## THERMAL SCIENCE

International Scientific Journal

### AN ENERGY-STABLE PSEUDOSPECTRAL SCHEME FOR SWIFT-HOHENBERG EQUATION WITH ITS LYAPUNOV FUNCTIONAL

**ABSTRACT**

We analyze a first order in time Fourier pseudospectral scheme for Swift-Hohenberg equation. One major challenge for the higher order diffusion non-linear systems is how to ensure the unconditional energy stability and we propose an efficient scheme for the equation based on the convex splitting of the energy. The¬oretically, the energy stability of the scheme is proved. Moreover, following the derived aliasing error estimate, the convergence analysis in the discrete l2-norm for the proposed scheme is given.

**KEYWORDS**

PAPER SUBMITTED: 2018-06-12

PAPER REVISED: 2018-09-20

PAPER ACCEPTED: 2018-11-25

PUBLISHED ONLINE: 2019-06-08

**THERMAL SCIENCE** YEAR

**2019**, VOLUME

**23**, ISSUE

**Supplement 3**, PAGES [S975 - S982]

- Swift J., et al., Hydrodynamic Fluctuations at the Convective Instability, Physical Review A, 15(1977), 1, pp. 319-328
- Bordeu I., et al., From Localized Spots to the Formation of Invaginated Labyrinthine Structures in a Swift-Hohenberg Model, Communications in Nonlinear Science and Numerical Simulation, 29(2015), 1-3, pp. 482-4874
- Lloyd D., et al., Localized Radial Solutions of the Swift-Hohenberg Equation, Nonlinearity, 22(2009), 2, pp. 485-524
- Mccalla S., et al., Snaking of Radial Solutions of the Multi-Dimensional Swift-Hohenberg Equation: A Numerical Study, Physica D, 239(2010), 16, pp. 1581-1592
- Mccalla S., et al., Spots in the Swift-Hohenberg Equation, SIAM Journal on Applied Dynamical Systems, 12(2013), 2, pp. 831-877
- Gomez H., et al., A New Space-Time Discretization for the Swift-Hohenberg Equation that Strictly Respects the Lyapunov Functional, Communications in Nonlinear Science and Numerical Simulation, 17(2012), 12, pp. 4930-4946
- Christov C. I., et al., Implicit Time Splitting for Fourth-Order Parabolic Equations, Computer Methods in Applied Mechanics and Engineering, 148(1997), 3-4, pp. 209-224
- Christov C. I., et al., Numerical Scheme for Swift-Hohenberg Equation with Strict Implementation of Lyapunov Functional, Mathematical and Computer Modelling, 35(2002), 1, pp. 87-99
- Cheng K., et al., A Second-Order, Weakly Energy-Stable Pseudo-Spectral Scheme for the Cahn-Hilliard Equation and its Solution by the Homogeneous Linear Iteration Method, Journal of Scientific Computing, 69(2016), 3, pp. 1083-1114
- Cheng K., et al., An Energy Stable Fourth Order Finite Difference Scheme For The Cahn-Hilliard Equation, Journal of Computational and Applied Mathematics, 2018, DOI 10.1016/j.cam.2018.05.039
- Elsey M., et al., A Simple and Efficient Scheme for Phase Field Crystal Simulation, ESAIM: Mathematical Modelling and Numerical Analysis, 47(2013), 5, pp. 1413-1432
- Wise S. M., et al., An Energy Stable and Convergent Finite-Difference Scheme for the Phase Field Crystal Equation, SIAM Journal on Numerical Analysis, 47(2009), 3, pp. 2269-2288
- Eyre D. J., Unconditionally Gradient Stable Time Marching the Cahn-Hilliard Equation, MRS Online Proceeding Library Archive, 529(2011), 1, pp. 39-46
- Gottlieb S., et al., Stability and Convergence Analysis of Fully Discrete Fourier Collocation Spectral Method for 3-D Viscous Burgers' Equation, Journal of Scientific Computing, 53(2012), 1, pp.102-128