Stability Analysis of Large Time-Stepping Methods for Epitaxial Growth Models

By Chuanju Xu, Tao Tang. SIAM Journal on Numerical Analysis, 2006

411 citations recorded by the source corpus.

Subfields: Computational Mechanics, Condensed Matter Physics, Materials Chemistry

View Paper: https://www.semanticscholar.org/paper/7e2edac9c33ce4c77a2d99691de3410498d01859

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Citations in the corpus

Efficient linear schemes with unconditional energy stability for the phase field elastic bending energy model (2017, background): “the convex splitting approach (implicit) [41] or the stabilization approach (explicit) [31,39,38].”
Decoupled, Energy Stable Schemes for Phase-Field Models of Two-Phase Incompressible Flows (2015, background): “The difficulty (iii) is also well studied recently: two classes of methods, one based on stabilization [44, 40, 33, 42] and the other based on convex splitting [8, 9, 39, 6], have proved to be effective.”
On large time-stepping methods for the Cahn--Hilliard equation (2007, methodology): “The numerical experiments are used to demonstrate the effectiveness of the large time-stepping approaches.”
Unconditionally stable schemes for equations of thin film epitaxy (2010, positive): “Xu and Tang [?] proposed the first-order linear scheme φ n+1 − φ n k = A ∆φ n+1 − ∆φ n + ∇ x · ∇ y F (∇ x φ n ) − ǫ 2 ∆ 2 φ n+1 , (25) where A is a linear splitting parameter.”
Numerical approximations of Allen-Cahn and Cahn-Hilliard equations (2010, methodology): “2 for the Cahn-Hilliard equation below), we can prove that the stabilized scheme for the epitaxial growth model consider in [23] satisfies an energy law and is also unconditionally stable.”
The phase field method for geometric moving interfaces and their numerical approximations (2019, neutral): “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”
Implicit-Explicit Scheme for the Allen-Cahn Equation Preserves the Maximum Principle (2016, methodology): “, [2, 5, 7, 8, 11, 13], and there have no rigorous l∞-stability analysis for the numerical methods.”
Error analysis of stabilized semi-implicit method of Allen-Cahnequation (2009, methodology): “The similar technique of the stabilizer also appeared in [17].”
An Energy-Stable and Convergent Finite-Difference Scheme for the Phase Field Crystal Equation (2009, methodology): “We mention that an alternate approach to the nonlinear splitting scheme proposed here is a linear splitting scheme, as was suggested by Eyre [8] for the Cahn-Hilliard equation and by Xu and Tang [20] for a bistable epitaxial thin film equation.”
Stabilized Crank-Nicolson/Adams-Bashforth Schemes for Phase Field Models (2013, methodology): “He, Y. Liu, T. Tang, On large time-stepping methods for the Cahn-Hilliard equation, Appl.”
Second-order Convex Splitting Schemes for Gradient Flows with Ehrlich-Schwoebel Type Energy: Application to Thin Film Epitaxy (2012, background): “We also point out that efficient first-, second-, and third-order in time accurate schemes have been investigated in [32], where the authors derived energy stability for their first-order scheme under certain assumption on the numerically computed solution itself.”
Stable and efficient finite-difference nonlinear-multigrid schemes for the phase field crystal equation (2009, methodology): “[6] for the Cahn–Hilliard equation and by Xu and Tang [20] for a bistable epitaxial thin film equation.”
An Adaptive Time-Stepping Strategy for the Cahn-Hilliard Equation (2012, methodology): “In [20], the same large time-stepping method was applied to the epitaxial growth models.”
Numerical approximations for the molecular beam epitaxial growth model based on the invariant energy quadratization method (2017, methodology): “[18,37,38]), W (t) = √ 1 | | ∫ (φ(x, t) − φ̄(x, t))2dx, where φ̄(t) = 1 ∫ φ(x, t)dx is the average.”
Linear, first and second-order, unconditionally energy stable numerical schemes for the phase field model of homopolymer blends (2016, background): “[5,31,34]), where the term from the nonlinear potential is simply treated explicitly and some linear stabilizing terms are added to improve the stability.”