# On Efficient Second Order Stabilized Semi-implicit Schemes for the Cahn–Hilliard Phase-Field Equation

On Efficient Second Order Stabilized Semi-implicit Schemes for the Cahn–Hilliard Phase-Field... Efficient and energy stable high order time marching schemes are very important but not easy to construct for the study of nonlinear phase dynamics. In this paper, we propose and study two linearly stabilized second order semi-implicit schemes for the Cahn–Hilliard phase-field equation. One uses backward differentiation formula and the other uses Crank–Nicolson method to discretize linear terms. In both schemes, the nonlinear bulk forces are treated explicitly with two second-order stabilization terms. This treatment leads to linear elliptic systems with constant coefficients, for which lots of robust and efficient solvers are available. The discrete energy dissipation properties are proved for both schemes. Rigorous error analysis is carried out to show that, when the time step-size is small enough, second order accuracy in time is obtained with a prefactor controlled by a fixed power of $$1/\varepsilon$$ 1 / ε , where $$\varepsilon$$ ε is the characteristic interface thickness. Numerical results are presented to verify the accuracy and efficiency of proposed schemes. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Scientific Computing Springer Journals

# On Efficient Second Order Stabilized Semi-implicit Schemes for the Cahn–Hilliard Phase-Field Equation

, Volume 77 (2) – May 31, 2018
25 pages

/lp/springer_journal/on-efficient-second-order-stabilized-semi-implicit-schemes-for-the-xqN28S9dyx
Publisher
Springer Journals
Subject
Mathematics; Algorithms; Computational Mathematics and Numerical Analysis; Mathematical and Computational Engineering; Theoretical, Mathematical and Computational Physics
ISSN
0885-7474
eISSN
1573-7691
D.O.I.
10.1007/s10915-018-0746-2
Publisher site
See Article on Publisher Site

### Abstract

Efficient and energy stable high order time marching schemes are very important but not easy to construct for the study of nonlinear phase dynamics. In this paper, we propose and study two linearly stabilized second order semi-implicit schemes for the Cahn–Hilliard phase-field equation. One uses backward differentiation formula and the other uses Crank–Nicolson method to discretize linear terms. In both schemes, the nonlinear bulk forces are treated explicitly with two second-order stabilization terms. This treatment leads to linear elliptic systems with constant coefficients, for which lots of robust and efficient solvers are available. The discrete energy dissipation properties are proved for both schemes. Rigorous error analysis is carried out to show that, when the time step-size is small enough, second order accuracy in time is obtained with a prefactor controlled by a fixed power of $$1/\varepsilon$$ 1 / ε , where $$\varepsilon$$ ε is the characteristic interface thickness. Numerical results are presented to verify the accuracy and efficiency of proposed schemes.

### Journal

Journal of Scientific ComputingSpringer Journals

Published: May 31, 2018

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