The main aim of this paper is to develop a new framework of a meshless approximation for solving numerically three nonlinear partial differential equations in biology, i.e., the chemotaxis models defined on the smooth, closed manifolds embedded in ${\mathbb{R}}^3$. The radial basis function-generated finite difference scheme is considered to deal with the spatial variables, which depends only on the location of nodes and the value of normal vector at each point per spherical cap. The robust artificial hyperviscosity formulation is derived for each model, which has been used for preventing the spurious growth modes in the numerical solution. An implicit–explicit time discretization is employed to deal with the time variable. The resulting fully discrete scheme is solved via the biconjugate gradient stabilized method with a zero-fill incomplete lower upper preconditioner per time step, where a positivity-preserving filter is used to prevent the negative sign of the cell density variable. Besides, to reduce the used central processor unit (CPU) time, the proper orthogonal decomposition is considered for constructing a set of new orthogonal basis vectors based on the singular value decomposition. The developed numerical method is called a proper orthogonal decomposition-radial basis function-generated finite difference (POD-RBF-FD) scheme. Finally, the ability of the proposed method is investigated by simulation results showing the blowing-up, pattern formulation (perforated stripe) and aggregations of bacteria on some surfaces.

本文的主要目的是开发一个无网格近似的新框架，用于数值求解生物学中的三个非线性偏微分方程，即定义在嵌入在${\mathbb{R}}^3$. 考虑径向基函数生成的有限差分格式来处理空间变量，它仅取决于节点的位置和每个球冠每个点的法向量值。为每个模型导出了鲁棒的人工高粘度公式，用于防止数值解中的虚假增长模式。采用隐式-显式时间离散化来处理时间变量。得到的完全离散方案通过双共轭梯度稳定方法求解，每个时间步都有一个零填充不完整的下上预条件子，其中使用正性保持滤波器来防止细胞密度变量的负号。此外，为了减少使用的中央处理器（CPU）时间，在奇异值分解的基础上，考虑适当的正交分解来构造一组新的正交基向量。所开发的数值方法称为适当的正交分解-径向基函数生成的有限差分（POD-RBF-FD）方案。最后，通过模拟结果研究了所提出方法的能力，该结果显示了一些表面上细菌的膨胀、图案形成（穿孔条纹）和聚集。