Second-Order Flows for Approaching Stationary Points via the SAV-E Approach

Abstract

Second-order flows represent an emerging computational framework for approaching stationary points of variational energies, characterized by damped hyperbolic partial differential equations. In this paper, we develop a scalar auxiliary variable (SAV) method for this class of flows, referred to as SAV-E. The resulting scheme is linear and decoupled, and we prove unconditional modified-energy stability. We further establish subsequential convergence to stationary points for the Ginzburg-Landau and conserved Swift-Hohenberg models under suitable assumptions. Numerical experiments support the analysis and demonstrate robustness and efficiency compared with representative baselines and the corresponding gradient flows.