Structure-Preserving SAV-RK$s$ Spectral Methods for Nonlinear Hamiltonian System

Abstract

It is well-established that high-order symplectic Runge–Kutta methods cannot preserve polynomial energy of Hamiltonian systems beyond degree two. However, the scalar auxiliary variable (SAV) approach, which was originally proposed for gradient flow systems, offers an effective strategy to overcome this limitation, albeit with a modified energy. In this paper, we study high-order Runge–Kutta methods for solving general Hamiltonian systems without imposing a lower bound on the high-order part of the energy functional. By integrating a symplectic Runge–Kutta method with a new SAV strategy, we develop a class of high-order nonlinear structure-preserving $s$-stage Runge–Kutta SAV (SAV-RK$s$) spectral methods, and prove that the nonlinear semi-discrete scheme can preserve the Hamiltonian energy and other conservative quantities such as mass/momentum (if they exist). In addition, we design a tailored fast Newton iteration algorithm to efficiently solve the resulting nonlinear algebraic system. Finally, we carry out extensive numerical simulations on several benchmark problems, including cases where the energy functional is either bounded or unbounded, to validate the accuracy and robustness of the proposed algorithms.