Improving the accuracy of convexity splitting methods for gradient flow equations

Glasner, Karl; Orizaga, Saulo

doi:10.1016/j.jcp.2016.03.042

Cited by 61 publications

(36 citation statements)

References 22 publications

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“…The latter were found by evolving the dynamic equations toward a steady state, using a periodic domain much larger than the bilayer width. Other details of the algorithm can be found elsewhere [42]. The numerical comparison uses the specific potential (5) with χ AS = 1, from which one finds that β = 4 and eigenvalues of H 0 are both equal to λ = 2.…”

Section: A Bilayer Equilibriamentioning

confidence: 99%

Theoretical prediction of morphological selection in amphiphilic systems

Glasner

2019

Phys. Rev. E

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Biological and synthetic amphiphilic systems exhibit a wide range of morphologies. A density functional model for amphiphilic polymer phase mixtures is utilized to quantify localized equilibria and their stability, and ultimately predict and explain morphological preference. This is done by utilizing matched asymptotic expansions, which produces explicit connections between model parameters and macroscopic properties of equilibrium structures. Bilayers, cylindrical, and spherical micelle and vesicle configurations are found, and formulas which connect their geometry to ambient chemical potential are derived. Dynamics are studied in the context of a free boundary problem which describes the evolution of the hydrophobic-solvent domain interface. Linearization of this problem is used to explicitly determine growth rates and parameter regions of stability. All equilibria are found to have two branches of solutions terminating at a fold in the bifurcation diagram which signals the crossover from competitive stability to instability leading to ripening behavior. Ideally flat bilayers are determined to always possess a long wavelength buckling instability, suggesting that curved structures should be generically preferred. Spherical micelles exhibit morphological instabilities which are suppressed by large enough surface tension. Cylindrical micelles may have short-wavelength pearling and long-wavelength Rayleigh-Plateau-type instabilities. In addition, ideally infinite cylinders have an undulatory instability, suggesting that only finite length structures should be observed. A morphological phase diagram can be assembled which takes into account both existence and stability of different geometries. Consistent with experimental evidence, a bifurcation sequence from spheres to cylinders to vesicles is found as either surface tension or polymer composition increases. Coexistence of different stable morphologies is also observed.

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Section: A Bilayer Equilibriamentioning

confidence: 99%

Theoretical prediction of morphological selection in amphiphilic systems

Glasner

2019

Phys. Rev. E

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“…Various numerical approaches have been proposed to solve the equation efficiently. We consider a convexity splitting approach, e.g., [7][8][9][10][11]. The idea is to split the double well potential B(φ) = B c (φ) − B e (φ), such that both parts are convex and to consider the time discretization as…”

Section: Methodsmentioning

confidence: 99%

“…with discrete time derivative d τ φ n+1 = (φ n+1 − φ n )/τ n . The resulting scheme is unconditionally energy stable, unconditionally solvable and converges optimally in the energy norm [9]. To solve the above systems, we consider a linearization of B c (φ n+1 ) ≈ B c (φ n ) + B c (φ n )(φ n+1 − φ n ).…”

Section: Methodsmentioning

confidence: 99%

Towards Infinite Tilings with Symmetric Boundaries

Stenger

Voigt

2019

Symmetry

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Large-time coarsening and the associated scaling and statistically self-similar properties are used to construct infinite tilings. This is realized using a Cahn–Hilliard equation and special boundaries on each tile. Within a compromise between computational effort and the goal to reduce recurrences, an infinite tiling has been created and software which zooms in and out evolve forward and backward in time as well as traverse the infinite tiling horizontally and vertically. We also analyze the scaling behavior and the statistically self-similar properties and describe the numerical approach, which is based on finite elements and an energy-stable time discretization.

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“…To improve the accuracy as well as stability of Scheme 3.1, we propose a prediction-correction scheme motivated by the works in [14,19,29]. Employing the prediction-correction strategy to Scheme 3.1, we obtain the following prediction-correction method: For given (Φ n , q n ) and Φ N (t n +c i ∆t), Q N (t n +c i ∆t), ∀i, the following intermediate values are first calculated by the following prediction-correction strategy 1.…”

Section: Leqrk-pc Schemesmentioning

confidence: 99%

Arbitrarily high-order unconditionally energy stable SAV schemes for gradient flow models

Gong

Zhao

Wang

2020

Computer Physics Communications

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We present a paradigm for developing arbitrarily high order, linear, unconditionally energy stable numerical algorithms for gradient flow models. We apply the energy quadratization (EQ) technique to reformulate the general gradient flow model into an equivalent gradient flow model with a quadratic free energy and a modified mobility. Given solutions up to t n = n∆t with ∆t the time step size, we linearize the EQ-reformulated gradient flow model in (t n , t n+1 ] by extrapolation. Then we employ an algebraically stable Runge-Kutta method to discretize the linearized model in (t n , t n+1 ]. Then we use the Fourier pseudo-spectral method for the spatial discretization to match the order of accuracy in time. The resulting fully discrete scheme is linear, unconditionally energy stable, uniquely solvable, and may reach arbitrarily high order. Furthermore, we present a family of linear schemes based on predictioncorrection methods to complement the new linear schemes. Some benchmark numerical examples are given to demonstrate the accuracy and efficiency of the schemes.solutions [13]. This is because non-energy-stable schemes may introduce truncation errors that destroy the physical law numerically. Thus, developing energy stable numerical algorithms is necessary for accurately resolving the dynamics of gradient flow models [20,21,23,30,44].Over the years, the development of numerical algorithms has been done primarily on a specific gradient flow model, exploiting its specific mathematical properties and structures. The noticeable ones include the Allen-Cahn and Cahn-Hilliard equation [1,9,11,20,24,32,36,42] as well as the molecular beam epitaxy model [6,12,[26][27][28]41]. As a result, most numerical algorithms developed for a specific gradient flow model can hardly be applied to another gradient flow model with a different free energy and mobility. The statusquo did not change until the energy quaratization (EQ) approach was introduced a few years ago [39,44], which turns out to be rather general so that it can be readily applied to general gradient flow models with no restrictions on the specific form of the mobility and free energy [16,40] (so long as it is bounded below, which is usually the case). Based on the idea of EQ, the scalar auxiliary variable (SAV) approach was introduced later [31], where each time step only linear systems with constant coefficients need to be solved. Several other extensions of EQ and SAV approaches have been explored further. For instance, regularization terms [5] and stabilization terms [38] are added, and the modified energy quadratization technique [22,25] are introduced to improve the EQ methodology. We notice that most existing schemes related to the EQ methodology are up to 2nd order accurate in time. Some higher-order ETD schemes have been introduced to solve the MBE model and Cahn-Hilliard models recently, but their theoretical proofs of energy stability are still missing. Shen et al.[30] remarked on higher-order BDF schemes using the SAV approach, but no rigorous theoretical p...

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Improving the accuracy of convexity splitting methods for gradient flow equations

Cited by 61 publications

References 22 publications

Theoretical prediction of morphological selection in amphiphilic systems

Theoretical prediction of morphological selection in amphiphilic systems

Towards Infinite Tilings with Symmetric Boundaries

Arbitrarily high-order unconditionally energy stable SAV schemes for gradient flow models

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