An Energy-Stable and Convergent Finite-Difference Scheme for the Phase Field Crystal Equation

By Steven M. Wise, Cheng Wang, John S. Lowengrub. SIAM Journal on Numerical Analysis, 2009

521 citations recorded by the source corpus.

Subfields: Aerospace Engineering, Computational Theory and Mathematics, Materials Chemistry

View Paper: https://www.semanticscholar.org/paper/9667b54f15895d1a728aa3ad5a69e4efe2fcdf6a

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Citations in the corpus

Two‐phase flows in karstic geometry (2014, background): “For system that enjoys variational structure, a key idea in the development of unconditionally stable schemes is convex splitting (see [55, 85, 86, 87] among many others).”
Stabilized Crank-Nicolson/Adams-Bashforth Schemes for Phase Field Models (2013, unknown): “[13,20].”
Second-order Convex Splitting Schemes for Gradient Flows with Ehrlich-Schwoebel Type Energy: Application to Thin Film Epitaxy (2012, methodology): “We note that second-order, uniquely solvable, and unconditionally energy stable schemes have been constructed recently in [31] for Swift–Hohenberg and phase field crystal type equations.”
Unconditionally stable schemes for equations of thin film epitaxy (2010, methodology): “Although the theory herein is presented for energy functionals suitable for thin film epitaxy, it is easily generalized to a wider class of gradient flow equations, such as Swift-Hohenberg and phase field crystal (PFC) equations [23] (see also [2]).”
A MIXED DISCONTINUOUS GALERKIN, CONVEX SPLITTING SCHEME FOR A MODIFIED CAHN-HILLIARD EQUATION AND AN EFFICIENT NONLINEAR MULTIGRID SOLVER (2013, methodology): “For the time discretization we use the convex-splitting (CS) approach popularized by David Eyre [16] and used in several other settings [12, 18, 29, 33].”
A Second-Order Energy Stable BDF Numerical Scheme for the Cahn-Hilliard Equation (2018, background): “See the related works for the phase field crystal (PFC) equation and the modified phase field crystal (MPFC) equation [3, 4, 28, 35, 37]; epitaxial thin film growth models [8, 10, 32, 34]; non-local Cahn-Hilliard-type models [22, 23]; the CHHS and related models [9, 13, 20, 36]; et cetera.”
A novel linear second order unconditionally energy stable scheme for a hydrodynamic Q-tensor model of liquid crystals (2017, methodology): “In [47], we have designed a first and a second order semi-discrete scheme in time employing both the convex-splitting strategy [12,31,40] and the stabilizer approach [32,33], and showed the developed numerical schemes are unconditionally energy stable.”
Phase-field-crystal models for condensed matter dynamics on atomic length and diffusive time scales: an overview (2012, methodology): “Energy stable large time-step implicit FD methods have been developed for the PFC [160] and MPFC [161] equations that lead to large sets of sparse algebraic equations.”
The phase field method for geometric moving interfaces and their numerical approximations (2019, methodology): “Although for different models, the norms used to measure errors and the orders of errors (indicated by l1 and l2 in (62) ) may be different, the nature of the exponential dependence on 1 ε of the constants C1 and C2 remains the same for all these methods and models when the above standard perturbati”