Asymptotically compatible energy of variable-step fractional BDF2 formula for time-fractional Cahn-Hilliard model

TL;DR

The paper addresses time-fractional Cahn-Hilliard dynamics with Caputo derivatives of order $α∈(0,1)$ by designing a high-order, variable-step FBDF2 scheme that preserves a discrete energy law. A local-nonlocal splitting yields a discrete gradient structure for both local and nonlocal components, enabling energy dissipation at each time step and asymptotic compatibility with the classical CH energy as $α\to1^-$. Under step-ratio constraints, the scheme is unconditionally solvable with a convex energy framework, and numerical tests show $O(τ^{3-α})$ accuracy and effective adaptive stepping for long-time simulations. The discrete energy $E_α$ converges to the Ginzburg-Landau energy $E$ and the discrete energy law converges to its integer-order counterpart, ensuring practical reliability and relevance for multiscale diffusion processes.

Abstract

A new discrete energy dissipation law of the variable-step fractional BDF2 (second-order backward differentiation formula) scheme is established for time-fractional Cahn-Hilliard model with the Caputo's fractional derivative of order $α\in(0,1)$, under a weak step-ratio constraint $0.4753\le τ_k/τ_{k-1}<r^*(α)$, where $τ_k$ is the $k$-th time-step size and $r^*(α)\ge4.660$ for $α\in(0,1)$.We propose a novel discrete gradient structure by a local-nonlocal splitting technique, that is, the fractional BDF2 formula is split into a local part analogue to the two-step backward differentiation formula of the first derivative and a nonlocal part analogue to the L1-type formula of the Caputo's derivative. More interestingly, in the sense of the limit $α\rightarrow1^-$, the discrete energy and the corresponding energy dissipation law are asymptotically compatible with the associated discrete energy and the energy dissipation law of the variable-step BDF2 method for the classical Cahn-Hilliard equation, respectively. Numerical examples with an adaptive stepping procedure are provided to demonstrate the accuracy and the effectiveness of our proposed method.

Figures (5)

• Figure 1: Solution curves of $g_1(r^*,\alpha)=0$ and $g_1(2^{\gamma_{\max}}-1,\alpha)=0$.
• Figure 2: Log-log plots of time accuracy for two different fractional orders.
• Figure 3: Energy curves by uniform step and adaptive strategy with different parameters $\eta$.
• Figure 4: Energy curves of Example 2 with different fractional orders $\alpha$.
• Figure 5: Snapshots of dynamic coarsening processes for different fractional orders $\alpha$.

Theorems & Definitions (28)

• Lemma 2.1
• proof
• Remark 1
• Remark 2
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Theorem 2.1
• proof