An efficient numerical method for reaction–diffusion equation on the general curved surfaces

In this paper, we propose an efficient numerical algorithm for reaction–diffusion equation on the general curved surface. The surface is discretized by a mesh consisting of triangular grids. The partial differential operators are defined based on the surface mesh and its dual surface polygonal tessellation. The proposed method has three advantages including intrinsic geometry, conservation law, and convergence property. The proposed method only needs the information of 1-ring of neighboring vertices for the divergence of a vector field and the Laplace–Beltrami operators, while the numerical conservation laws still hold. The proposed method avoids the global surface triangulation and its implementation is simple since we can explicitly define the Laplace–Beltrami operator by using the information of the neighborhood of each triangular grid. In order to obtain second-order temporal accuracy, we utilize the Crank–Nicolson formula to the reaction–diffusion system. The discrete system is solved by the biconjugate gradient stabilized method. The proposed algorithm is simple to implement and is second-order accurate both in space and time. Various numerical experiments are presented to demonstrate the efficiency of our algorithm.