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.

在本文中，我们提出了一种有效的一般曲面上反应扩散方程的数值算法。表面由三角形网格组成的网格离散化。偏微分算子是基于曲面网格及其对偶曲面多边形细分定义的。该方法具有三个优点，包括内在几何、守恒定律和收敛性。所提出的方法只需要向量场的散度和拉普拉斯-贝尔特拉米算子的相邻顶点的一环信息，而数值守恒定律仍然成立。所提出的方法避免了全局表面三角剖分并且其实现很简单，因为我们可以通过使用每个三角形网格的邻域信息来明确定义拉普拉斯-贝尔特拉米算子。为了获得二阶时间精度，我们将 Crank-Nicolson 公式用于反应-扩散系统。离散系统采用双共轭梯度稳定法求解。所提出的算法实现简单，在空间和时间上都具有二阶精度。提出了各种数值实验来证明我们算法的效率。