Numerical scheme for solving the nonuniformly forced cubic and quintic Swift–Hohenberg equations strictly respecting the Lyapunov functional

Coelho, Daniel Menezes; Vitral, Eduardo; Pontes, J.; Mangiavacchi, Norberto

doi:10.1016/j.cam.2021.114005

Cited by 2 publications

(8 citation statements)

References 32 publications

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“…The scheme preserves the monotonically decreasing nature of the associated Lyapunov functional. A description of the scheme is given in [23] and summarized in the Appendix A.…”

Section: The Swift-hohenberg Equationmentioning

confidence: 99%

“…III. The study was made through a numerical integration of the SH3 equation, using a finitedifference semi-implicit time splitting scheme that has been previously adopted for Swift-Hohenberg [23,26] and other nonlinear parabolic equations [27]. The common parameters used were α = 1, and q 0 = 1.…”

Section: Competition Between the Gradient Boundary And Bulk Effects -Sh3mentioning

confidence: 99%

“…Space and time steps were chosen following [23] for a good compromise between accuracy and computational cost (performance). The grid resolution represents the number of nodes per critical wavelength.…”

Section: Parameters Adopted In the Simulationsmentioning

confidence: 99%

“…The scheme is unconditionally stable, features truly second order representation of time and space derivatives, and strict representation of the associated Lyapunov functional, which monotonically decays. Details of the scheme are given by Christov and Pontes (2002) [24] and by Coelho et al (2020) [23]. The scheme is summarized below, for the sake of completeness.…”

Section: Appendix A: Finite Difference Scheme For Solving the Bidimen...mentioning

confidence: 99%

“…Appendix A, describes the finite differences scheme adopted for solving the Swift-Hohenberg equation and lists the parameters used in the simulations. Further details of the proceedure are given in our recent work [23]. Appendix B describes the spectral method employed for obtaining steady state amplitude profiles in cross sections of the patterns.…”

Section: Introductionmentioning

confidence: 99%

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Stripe patterns orientation resulting from nonuniform forcings and other competitive effects in the Swift-Hohenberg dynamics

Coelho,

Vitral,

Pontes

et al. 2020

Preprint

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Spatio-temporal pattern formation in natural systems presents rich nonlinear dynamics which lead to the emergence of periodic nonequilibrium structures. One of the most successful equations currently available for theoretically investigating the behavior of these structures is the Swift-Hohenberg (SH), which contains a bifurcation parameter (forcing) that controls the dynamics by changing the energy landscape of the system. Though a large part of the literature on pattern formation addresses uniformly forced systems, nonuniform forcings are also observed in several natural systems, for instance, in developmental biology and in soft matter applications. In these cases, an orientation effect due to a forcing gradient is a new factor playing a role in the development of patterns, particularly in the class of stripe patterns, which we investigate through the nonuniformly forced SH dynamics. The present work addresses the stability of stripes, and the competition between the orientation effect of the gradient and other bulk, boundary, and geometric effects taking part in the selection of the emerging patterns. A weakly nonlinear analysis shows that stripes tend to align with the gradient, and become unstable when perpendicular to the preferred direction. This analysis is complemented by a numerical work that accounts for other effects, confirming that stripes align, or even reorient from preexisting conditions. However, we observe that the orientation effect does not always prevail in face of competing effects. Other than the cubic SH equation (SH3), the quadratic-cubic (SH23) and cubic-quintic (SH35) equations are also studied. In the SH23 case, not only do forcing gradients lead to stripe orientation, but also interfere in the transition from hexagonal patterns to stripes.

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“…The scheme preserves the monotonically decreasing nature of the associated Lyapunov functional. A description of the scheme is given in [23] and summarized in the Appendix A.…”

Section: The Swift-hohenberg Equationmentioning

confidence: 99%

Section: Competition Between the Gradient Boundary And Bulk Effects -Sh3mentioning

confidence: 99%

Section: Parameters Adopted In the Simulationsmentioning

confidence: 99%

Section: Appendix A: Finite Difference Scheme For Solving the Bidimen...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Stripe patterns orientation resulting from nonuniform forcings and other competitive effects in the Swift-Hohenberg dynamics

Coelho,

Vitral,

Pontes

et al. 2020

Preprint

View full text Add to dashboard Cite

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Perspective: New directions in dynamical density functional theory

Vrugt

Wittkowski

2022

J. Phys.: Condens. Matter

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Classical dynamical density functional theory (DDFT) has become one of the central modeling approaches in nonequilibrium soft matter physics. Recent years have seen the emergence of novel and interesting fields of application for DDFT. In particular, there has been a remarkable growth in the amount of work related to chemistry. Moreover, DDFT has stimulated research on other theories such as phase field crystal models and power functional theory. In this perspective, we summarize the latest developments in the field of DDFT and discuss a variety of possible directions for future research.

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Numerical scheme for solving the nonuniformly forced cubic and quintic Swift–Hohenberg equations strictly respecting the Lyapunov functional

Cited by 2 publications

References 32 publications

Stripe patterns orientation resulting from nonuniform forcings and other competitive effects in the Swift-Hohenberg dynamics

Stripe patterns orientation resulting from nonuniform forcings and other competitive effects in the Swift-Hohenberg dynamics

Perspective: New directions in dynamical density functional theory

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