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.
Highlights
A Swift–Hohenberg (SH) type of partial differential equation [1] has been used to study pattern formation [2,3,4,5]: ∂φ= − φ3 − gφ2 + −e + (1 + ∆)2 φ,∂t where φ is the density field and g ≥ 0 and e > 0 are constants
We present an efficient linear second-order method for the SH type of equation on surfaces, which is based on the closest point method [14,15]
We introduce a narrow band domain of a surface and apply a pseudo-Neumann boundary condition on the boundary of the narrow band domain through the closest point [16]
Summary
A Swift–Hohenberg (SH) type of partial differential equation [1] has been used to study pattern formation [2,3,4,5]:. We present an efficient linear second-order method for the SH type of equation on surfaces, which is based on the closest point method [14,15]. We introduce a narrow band domain of a surface and apply a pseudo-Neumann boundary condition on the boundary of the narrow band domain through the closest point [16]. This results in a constant value of φ in the direction normal to the surface, the Laplace–Beltrami operator can be replaced by the standard Laplacian operator.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
