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The Exponential Scalar Auxiliary Variable (E-SAV) Approach for Phase Field Models and Its Explicit Computing
SIAM Journal on Scientific Computing ( IF 3.0 ) Pub Date : 2020-05-13 , DOI: 10.1137/19m1305914 Zhengguang Liu , Xiaoli Li
SIAM Journal on Scientific Computing ( IF 3.0 ) Pub Date : 2020-05-13 , DOI: 10.1137/19m1305914 Zhengguang Liu , Xiaoli Li
SIAM Journal on Scientific Computing, Volume 42, Issue 3, Page B630-B655, January 2020.
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 the exponential form of the nonlinear free energy potential is more effective and applicable than the traditional SAV method, which is very popular in constructing 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 semidiscrete schemes carefully and rigorously. Another contribution is that we provide a total and explicit discretization of the auxiliary variable combined with the nonlinear term. Such a modification is very efficient for fast calculations. Furthermore, the positivity preserving property of $r$ can be guaranteed, which is very important and reasonable for the models' equivalence. In addition, 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.
中文翻译：
相场模型的指数标量辅助变量（E-SAV）方法及其显式计算
SIAM科学计算杂志，第42卷，第3期，第B630-B655页，2020年1月。
在本文中，我们考虑采用指数标量辅助变量（E-SAV）方法来获得一类相场模型的能量稳定方案。这种新颖的基于非线性自由能势指数形式的辅助变量方法比传统的SAV方法更有效和适用，传统的SAV方法在构造能量稳定方案中非常流行。第一个贡献是，没有平方根的辅助变量消除了非线性自由能势的从下至下的限制。然后，我们仔细，严格地证明了半离散方案的无条件能量稳定性。另一个贡献是，我们提供了与非线性项结合的辅助变量的整体和显式离散化。这种修改对于快速计算非常有效。此外，可以保证$ r $的正保持性，这对于模型的等效性非常重要且合理。此外，对于具有两个或多个未知变量和非线性项的复杂相场模型，我们构造了多重E-SAV（ME-SAV）方法来增强所提出的E-SAV方法的适用性。对经典SAV和E-SAV方法的比较研究被认为可以显示准确性和效率。最后，我们提出各种2D数值模拟以证明稳定性和准确性。我们构建了多种E-SAV（ME-SAV）方法，以增强所提出的E-SAV方法的适用性。对经典SAV和E-SAV方法的比较研究被认为可以显示准确性和效率。最后，我们提出各种2D数值模拟以证明稳定性和准确性。我们构建了多种E-SAV（ME-SAV）方法，以增强所提出的E-SAV方法的适用性。对经典SAV和E-SAV方法的比较研究被认为可以显示准确性和效率。最后，我们提出各种2D数值模拟以证明稳定性和准确性。
更新日期：2020-05-13
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 the exponential form of the nonlinear free energy potential is more effective and applicable than the traditional SAV method, which is very popular in constructing 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 semidiscrete schemes carefully and rigorously. Another contribution is that we provide a total and explicit discretization of the auxiliary variable combined with the nonlinear term. Such a modification is very efficient for fast calculations. Furthermore, the positivity preserving property of $r$ can be guaranteed, which is very important and reasonable for the models' equivalence. In addition, 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.
中文翻译：
相场模型的指数标量辅助变量（E-SAV）方法及其显式计算
SIAM科学计算杂志，第42卷，第3期，第B630-B655页，2020年1月。
在本文中，我们考虑采用指数标量辅助变量（E-SAV）方法来获得一类相场模型的能量稳定方案。这种新颖的基于非线性自由能势指数形式的辅助变量方法比传统的SAV方法更有效和适用，传统的SAV方法在构造能量稳定方案中非常流行。第一个贡献是，没有平方根的辅助变量消除了非线性自由能势的从下至下的限制。然后，我们仔细，严格地证明了半离散方案的无条件能量稳定性。另一个贡献是，我们提供了与非线性项结合的辅助变量的整体和显式离散化。这种修改对于快速计算非常有效。此外，可以保证$ r $的正保持性，这对于模型的等效性非常重要且合理。此外，对于具有两个或多个未知变量和非线性项的复杂相场模型，我们构造了多重E-SAV（ME-SAV）方法来增强所提出的E-SAV方法的适用性。对经典SAV和E-SAV方法的比较研究被认为可以显示准确性和效率。最后，我们提出各种2D数值模拟以证明稳定性和准确性。我们构建了多种E-SAV（ME-SAV）方法，以增强所提出的E-SAV方法的适用性。对经典SAV和E-SAV方法的比较研究被认为可以显示准确性和效率。最后，我们提出各种2D数值模拟以证明稳定性和准确性。我们构建了多种E-SAV（ME-SAV）方法，以增强所提出的E-SAV方法的适用性。对经典SAV和E-SAV方法的比较研究被认为可以显示准确性和效率。最后，我们提出各种2D数值模拟以证明稳定性和准确性。
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