A New Class of Efficient and Robust Energy Stable Schemes for Gradient Flows

2019/01/01 by Jie Shen, Jie Xu, Jiang Yang · 698 citations
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Applied mathematics #Artificial intelligence #Class (philosophy) #Computer science #Constant (computer programming) #Construct (python library) #Energy (signal processing) #Fluid Dynamics and Turbulent Flows #Geometry #Mathematical optimization #Mathematics #Nanofluid Flow and Heat Transfer #Nonlinear system #Physics #Scalar (mathematics)

paper · pdf · doi:10.1137/17m1150153

published in SIAM Review 61(3), 474-506 (Society for Industrial and Applied Mathematics)

openalex publication_date 2019/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/06

Abstract

We propose a new numerical technique to deal with nonlinear terms in gradient flows. By introducing a scalar auxiliary variable (SAV), we construct efficient and robust energy stable schemes for a large class of gradient flows. The SAV approach is not restricted to specific forms of the nonlinear part of the free energy, and only requires to solve \it decoupled linear equations with \it constant coefficients. We use this technique to deal with several challenging applications which can not be easily handled by existing approaches, and present convincing numerical results to show that our schemes are not only much more efficient and easy to implement, but can also better capture the physical properties in these models. Based on this SAV approach, we can construct unconditionally second-order energy stable schemes; and we can easily construct even third or fourth order BDF schemes, although not unconditionally stable, which are very robust in practice. In particular, when coupled with an adaptive time stepping strategy, the SAV approach can be extremely efficient and accurate.

Citations

• Reformulation of elasticity theory for discontinuities and long-range forces
• Exponential Time Differencing for Stiff Systems

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