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The exponential scalar auxiliary variable (E-SAV) approach for phase field models and its explicit computing
Zhengguang Liu, Xiaoli Li
TL;DR
The paper addresses SAV’s bounded-below energy assumption and develops an exponential scalar auxiliary variable approach for energy-stable phase-field schemes. E-SAV supports unconditional stability and fully explicit treatment of the auxiliary variable, while numerical comparisons report smaller errors and roughly half the CPU time of SAV.
Problem
The SAV approach requires the nonlinear free energy E1(φ) to be bounded from below, and the chosen constant C can influence simulation accuracy.
Method
The paper introduces an exponential scalar auxiliary variable approach, including modified and multiple E-SAV formulations for phase-field models.
Results
E-SAV schemes are unconditionally energy stable and can discretize the auxiliary variable with the nonlinear term totally explicitly; for Allen–Cahn, E-SAV preserves first-order accuracy with smaller error and about half the CPU time of SAV.
Takeaways & Limitations
E-SAV provides a more applicable and efficient route to energy-stable phase-field schemes than SAV, including explicit computation and extensions to models with multiple unknown variables.
Takeaways & Limitations
The exponential function is rapidly increasing, creating a risk of failure that motivates adding a positive constant C in the modified E-SAV variable.
Abstract
from arXiv · showhide
In this paper, we consider an exponential scalar auxiliary variable (E-SAV) approach to obtain energy stable schemes for a class of phase field models. This novel auxiliary variable method based on exponential form of nonlinear free energy potential is more effective and applicable than the traditional SAV method which is very popular to construct energy stable schemes. The first contribution is that the auxiliary variable without square root removes the bounded from below restriction of the nonlinear free energy potential. Then, we prove the unconditional energy stability for the semi-discrete schemes carefully and rigorously. Another contribution is that we can discrete the auxiliary variable combined with the nonlinear term totally explicitly. Such modification is very efficient for fast calculation. Furthermore, the positive property of $r$ can be guaranteed which is very important and reasonable for the models' equivalence. Besides, for complex phase field models with two or more unknown variables and nonlinear terms, we construct a multiple E-SAV (ME-SAV) approach to enhance the applicability of the proposed E-SAV approach. A comparative study of classical SAV and E-SAV approaches is considered to show the accuracy and efficiency. Finally, we present various 2D numerical simulations to demonstrate the stability and accuracy.
1. Introduction.
The paper introduces E-SAV schemes for phase field models to preserve energy dissipation while removing key SAV restrictions and enabling efficient explicit computation. It establishes unconditional stability and extends the approach to complex models through multiple auxiliary variables.
- Energy dissipation is essential for phase-field numerical schemes, motivating methods that preserve dissipative properties after time discretization.
- E-SAV removes the bounded-below assumption on the nonlinear free energy potential.
- SAV schemes are linear, second-order, and unconditionally energy stable, but require the nonlinear free energy to be bounded from below.SAV also requires choosing a sufficiently large positive constant, which can influence simulation accuracy.
- E-SAV guarantees positivity of its auxiliary variables r and r^n, unlike the SAV approach.
- E-SAV supports totally explicit treatment of the auxiliary variable and nonlinear term while retaining unconditional energy stability.
- E-SAV computes φ and the auxiliary variable step-by-step and dissipates the original energy rather than a modified energy.
- The paper develops multiple E-SAV for complex phase-field models and uses numerical simulations to assess stability, accuracy, and efficiency.The paper also introduces a modified technique because the exponential function can grow rapidly and risk E-SAV failure.
2. E-SAV approach for phase field models.
The E-SAV reformulation uses an exponential auxiliary variable to produce linear, unconditionally energy-stable schemes while preserving positivity and simplifying computation. First- and second-order schemes can be implemented sequentially, with explicit nonlinear evaluations and decoupled updates.
- E-SAV formulation: The exponential auxiliary variable r remains positive and makes the modified energy law equal to the original energy dissipation law.The equivalence follows from ln(r) = E1(φ).
- Energy-stable schemes: E-SAV schemes can be linear, unconditionally energy stable, and constructed at both first and second order.The second-order Crank–Nicolson scheme is reported as second-order accurate and unconditionally energy stable.
- Efficient implementation: The first-order scheme computes φn+1 directly and then obtains rn+1, decoupling the phase variable and auxiliary variable updates.This avoids the inner-product calculation required by the classical SAV algorithm.
- Second-order discretization: The second-order scheme uses explicit O(∆t^2) approximations at the time midpoint and admits sequential computation of φ and r.The construction includes explicit midpoint approximations for both variables.
3. E-SAV approach for phase field models of several functions.
For phase-field systems with several functions, the E-SAV approach extends to coupled nonlinear terms and yields sequential, linear, second-order, unconditionally stable schemes. The resulting updates avoid the inner products needed by classical SAV computations.
