Next Article in Journal
On Formality of Some Homogeneous Spaces
Previous Article in Journal
Cosmological Consequences of a Parametrized Equation of State
Previous Article in Special Issue
A Numerical Solution of Fredholm Integral Equations of the Second Kind Based on Tight Framelets Generated by the Oblique Extension Principle
Open AccessArticle

Numerical Simulation of Pattern Formation on Surfaces Using an Efficient Linear Second-Order Method

Department of Mathematics, Kwangwoon University, Seoul 01897, Korea
Symmetry 2019, 11(8), 1010; https://doi.org/10.3390/sym11081010
Received: 4 June 2019 / Revised: 24 July 2019 / Accepted: 25 July 2019 / Published: 5 August 2019
(This article belongs to the Special Issue Numerical Analysis or Numerical Method in Symmetry)
  |  
PDF [1315 KB, uploaded 5 August 2019]
  |  

Abstract

We present an efficient linear second-order method for a Swift–Hohenberg (SH) type of a partial differential equation having quadratic-cubic nonlinearity on surfaces to simulate pattern formation on surfaces numerically. The equation is symmetric under a change of sign of the density field if there is no quadratic nonlinearity. We introduce a narrow band neighborhood of a surface and extend the equation on the surface to the narrow band domain. By applying a pseudo-Neumann boundary condition through the closest point, the Laplace–Beltrami operator can be replaced by the standard Laplacian operator. The equation on the narrow band domain is split into one linear and two nonlinear subequations, where the nonlinear subequations are independent of spatial derivatives and thus are ordinary differential equations and have closed-form solutions. Therefore, we only solve the linear subequation on the narrow band domain using the Crank–Nicolson method. Numerical experiments on various surfaces are given verifying the accuracy and efficiency of the proposed method. View Full-Text
Keywords: Swift–Hohenberg type of equation; surfaces; narrow band domain; closest point method; operator splitting method Swift–Hohenberg type of equation; surfaces; narrow band domain; closest point method; operator splitting method
Figures

Figure 1

This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
SciFeed

Share & Cite This Article

MDPI and ACS Style

Lee, H.G. Numerical Simulation of Pattern Formation on Surfaces Using an Efficient Linear Second-Order Method. Symmetry 2019, 11, 1010.

Show more citation formats Show less citations formats

Note that from the first issue of 2016, MDPI journals use article numbers instead of page numbers. See further details here.

Related Articles

Article Metrics

Article Access Statistics

1

Comments

[Return to top]
Symmetry EISSN 2073-8994 Published by MDPI AG, Basel, Switzerland RSS E-Mail Table of Contents Alert
Back to Top