-Kutta-NystrÖm (csRKN) methods in this paper. By leading weight function into the formalism of csRKN methods and modifying the original pattern of continuous-stage methods, we establish a new and larger framework for csRKN methods and it enables us to derive more effective RKN-type methods. Particularly, a variety of classical weighted orthogonal polynomials can be used in the construction of RKN-type methods. As an important application, new families of symmetric and symplectic integrators can be easily acquired in such framework. Numerical experiments have verified the effectiveness of the new integrators presented in this paper.

Motivation & Objective

• To extend the classical csRKN framework by introducing a weight function and generalizing the integration interval to any finite or infinite domain.
• To enable the use of classical weighted orthogonal polynomials in constructing RKN-type methods for improved accuracy and stability.
• To develop a systematic approach for deriving high-order symplectic and symmetric integrators using simplifying assumptions and orthogonal expansions.
• To demonstrate the effectiveness of the new integrators in preserving geometric invariants in Hamiltonian systems through numerical experiments.

Proposed method

• Introduce a weight function into the csRKN formalism, transforming the discrete stage set into a continuous interval with a measure defined by the weight function.
• Define the new csRKN method using a general interval $ I $, allowing for the use of classical weighted orthogonal polynomials such as Hermite, Legendre, and Jacobi polynomials.
• Apply simplifying assumptions to the order conditions, enabling systematic derivation of high-order methods without solving complex nonlinear algebraic equations.
• Utilize orthogonal polynomial expansions to represent Butcher coefficients as smooth functions, facilitating analytical treatment and integration of geometric properties.
• Construct symplectic and symmetric integrators by ensuring the coefficient functions satisfy specific algebraic conditions derived from the simplifying assumptions.
• Implement the methods numerically using collocation and Galerkin-type approaches, with error analysis based on maximum and Euclidean norms.

Experimental results

Research questions

• RQ1 How can the classical csRKN framework be generalized to allow for a broader class of orthogonal polynomial bases?
• RQ2 Can the incorporation of a weight function in the csRKN formulation lead to more effective and flexible RKN-type integrators?
• RQ3 To what extent can symplectic and symmetric integrators be systematically derived using orthogonal polynomial expansions and simplifying assumptions?
• RQ4 How well do the new integrators preserve geometric invariants such as energy, angular momentum, and Runge-Lenz-Pauli vector in Hamiltonian systems?
• RQ5 What is the numerical performance of the new integrators in long-time simulations of chaotic and non-chaotic Hamiltonian problems?

Key findings

• The new csRKN framework successfully incorporates classical weighted orthogonal polynomials—such as Hermite, Legendre, and Jacobi—into the construction of RKN-type methods.
• The Hermite-4 method demonstrated superior accuracy among the tested integrators, while the Hermite-3 method showed lower performance due to its lower order.
• All symplectic integrators exhibited near-conservation of the Hamiltonian and Runge-Lenz-Pauli invariant, with the quadratic angular momentum preserved up to machine precision.
• Numerical orbits remained bounded within the expected region for the Hénon-Heiles problem, confirming the correct qualitative behavior without escape from the equilateral triangle domain.
• Solution errors in both position and momentum variables grew linearly over time, consistent with expected behavior for symplectic integrators.
• The framework enables the derivation of high-order symplectic and symmetric integrators without solving complex nonlinear order conditions, thanks to simplifying assumptions and orthogonal expansions.