[Paper Review] Energy-preserving continuous-stage partitioned Runge-Kutta methods

Wensheng Tang|arXiv (Cornell University)|Aug 7, 2018

Numerical methods for differential equations42 references3 citations

TL;DR

This paper proposes energy-preserving continuous-stage partitioned Runge-Kutta (csPRK) methods for Hamiltonian systems by deriving a sufficient condition for energy preservation. The method ensures exact energy conservation by requiring both Butcher weight coefficients $B_\tau$ and $\widehat{B}_\tau$ to equal 1 when using normalized shifted Legendre polynomials and simplifying assumptions, leading to high-order integrators verified through numerical experiments on nonlinear systems with near-machine-precision energy conservation.

ABSTRACT

In this paper, we present continuous-stage partitioned Runge-Kutta (csPRK) methods for energy-preserving integration of Hamiltonian systems. A sufficient condition for the energy preservation of the csPRK methods is derived. It is shown that the presented condition contains the existing condition for energy-preserving continuous-stage Runge-Kutta methods as a special case. A noticeable and interesting result is that when we use the simplifying assumptions of order conditions and the normalized shifted Legendre polynomials for constructing high-order energy-preserving csPRK methods, both the Butcher "weight" coefficients $B_τ$ and $\widehat{B}_τ$ must be equal to $1$. As illustrative examples, new energy-preserving integrators are acquired by virtue of the presented condition, and for the sake of verifying our theoretical results, some numerical experiments are reported.

Motivation & Objective

• To develop energy-preserving numerical integrators for Hamiltonian systems that maintain the total energy exactly over long-time simulations.
• To extend the theory of continuous-stage Runge-Kutta methods to partitioned formulations for better geometric structure preservation.
• To establish a sufficient condition for energy preservation in csPRK methods that generalizes existing results for continuous-stage Runge-Kutta methods.
• To construct high-order energy-preserving integrators using simplifying assumptions and normalized shifted Legendre polynomials, ensuring both $B_\tau = 1$ and $\widehat{B}_\tau = 1$.
• To numerically validate the theoretical findings through experiments on nonlinear Hamiltonian systems, comparing energy and invariant preservation with symplectic methods.

Proposed method

• Derives a sufficient condition for energy preservation in continuous-stage partitioned Runge-Kutta (csPRK) methods using series expansion techniques.
• Applies simplifying assumptions for order conditions and uses normalized shifted Legendre polynomials to construct high-order integrators.
• Imposes the constraint that both Butcher weight coefficients $B_\tau$ and $\widehat{B}_\tau$ must equal 1 to ensure energy preservation.
• Reformulates the method as a nonlinear Hamiltonian system to enable numerical testing and validation.
• Employs 3-point Gaussian quadrature for numerical integration and compares results with the 2-stage 4th-order Gauss-Legendre RK method (GLRK-4).
• Evaluates performance using maximum norm errors for global solution error and invariant errors in $H$, $I$, and $L$.

Experimental results

Research questions

• RQ1 Can a sufficient condition for energy preservation be derived for continuous-stage partitioned Runge-Kutta methods in Hamiltonian systems?
• RQ2 How do the Butcher weight coefficients $B_\tau$ and $\widehat{B}_\tau$ behave under simplifying assumptions and Legendre polynomial-based construction for high-order energy-preserving methods?
• RQ3 To what extent do the proposed csPRK methods preserve energy and other invariants compared to symplectic methods like GLRK-4?
• RQ4 Can the derived energy-preserving csPRK methods achieve linear error growth and accurate long-term dynamics in chaotic or stiff systems?
• RQ5 Is the proposed method applicable beyond standard Runge-Kutta frameworks, particularly in the context of P-series integrators?

Key findings

• The derived sufficient condition for energy preservation in csPRK methods generalizes the existing condition for continuous-stage Runge-Kutta methods.
• When using simplifying assumptions and normalized shifted Legendre polynomials, both $B_\tau$ and $\widehat{B}_\tau$ must be exactly 1 for energy preservation.
• Numerical experiments show that all proposed energy-preserving methods conserve the Hamiltonian $H$ up to machine precision, while symplectic GLRK-4 preserves the quadratic angular momentum $I$ exactly.
• The energy-preserving methods exhibit linear error growth in global solution error and invariant errors, comparable to the symplectic method.
• All numerical orbits from the energy-preserving methods closely approximate the exact elliptical trajectory, indicating high accuracy and stability.
• The proposed csPRK methods perform comparably to symplectic methods in preserving geometric and dynamical properties, despite not being symplectic.