- Multiple-function formulation: The multiple-function E-SAV formulation uses a shared exponential auxiliary variable for energies involving several phase fields.The formulation assumes a symmetric positive definite coupling matrix and a self-adjoint non-negative linear operator.
- Energy-stable discretization: A linear, second-order, sequentially solved, unconditionally stable Crank–Nicolson E-SAV scheme is constructed for the multiple-function system.Its discrete energy law is non-increasing.
- Implementation: The multiple-function scheme computes the phase variables and auxiliary variable sequentially rather than requiring coupled inner-product calculations.The implementation proceeds through initial values, midpoint quantities, phase updates, and auxiliary-variable updates.
- Comparison with SAV: The authors state that E-SAV simplifies calculation for multiple-function models by avoiding inner products needed before obtaining the new phase state in classical SAV.This is presented as beneficial for efficient computation.
- Scope extension: The rapidly increasing exponential function creates a failure risk, motivating a modified E-SAV variable with an added positive constant.This modification is introduced to improve the approach’s scope of application.
4. Modified E-SAV approach.
The modified E-SAV approach addresses numerical risk from exponential growth by adding a positive constant to the auxiliary-variable definition. The paper supplies a practical constant choice and retains an energy-dissipation relation.
- Energy law: The modified equivalent system satisfies a modified energy dissipation law after combining the evolution equations.The derivation uses inner products with the chemical potential and phase-field time derivative.
- Choice of C: The dissipative energy law provides a sufficient basis for choosing C from the initial nonlinear or total energy, including C = |E1(φ(x, 0))|.The paper also notes that a very large positive constant satisfies the requirements.
5. Multiple E-SAV approach.
ME-SAV extends the exponential auxiliary-variable strategy to phase-field models with multiple unknowns and nonlinear terms, including a coupled binary fluid–surfactant system. The resulting schemes are linear, sequential, and unconditionally energy stable.
- Formulation: The general ME-SAV formulation introduces exponential auxiliary variables for multiple nonlinear energy contributions.The construction uses a large constant so each auxiliary variable remains not very large.
- First-order scheme: The first-order ME-SAV scheme is linear, first-order accurate, and unconditionally energy stable.This property is stated in Theorem 5.1 through a discrete energy inequality.
- Second-order scheme: The second-order ME-SAV scheme is linear, second-order accurate, and unconditionally energy stable.Theorem 5.1 states the corresponding discrete energy inequality for the second-order scheme.
- Binary fluid–surfactant application: For the binary fluid–surfactant model, the computations of φ, ρ, r, and q are totally decoupled across sequential steps.The scheme first updates ρ and q, then updates φ and r.
- Numerical evaluation: Numerical examples use periodic boundary conditions and Fourier spectral discretization to assess accuracy, energy stability, and efficiency.The examples include classical models such as Allen–Cahn, Cahn–Hilliard, and phase-field crystal equations.
- Motivation: ME-SAV is introduced because a single auxiliary variable cannot adequately represent complex models with two or more evolution processes and nonlinear terms.The approach targets broader applicability to multi-variable phase-field systems.
6. Examples and discussion.
The examples demonstrate that E-SAV schemes reproduce expected phase-field dynamics, preserve energy dissipation across time steps, and reduce computational cost relative to SAV while retaining comparable accuracy.
- Allen-Cahn and Cahn-Hilliard equations: E-SAV and SAV are compared for classical Allen–Cahn and Cahn–Hilliard models, which are widely used phase-field equations.
- Allen-Cahn and Cahn-Hilliard equations: Both first-order Allen–Cahn schemes achieve first-order temporal accuracy, while E-SAV produces smaller errors and takes about half the CPU time of SAV.
- Allen-Cahn and Cahn-Hilliard equations: For the second-order Cahn–Hilliard comparison, SAV and E-SAV have almost identical errors and convergence rates, but E-SAV saves half the computation time.
- Allen-Cahn and Cahn-Hilliard equations: In the Allen–Cahn example, two initially kissing bubbles merge, shrink, and disappear, while the E-SAV energy decreases monotonically for Δt = 0.001, 0.01, 0.1, 1, and 2.
- Allen-Cahn and Cahn-Hilliard equations: The Cahn–Hilliard example shows phase separation and coarsening at t = 0.02, 0.5, 3, and 20 with Δt = 0.01.
- Phase field crystal equations: For the phase field crystal model, E-SAV simulations show crystalline-phase growth and decreasing energy across δt = 0.01, 0.1, and 1, with no visible difference from SAV configurations.
- Phase field crystal equations: The reported convergence rates for both phase variables φ and ρ in the phase field crystal computation are first order.