## THERMAL SCIENCE

International Scientific Journal

### Thermal Science - Online First

online first only
### 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 nonlinear 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. Theoretically, 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

- 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