Energy-conserving successive secant-difference method for nonlinear wave equations

Abstract

We present a high-order time-discretization method that conserves energy in nonlinear wave equations. The method builds on the secant-difference approach, which inherently ensures energy conservation. To achieve higher temporal accuracy while preserving the structural properties necessary for energy preservation, we propose a novel high-order multi-stage method based on a successive strategy. While the successive secant-difference (SSD) method deviates from the standard Runge–Kutta framework, we derive the order conditions for high-order accuracy using classical Taylor expansion techniques. We further derive the full set of order conditions up to sixth-order and determine the minimal independent constraints by examining their linear dependencies. This derivation provides detailed insights and introduces a novel perspective distinct from traditional composition methods. We validate the temporal accuracy and unconditional energy stability of our proposed method through numerical examples, including the sine-Gordon and Boussinesq equations. We also provide numerical comparisons with fourth-order averaged vector field methods. © 2026 IMACS.