Spectral approximation of pattern-forming nonlinear evolution equations with double-well potentials of quadratic growth

Condette, Nicolas; Melcher, Christof; Süli, Endre

doi:10.1090/s0025-5718-10-02365-3

Cited by 61 publications

(66 citation statements)

References 11 publications

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“…Therefore, it has been a common practice (cf. [21,8,42]) to consider the Allen-Cahn and Cahn-Hilliard equations with a truncated double-well potentialF (φ). Similarly, we can truncate G(d) to satisfy (3.20).…”

Section: Where H(d) Is the Hessian Matrix Of G(d)mentioning

confidence: 99%

Decoupled Energy Stable Schemes for Phase-Field Models of Two-Phase Complex Fluids

Shen¹,

Yang²

2014

SIAM J. Sci. Comput.

107

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Abstract. We consider in this paper numerical approximations of phase-field models for twophase complex fluids. We first reformulate the phase-field models derived from an energetic variational formulation into a form which is suitable for numerical approximation and establish their energy laws. Then, we construct two classes, stabilized and convex-splitting, of decoupled time discretization schemes for the coupled nonlinear systems. These schemes are unconditionally energy stable and lead to decoupled elliptic equations to solve at each time step. Furthermore, these elliptic equations are linear for the stabilized version. Stability analysis and ample numerical simulations are presented.Key words. phase-field, liquid crystals, multiphase flows, Navier-Stokes, Allen-Cahn, CahnHilliard, stability AMS subject classifications. 65M12, 65M70, 65Z05 DOI. 10.1137/130921593 1. Introduction. Complex fluids have great practical utility since the microstructure can be manipulated to produce useful mechanical, optical, or thermal properties [4,23]. Numerical studies of multiphase complex fluids is a rich area for research with important applications that can be addressed. Some numerical difficulties are (i) the moving interfaces between various components; (ii) open issues regarding wellposedness of many multiphase models leading to confusion as to whether numerical pathologies are due to the model or the methods; and most important, (iii) nonlinear coupling (hydrodynamics, interface, and microstructure) makes it very difficult to design easy-to-implement, energy stable numerical schemes.For the first two issues above, the diffuse-interface/phase-field models, whose origin can be traced back to [38] and [46], have been proved efficient with much success. A particular advantage of the phase-field approach is that models can often be derived from an energy-based variational formalism (energetic variational approaches), leading to well-posed nonlinear coupled systems that satisfy thermodynamics-consistent energy dissipation laws. This makes it possible to carry out mathematical analysis and design numerical schemes which satisfy a corresponding discrete energy dissipation law. In recent years, the phase-field method has become one of the major tools to study a variety of interfacial phenomena (cf. [17,2,36,19,32,49] and recent review papers [44,22] and the references therein).For phase-field models which satisfy a dissipative energy law, it is especially desirable to design numerical schemes that preserve the energy dissipation law at the

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Section: Where H(d) Is the Hessian Matrix Of G(d)mentioning

confidence: 99%

Decoupled Energy Stable Schemes for Phase-Field Models of Two-Phase Complex Fluids

Shen¹,

Yang²

2014

SIAM J. Sci. Comput.

107

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“…The Fourier collocation method is used for spatial discretization, in situations similar to ours (e.g. [6]), as an alternative to the Fourier-Galerkin method. Fourier collocation methods define spatial discretization by sampling the differential equation, with the analytical solution replaced by the numerical solution, at equally spaced collocation points.…”

Section: Fourier-galerkin Approximation In Briefmentioning

confidence: 99%

“…In addition to the bounds outlined in [10,6], we exploit the monotonicity of the nonlinearity. This, combined with the fact that we are using a spatial discretization based on a Fourier spectral method, allows us to avoid assuming uniform bounds on the sequence of approximate solutions, which were essential, for example, in [10].…”

Section: Convergence Analysismentioning

confidence: 99%

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An Implicit Midpoint Spectral Approximation of Nonlocal Cahn--Hilliard Equations

Benešová¹,

Melcher²,

Süli³

2014

SIAM J. Numer. Anal.

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Abstract. The paper is concerned with the convergence analysis of a numerical method for nonlocal CahnHilliard equations. The temporal discretization is based on the implicit midpoint rule and a Fourier spectral discretization is used with respect to the spatial variables. The sequence of numerical approximations in shown to be bounded in various norms, uniformly with respect to the discretization parameters, and optimal order bounds on the global error of the scheme are derived. The uniform bounds on the sequence of numerical solutions as well as the error bounds hold unconditionally, in the sense that no restriction on the size of the time step in terms of the spatial discretization parameter needs to be assumed.

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“…Ohta-Kawasaki model, including the limiting case of σ = 0 which corresponds to the classical Cahn-Hilliard model. However, the numerical method proposed here can be adapted to more general functions ˆσ in (1.1), including, e.g., dipolar stray-field interaction in magnetic garnet films (Condette, Melcher, & Süli, 2011) …”

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confidence: 99%

Heat Treatment of Duplex Stainless Steel 2205 by Inserting Nano Nd2FeB14 in HIP Manifolds Under the Scope of Category Theory

Gebril¹

2016

JMSR

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In this paper, improving mechanical properties of duplex stainless steel by spinodal reaction inhibitors has been discussed. Spinodal gaps can be minimized when the host particles are minimized to the nanoscale. The Cahn-Hilliard Equation and the Allen-Cahn Equation on Manifolds with Conical Singularities can be related to manifolds produced by hot isostatic pressing (PM HIP) are used to improve mechanical strength. Spinodal reactions will be Inhibited by magnetic treatment and Nd-Fe-B Magnets. Grain refiners will be used to retard spinodal reactions. Mathematically, category theory is used to establish links between these different concepts.

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Spectral approximation of pattern-forming nonlinear evolution equations with double-well potentials of quadratic growth

Cited by 61 publications

References 11 publications

Decoupled Energy Stable Schemes for Phase-Field Models of Two-Phase Complex Fluids

Decoupled Energy Stable Schemes for Phase-Field Models of Two-Phase Complex Fluids

An Implicit Midpoint Spectral Approximation of Nonlocal Cahn--Hilliard Equations

Heat Treatment of Duplex Stainless Steel 2205 by Inserting Nano Nd2FeB14 in HIP Manifolds Under the Scope of Category Theory

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