Phase-field modeling and linearly energy-stable Runge–Kutta algorithm of colloidal crystals on curved surfaces

Abstract

In this work, we develop an efficient and unconditionally energy-stable numerical method for solving a generalized H−1-gradient flow based Swift–Hohenberg equation for colloidal crystal on surfaces. By introducing an appropriate truncated nonlinear potential, we construct linear and temporally second-order accurate scheme based on the linear convex splitting and implicit-explicit Runge–Kutta approaches. The surface is discretized by a set of triangles. The spatial approximation is performed based on the finite volume type method. During every time step, our task simply involves solving a set of elliptic-type equations that have constant coefficients. We analytically prove the energy dissipation law and unique solvability. Numerical examples indicate that the proposed scheme not only works well on various surfaces but also has desired accuracy and energy stability. We also provide the MATLAB code for generating triangular mesh on the curved surface in the link: http://github.com/yang521/MATLAB-distmesh.