Numerical solution of the Swift-Hohenberg equation in two dimensions

Hao wen Xi; Jorge Viñals; J. D. Gunton

doi:10.1016/0378-4371(91)90173-A

Numerical solution of the Swift-Hohenberg equation in two dimensions

Hao wen Xi, Jorge Viñals, J. D. Gunton

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Abstract

The Swift-Hohenberg equation with either a stochastic or a constant forcing term has been solved numerically in two spatial dimensions. The parameters that enter the equation have been chosen to match the experiments on Rayleigh-Bénard convection by Meyer et al. [C.W. Meyer, G. Ahlers and D.S. Cannell, Phys. Rev. Lett. 59 (1987) 1577]. Our numerical results for the convective heat current as a function of time fit the experiments well (the fitting parameter is the amplitude of the forcing term). We find a value of F = 5.52 × 10^-6 for the stochastic case, compared to F_th = 1.06 × 10^-10, the value obtained from fluctuation theory. The structure of the convective pattern also depends on the type of forcing considered. A constant forcing induces a roll-like pattern that reflects the geometry of the sidewalls. A stochastic forcing is seen to induce a random, cellular pattern.

Original language	English (US)
Pages (from-to)	356-365
Number of pages	10
Journal	Physica A: Statistical Mechanics and its Applications
Volume	177
Issue number	1-3
DOIs	https://doi.org/10.1016/0378-4371(91)90173-A
State	Published - Sep 15 1991
Externally published	Yes

Bibliographical note

Funding Information:
We would like to thank G. Ahlers and P.C. Hohenberg for suggesting the numerical investigation of the stochastic Swift-Hohenberg equation and G. Ahlers, C. Meyer and P.C. Hohenberg for many stimulating conversations and comments. We also thank H. Greenside for useful suggestions concerning the numerical code. This work was supported in part by the National Science Foundation under Grant No. DMR-9100245. This work is also supported in part by the Supercomputer Computations Research Institute, which is partially funded by the U.S. Department of Energy contract No. DE-FC05-85ER25000. Most of the calculations reported here have been performed on the 64K-node Connection Machine at the Supercomputer Computations Research Institute. We also thank Jim Hudgens of SCRI for his assistance with the graphics packages used to analyze the data.

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title = "Numerical solution of the Swift-Hohenberg equation in two dimensions",

abstract = "The Swift-Hohenberg equation with either a stochastic or a constant forcing term has been solved numerically in two spatial dimensions. The parameters that enter the equation have been chosen to match the experiments on Rayleigh-B{\'e}nard convection by Meyer et al. [C.W. Meyer, G. Ahlers and D.S. Cannell, Phys. Rev. Lett. 59 (1987) 1577]. Our numerical results for the convective heat current as a function of time fit the experiments well (the fitting parameter is the amplitude of the forcing term). We find a value of F = 5.52 × 10-6 for the stochastic case, compared to Fth = 1.06 × 10-10, the value obtained from fluctuation theory. The structure of the convective pattern also depends on the type of forcing considered. A constant forcing induces a roll-like pattern that reflects the geometry of the sidewalls. A stochastic forcing is seen to induce a random, cellular pattern.",

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note = "Funding Information: We would like to thank G. Ahlers and P.C. Hohenberg for suggesting the numerical investigation of the stochastic Swift-Hohenberg equation and G. Ahlers, C. Meyer and P.C. Hohenberg for many stimulating conversations and comments. We also thank H. Greenside for useful suggestions concerning the numerical code. This work was supported in part by the National Science Foundation under Grant No. DMR-9100245. This work is also supported in part by the Supercomputer Computations Research Institute, which is partially funded by the U.S. Department of Energy contract No. DE-FC05-85ER25000. Most of the calculations reported here have been performed on the 64K-node Connection Machine at the Supercomputer Computations Research Institute. We also thank Jim Hudgens of SCRI for his assistance with the graphics packages used to analyze the data.",

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Numerical solution of the Swift-Hohenberg equation in two dimensions

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