In this work, a Generalized Finite Difference (GFD) scheme is presented for effectively computing the numerical solution of a parabolic–elliptic system modeling a bacterial strain with density-suppressed motility. The GFD method is a meshless method known for its simplicity for solving non-linear boundary value problems over irregular geometries. The paper first introduces the basic elements of the GFD method, and then an explicit–implicit scheme is derived. The convergence of the method is proven under a bound for the time step, and an algorithm is provided for its computational implementation. Finally, some examples are considered comparing the results obtained with a regular mesh and an irregular cloud of points.

中文翻译：

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抛物线-椭圆密度抑制运动系统的显式-隐式广义有限差分格式

在这项工作中，提出了一种广义有限差分（GFD）方案，用于有效计算抛物线-椭圆系统的数值解，该系统模拟具有密度抑制运动性的细菌菌株。 GFD 方法是一种无网格方法，以其解决不规则几何形状上的非线性边值问题的简单性而闻名。本文首先介绍了GFD方法的基本要素，然后推导了显式-隐式方案。在时间步有界下证明了该方法的收敛性，并给出了计算实现的算法。最后，考虑一些示例来比较使用规则网格和不规则点云获得的结果。