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

• Main Author: Hyun Geun Lee
• Format: Article
• Language: English
• Published: MDPI AG 2019-08-01
• Series: Symmetry
• Subjects: Swift–Hohenberg type of equation; surfaces; narrow band domain; closest point method; operator splitting method
• Online Access: https://www.mdpi.com/2073-8994/11/8/1010

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.