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Improving the Accuracy and Consistency of the Scalar Auxiliary Variable (SAV) Method with Relaxation
Maosheng Jiang, Zengyan Zhang, Jia Zhao
TL;DR
SAV schemes offer efficient, linear, unconditionally energy-stable discretizations, but their modified energy law can diverge from the original PDE energy law after discretization. This paper introduces relaxed-SAV methods that penalize auxiliary-variable inconsistency while preserving the baseline advantages, with numerical tests demonstrating accuracy and effectiveness.
Problem
SAV discretizations preserve a modified auxiliary-variable energy law, but numerical auxiliary variables may no longer match their continuous definitions or the original PDE energy law.
Method
The paper adds a relaxation step to SAV and MSAV schemes to penalize auxiliary-variable inconsistency while retaining linear, second-order, unconditionally energy-stable discretizations.
Results
The RSAV approach preserves the original energy when ξ reaches 0, while numerical experiments across several phase-field models highlight its accuracy and efficiency.
Takeaways & Limitations
RSAV provides a relaxation-based way to improve SAV accuracy and consistency without sacrificing its stated computational and stability properties.
Takeaways & Limitations
The analysis assumes periodic boundary conditions for simplicity, although the authors state that the results apply to thermodynamically consistent boundary conditions.
Abstract
from arXiv · showhide
The scalar auxiliary variable (SAV) method was introduced by Shen et al. and has been broadly used to solve thermodynamically consistent PDE problems. By utilizing scalar auxiliary variables, the original PDE problems are reformulated into equivalent PDE problems. The advantages of the SAV approach, such as linearity, unconditionally energy stability, and easy-to-implement, are prevalent. However, there is still an open issue unresolved, i.e., the numerical schemes resulted from the SAV method preserve a "modified" energy law according to the auxiliary variables instead of the original variables. Truncation errors are introduced during numerical calculations so that the numerical solutions of the auxiliary variables are no longer equivalent to their original continuous definitions. In other words, even though the SAV scheme satisfies a modified energy law, it does not necessarily satisfy the energy law of the original PDE models. This paper presents one essential relaxation technique to overcome this issue, which we named the relaxed-SAV (RSAV) method. Our RSAV method penalizes the numerical errors of the auxiliary variables by a relaxation technique. In general, the RSAV method keeps all the advantages of the baseline SAV method and improves its accuracy and consistency noticeably. Several examples have been presented to demonstrate the effectiveness of the RSAV approach.
1 Introduction
Gradient-flow models describe thermodynamically consistent physical systems, but nonlinear governing equations make exact and numerical solutions difficult. SAV and related approaches provide efficient, energy-stable discretizations, while the paper targets their mismatch between modified and original energy laws.
- Gradient-flow systems model diverse physical problems and agree with the second law of thermodynamics.
- Nonlinear terms make exact solutions and numerical solutions difficult to obtain.
- IEQ or EQ methods enable linear, second-order, unconditionally energy-stable schemes but typically produce coupled systems with time-dependent coefficients.
- SAV methods use scalar auxiliary variables to retain EQ advantages while usually producing decoupled constant-coefficient systems that are easier to implement.
- IEQ and SAV schemes preserve a modified auxiliary-variable energy law that may differ from the original PDE energy law after computation.
- The paper introduces relaxation to penalize inconsistency between numerical auxiliary variables and their continuous definitions.
2 A brief review of the SAV method
The SAV framework reformulates gradient-flow models with scalar auxiliary variables, enabling second-order schemes with unconditional energy stability. Its baseline and multiple-variable variants provide discrete modified energy laws for increasingly general free-energy forms.
- A gradient-flow model is specified by a state variable, free energy, and mobility or skew-symmetric operator.
- SAV introduces a scalar auxiliary variable and reformulates the gradient-flow model into an equivalent system.
- The SAV discretization satisfies a discrete modified energy law involving the auxiliary variable.
- The SAV-BDF2 and SAV-CN schemes are second-order discretizations with unconditional energy stability.
- MSAV introduces multiple scalar auxiliary variables for more complicated free-energy expressions with multiple bulk potentials.
- MSAV-BDF2 and MSAV-CN schemes likewise have discrete energy laws and unconditional energy stability.
3 Our Remedy: the relaxed SAV (RSAV) method
RSAV addresses the post-discretization mismatch between auxiliary variables and their intended definitions by adding a relaxation step. The resulting schemes retain second-order accuracy and unconditional energy stability, while extending the correction to MSAV.
- After temporal discretization, the numerical auxiliary variable need not equal its defining function, so modified-energy stability need not imply original-energy stability.
- RSAV penalizes the discrepancy between the numerical auxiliary variable and its defining function through a relaxation step.
- RSAV schemes: RSAV-BDF2 and RSAV-CN update the baseline SAV solution and then relax the scalar auxiliary variable.
- Stability: The RSAV-BDF2 and RSAV-CN schemes are unconditionally energy stable.
- Accuracy: The RSAV-BDF2 and RSAV-CN schemes are second-order accurate in time because the relaxation step does not change the temporal order.
- Relaxed MSAV: The relaxation technique is also applied to MSAV, producing RMSAV-BDF2 and RMSAV-CN schemes with unconditional energy-stability results.
4 Numerical results
Numerical tests apply RSAV-CN schemes to several phase-field models, evaluating temporal convergence, accuracy, consistency, and long-time dynamics. Across the tested Allen-Cahn, Cahn-Hilliard, MBE, PFC, and related models, the method shows second-order convergence and improved accuracy or consistency over baseline SAV schemes.
- Experimental setup: The experiments apply relaxed SAV algorithms to Allen-Cahn, Cahn-Hilliard, MBE, phase-field crystal, and diblock copolymer models.Periodic boundary conditions and Fourier pseudo-spectral spatial discretization are used in the numerical implementation.
- Temporal convergence: Second-order temporal convergence is observed for both φ and q in the Allen-Cahn, Cahn-Hilliard, and MBE tests.The MBE result is reported when the time step is not too large.
- Phase-field dynamics: The RSAV-CN simulations reproduce model-specific dynamics, including Allen-Cahn star-profile shrinking, Cahn-Hilliard phase evolution, and MBE coarsening.The Allen-Cahn star shape smooths into a disk and continues shrinking because the model does not preserve total phase volume.
- Other phase-field models: RSAV extends to other dissipative phase-field models, with PFC tests showing second-order convergence, accurate long-time dynamics, and parameter-dependent patterns.The reported PFC patterns include stripes for ˆφ0 = 0 and triangles for ˆφ0 = 0.2, while droplet counts scale with σ in another test.
5 Conclusion
The paper introduces relaxed-SAV (RSAV) schemes to improve the accuracy and consistency of SAV methods for dissipative PDE models. Numerical experiments, including a diblock copolymer example, support the effectiveness of the approach.
- RSAV uses relaxation to improve the accuracy and consistency of the baseline SAV method for dissipative PDE models.
- The proposed schemes are linear, second-order, and unconditionally energy stable, with original-energy preservation when the relaxation parameter ξ reaches 0.
- Comparisons across Allen–Cahn, Cahn–Hilliard, MBE, PFC, and diblock copolymer models report better accuracy and consistency than baseline SAV schemes.
- At t =500, stronger nonlocal interaction strength σ produces more droplets in the diblock copolymer model.