Global-in-time energy stability: a powerful analysis tool for the gradient flow problem without maximum principle or Lipschitz assumption

TL;DR

The paper tackles the challenge of proving energy dissipation for gradient flows without relying on Lipschitz or maximum-principle assumptions. It introduces global-in-time energy stability and establishes it for the Swift–Hohenberg equation using a second-order exponential Runge–Kutta scheme with linear stabilization $L_{\kappa}$, ensuring original energy dissipation under a suitable time-step bound. By deriving $\ell^2$, $H_h^2$, and $\ell^{\infty}$ estimates at all ERK stages and employing discrete Sobolev embeddings, the authors obtain a global-in-time energy-stability result and an optimal $L^2$-error estimate that is robust for long-time simulations. Numerical experiments demonstrate convergence, energy dissipation, and long-time dynamics, highlighting the method’s efficiency and reliability for SH-type gradient flows and suggesting broader applicability to related gradient-flow problems in materials science.

Abstract

Before proving (unconditional) energy stability for gradient flows, most existing studies either require a strong Lipschitz condition regarding the non-linearity or certain $L^{\infty}$ bounds on the numerical solutions (the maximum principle). However, proving energy stability without such premises is a very challenging task. In this paper, we aim to develop a novel analytical tool, namely global-in-time energy stability, to demonstrate energy dissipation without assuming any strong Lipschitz condition or $L^{\infty}$ boundedness. The fourth-order-in-space Swift-Hohenberg equation is used to elucidate the theoretical results in detail. We also propose a temporal second-order accurate scheme for efficiently solving such a strongly stiff equation. Furthermore, we present the corresponding optimal $L^2$ error estimate and provide several numerical simulations to demonstrate the dynamics.

Figures (3)

• Figure 1: Convergence of the first- and second-order Fourier pseudo-spectral schemes in time with fixed $\tau$ (left) and $\tau^2$ (right) for the 2-D SH equation. Top: $\varepsilon=0.25$. Bottom: $\varepsilon=0.025$. It is seen that the numerical error magnitude and computational cost of the ERK(2,2) scheme are smaller than those of both the ETDRK2 and IMEX-RK(2,2) schemes, although they share the same convergence order.
• Figure 2: The evolution of the original energy using ERK(2,2) with different time-step sizes (left) and a fixed size $\tau=0.1$ with various energy-stable methods (right) is shown. It is observed that, while only the red dashed line ($\tau=0.1$) approximately matches the reference line, the other two lines also exhibit the same energy-decreasing trend and eventually converge to the same steady state. From the right subplot, it can be seen that all methods maintain discrete energy stability over extended periods, but ERK(2,2) reaches the steady state more rapidly
• Figure 3: Evolution of 2-D polycrystal growth in a supercooled liquid at $T = 16,~40,~72,~96,~120$, and $160$ computed by ERK(2,2). It can be seen that three different nuclei grains grow and eventually become sufficiently large to form grain boundaries.

Theorems & Definitions (19)

• Theorem 1.1: Original energy stability
• Theorem 1.2: Global-in-time energy stability
• Definition 2.1
• Definition 2.2
• Proposition 2.1
• Remark 2.1
• Remark 2.2
• proof
• Lemma 3.1
• Proposition 3.1