Improving the accuracy and consistency of the energy quadratization method with an energy-optimized technique

TL;DR

The study addresses the mismatch between discrete and original energy dissipation in invariant energy quadratization (IEQ) methods for gradient-flow models. It introduces an energy-optimized IEQ (EOP-IEQ) framework that corrects auxiliary variables to align the discrete modified energy with the original energy, while preserving linearity and unconditional energy stability. The authors develop Crank-Nicolson and BDF$k$ time-stepping schemes within the EOP-IEQ and extend them to multi-variable energy forms, proving energy-dissipation guarantees and showing consistency with the original energy. Numerical experiments on Allen-Cahn, Cahn-Hilliard, phase-field crystal, and molecular-beam-epitaxy models demonstrate that EOP-IEQ yields higher accuracy and energy fidelity than baseline IEQ and REQ approaches, with accurate long-time dynamics and preserved mass. The approach provides a broadly applicable, efficient pathway to energy-stable simulations for a wide class of thermodynamically consistent gradient-flow problems.

Abstract

We propose an energy-optimized invariant energy quadratization method to solve the gradient flow models in this paper, which requires only one linear energy-optimized step to correct the auxiliary variables on each time step. In addition to inheriting the benefits of the baseline and relaxed invariant energy quadratization method, our approach has several other advantages. Firstly, in the process of correcting auxiliary variables, we can directly solve linear programming problem by the energy-optimized technique, which greatly simplifies the nonlinear optimization problem in the previous relaxed invariant energy quadratization method. Secondly, we construct new linear unconditionally energy stable schemes by applying backward Euler formulas and Crank-Nicolson formula, so that the accuracy in time can reach the first- and second-order. Thirdly, comparing with relaxation technique, the modified energy obtained by energy-optimized technique is closer to the original energy, meanwhile the accuracy and consistency of the numerical solutions can be improved. Ample numerical examples have been presented to demonstrate the accuracy, efficiency and energy stability of the proposed schemes.

Figures (12)

• Figure 4.1: Example 1A. Errors and convergence rates in the $L^2$ norm of the numerical solution $\phi$ and the auxiliary variable $q$ for AC equation by using various first- and second-order schemes.
• Figure 4.2: Example 2A. Profiles of the phase variable $\phi$ are taken at $t=0.1, 1.5, 9$.
• Figure 4.3: Example 2A. A comparison between the baseline IEQ/CN scheme, REQ/CN scheme and EOP-IEQ/CN scheme for solving the AC equation. (a) Normalized numerical energy comparisons of the three schemes; (b) Evolution of the the errors between modified energies and reference energy of the three schemes; (c) Time evolution of energy-optimized parameters $\lambda_1^n$ in EOP-IEQ/CN scheme and relaxation parameters $\xi^{n}$ in REQ/CN scheme.
• Figure 4.4: Example 2A. Errors and convergence rates in the $L^2$ norm of the numerical solution $\phi$ and auxiliary variable $q$ for CH equation using various first- and second-order schemes.
• Figure 4.5: Example 2A. Profiles of the phase variable $\phi$ are taken at $t=1, 10, 90$.

Theorems & Definitions (21)

• Remark 2.1
• Lemma 2.1: YANG2016294
• Remark 2.2: Optimal choice for $\xi^{n+1}$ ZHAO2021107331
• Lemma 2.2: ZHAO2021107331
• Remark 3.1
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3