[Paper Review] Stabilized exponential-SAV schemes preserving energy dissipation law and maximum bound principle for the Allen-Cahn type equations

Lili Ju, Xiao Li+1 more|arXiv (Cornell University)|Mar 14, 2022

Solidification and crystal growth phenomena4 citations

TL;DR

This paper proposes first- and second-order stabilized exponential scalar auxiliary variable (sESAV) schemes for Allen–Cahn type equations that simultaneously preserve the discrete energy dissipation law and the maximum bound principle (MBP). By combining exponential time integration with a novel stabilization technique, the schemes ensure unconditional energy stability and strict solution bounds, with optimal error estimates and extensive numerical validation.

ABSTRACT

It is well-known that the Allen-Cahn equation not only satisfies the energy dissipation law but also possesses the maximum bound principle (MBP) in the sense that the absolute value of its solution is pointwise bounded for all time by some specific constant under appropriate initial/boundary conditions. In recent years, the scalar auxiliary variable (SAV) method and many of its variants have attracted much attention in numerical solution for gradient flow problems due to their inherent advantage of preserving certain discrete analogues of the energy dissipation law. However, existing SAV schemes usually fail to preserve the MBP when applied to the Allen-Cahn equation. In this paper, we develop and analyze new first- and second-order stabilized exponential-SAV schemes for a class of Allen-Cahn type equations, which are shown to simultaneously preserve the energy dissipation law and MBP in discrete settings. In addition, optimal error estimates for the numerical solutions are rigorously obtained for both schemes. Extensive numerical tests and comparisons are also conducted to demonstrate the performance of the proposed schemes.

Motivation & Objective

• To develop numerical schemes that preserve both the energy dissipation law and the maximum bound principle (MBP) in time-discrete settings for Allen–Cahn type equations.
• To overcome the limitation of standard SAV schemes, which typically fail to preserve MBP, by introducing a stabilized exponential-SAV framework.
• To establish rigorous stability and convergence analysis for the proposed schemes, ensuring optimal error bounds.
• To demonstrate the effectiveness of the schemes through extensive numerical experiments on long-time coarsening dynamics and phase separation processes.
• To extend the applicability of SAV-based methods to gradient flows by ensuring both energy stability and solution boundedness via a novel stabilization mechanism.

Proposed method

• Adopt the exponential scalar auxiliary variable (ESAV) approach to reformulate the energy functional, ensuring a positive coefficient for the nonlinear term in the discrete scheme.
• Introduce a stabilization term by adding and subtracting a linear term in the scheme, which is crucial for proving MBP preservation and does not alter the original energy structure.
• Construct first- and second-order time discretizations using a one-step method framework, enabling the use of adaptive time-stepping strategies.
• Prove that the energy dissipation and MBP preservation are independent and can be established in parallel, unlike in classical stabilized schemes.
• Use a semi-implicit treatment of the nonlinear term with exponential time integration to improve stability and accuracy.
• Derive optimal error estimates in the $L^2$ and $L^rown$ norms through rigorous analysis, validated numerically.

Experimental results

Research questions

• RQ1 Can a SAV-based scheme be designed to preserve both the energy dissipation law and the maximum bound principle (MBP) simultaneously in discrete time?
• RQ2 How can the inherent instability of standard SAV schemes in preserving solution bounds be corrected while maintaining energy stability?
• RQ3 What is the role of the exponential time integration and stabilization term in enabling independent proofs of energy dissipation and MBP preservation?
• RQ4 Can optimal error estimates be rigorously derived for such stabilized exponential-SAV schemes?
• RQ5 How do the proposed schemes perform in long-time simulations of phase separation and coarsening dynamics?

Key findings

• The proposed sESAV schemes preserve the discrete energy dissipation law unconditionally for both first- and second-order schemes.
• The schemes strictly enforce the maximum bound principle (MBP), ensuring that the solution remains bounded by $\beta$ in absolute value for all time when the initial data satisfies $|u_{\text{init}}| \leq \beta$.
• Optimal error estimates of order $\mathcal{O}(\tau + h^2)$ for the first-order scheme and $\mathcal{O}(\tau^2 + h^2)$ for the second-order scheme are rigorously established.
• Numerical results show that the sESAV2 scheme with $\tau = 0.01$ accurately captures long-time coarsening dynamics, reaching steady states at $t \approx 604$ for the double-well potential and $t \approx 602$ for the Flory–Huggins potential.
• The solution remains bounded in the $L^\infty$ norm throughout the simulation, confirming MBP preservation, while the energy decays monotonically, verifying energy dissipation.
• The schemes are robust and produce results comparable to the IFRK4 scheme, with no nonphysical oscillations or violations of solution bounds.