Motivated by the vast number of applications of higher-order partial differential equations representing gradient flow, this contribution proposes an approach to solve perturbed linear systems that arise during an implicit time integration if Fourier-spectral methods are employed for the spatial discretization. The proposed approach is derived from the Lippmann–Schwinger framework, commonly applied to determine the effective response of composite materials with a Fourier spectral discretization [1]. An auxiliary problem is formulated by approximating the nonlinear components of the tangent matrix within the implicit solution of the gradient flow equations. A conjugate gradient (CG) solver is subsequently employed on the perturbed system to determine the updates. The perturbed system guarantees a better condition number and convergence acceleration while maintaining a low computational overhead. To characterize the performance gains, this study investigates the spectral properties of the tangent matrices, specifically their condition numbers and eigenvalue distributions in a standard one dimensional system. The requisite modifications to the update schemes are detailed for application to other higher-order gradient flow equations like the phase-field crystal equation and the Swift–Hohenberg equation. Furthermore, the framework’s extensibility to multiphase systems is demonstrated through the solution of the ternary Cahn–Hilliard equation. Finally, several numerical examples are presented to demonstrate the superior computational efficiency of the proposed scheme through a comparative analysis with existing state-of-the-art methods [2, 3]. REFERENCES [1] H. Moulinec, P. Suquet, A numerical method for computing the overall response of nonlinear composites with complex microstructure, Computer Methods in Applied Mechanics and Engineering, 157, p. 69–94, 1998. [2] A. Christlieb, J. Jones, K. Promislow, B. Wetton, and M. Willoughby, High accuracy solutions to energy gradient flows from material science models, Journal of Computational Physics, 257.PA, pp. 193–215, 2014. [3] A. Krischok, B. Yaraguntappa, and M. -A. Keip, Fast implicit update schemes for Cahn–Hilliard-type gradient flow in the context of Fourier-spectral method, Computer Methods in Applied Mechanics and Engineering, 431, p. 117220, 2024.