A family of second-order energy-stable schemes for Cahn–Hilliard type equations

Yang, Zhiguo; Lin, Li; Dong, Suchuan

doi:10.1016/j.jcp.2019.01.014

Cited by 42 publications

(27 citation statements)

References 41 publications

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“…The application of interest in this particular case is the study of two phase flows. The model consists of a constant mobility parameter and a polynomic double-well chemical free-energy [58,59]. The Cahn-Hilliard equation is defined as…”

Section: Cahn-hilliard Equation and Continuous Energy Estimatesmentioning

confidence: 99%

A free–energy stable p–adaptive nodal discontinuous Galerkin for the Cahn–Hilliard equation

Ntoukas

Manzanero

Rubio

et al. 2021

Journal of Computational Physics

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A novel free-energy stable discontinuous Galerkin method is developed for the Cahn-Hilliard equation with non-conforming elements. This work focuses on dynamic polynomial adaption (p-refinement) and constitutes an extension of the method developed by Manzanero et al. in Journal of Computational Physics 403:109072, 2020, which makes use of the summation-by-parts simultaneous-approximation term technique along with Gauss-Lobatto points and the Bassi-Rebay 1 (BR1) scheme. The BR1 numerical flux accommodates nonconforming elements, which are connected through the mortar method. The scheme has been analytically proven to retain its free-energy stability when transitioning to non-conforming elements. Furthermore, a methodology to perform the adaption is introduced based on the knowledge of the location of the interface between phases. The adaption methodology is tested for its accuracy and effectiveness through a series of steady and unsteady test cases. We solve a steady one-dimensional interface test case to initially examine the accuracy of the adaption. Furthermore, we study the formation of a static bubble in two dimensions and verify that the accuracy of the solver is maintained while the degrees of freedom decrease to less than half compared to the uniform solution. Lastly, we examine an unsteady case such as the spinodal decomposition and show that the same results for the free-energy are recovered with a 35% reduction of the degrees of freedom for the two-dimensional case considered and a 48% reduction for the three-dimensional case.

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Section: Cahn-hilliard Equation and Continuous Energy Estimatesmentioning

confidence: 99%

A free–energy stable p–adaptive nodal discontinuous Galerkin for the Cahn–Hilliard equation

Ntoukas

Manzanero

Rubio

et al. 2021

Journal of Computational Physics

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show abstract

“…Developing effective and efficient numerical algorithms for the Cahn-Hilliard equation can have important ramifications on the modeling and simulation of two-phase and multiphase flows. This problem has witnessed a sustained interest from the research community, and we refer to [34,36,38,42] for some examples.…”

Section: Introductionmentioning

confidence: 99%

“…In the past few years, the use of certain auxiliary functions or variables proves to be effective in devising linear energy-stable schemes. The invariant energy quadratization (IEQ) [39] and the scalar auxiliary variable (SAV) [32] are two prominent examples of such methods; see also [18,31,44,25,42,22,40], among others. The IEQ method introduces an auxiliary field function as an approximation of the square root of the potential energy density function together with a dynamic equation for this field function, and allows one to ensure the energy stability relatively easily.…”

Section: Introductionmentioning

confidence: 99%

gPAV-Based Unconditionally Energy-Stable Schemes for the Cahn-Hilliard Equation: Stability and Error Analysis

Qian,

Yang,

Wang

et al. 2020

Preprint

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We present several first-order and second-order numerical schemes for the Cahn-Hilliard equation with discrete unconditional energy stability. These schemes stem from the generalized Positive Auxiliary Variable (gPAV) idea, and require only the solution of linear algebraic systems with a constant coefficient matrix. More importantly, the computational complexity (operation count per time step) of these schemes is approximately a half of those of the gPAV and the scalar auxiliary variable (SAV) methods in previous works. We investigate the stability properties of the proposed schemes to establish stability bounds for the field function and the auxiliary variable, and also provide their error analyses. Numerical experiments are presented to verify the theoretical analyses and also demonstrate the stability of the schemes at large time step sizes.

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“…The gPAV procedure endows energy stability to the resultant scheme, and also can ensure the positivity of the computed values of the generalized auxiliary variable [33]. Compared with related works [21,35,34,26,32], the gPAV framework provides a more favorable way for treating the auxiliary variables, and it applies to very general dissipative systems.…”

Section: Introductionmentioning

confidence: 99%

An Energy-Stable Scheme for Incompressible Navier-Stokes Equations with Periodically Updated Coefficient Matrix

Lin,

Ni,

Yang

et al. 2019

Preprint

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We present an energy-stable scheme for simulating the incompressible Navier-Stokes equations based on the generalized Positive Auxiliary Variable (gPAV) framework. In the gPAV-reformulated system the original nonlinear term is replaced by a linear term plus a correction term, where the correction term is put under control by an auxiliary variable. The proposed scheme incorporates a pressure-correction type strategy into the gPAV procedure, and it satisfies a discrete energy stability property. The scheme entails the computation of two copies of the velocity and pressure within a time step, by solving an individual de-coupled linear equation for each of these field variables. Upon discretization the pressure linear system involves a constant coefficient matrix that can be pre-computed, while the velocity linear system involves a coefficient matrix that is updated periodically, once every k0 time steps in the current work, where k0 is a user-specified integer. The auxiliary variable, being a scalar-valued number, is computed by a well-defined explicit formula, which guarantees the positivity of its computed values. It is observed that the current method can produce accurate simulation results at large (or fairly large) time step sizes for the incompressible Navier-Stokes equations. The impact of the periodic coefficient-matrix update on the overall cost of the method is observed to be small in typical numerical simulations. Several flow problems have been simulated to demonstrate the accuracy and performance of the method developed herein.

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A family of second-order energy-stable schemes for Cahn–Hilliard type equations

Cited by 42 publications

References 41 publications

A free–energy stable p–adaptive nodal discontinuous Galerkin for the Cahn–Hilliard equation

A free–energy stable p–adaptive nodal discontinuous Galerkin for the Cahn–Hilliard equation

gPAV-Based Unconditionally Energy-Stable Schemes for the Cahn-Hilliard Equation: Stability and Error Analysis

An Energy-Stable Scheme for Incompressible Navier-Stokes Equations with Periodically Updated Coefficient Matrix

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