Purpose

This paper aims to develop a meshfree algorithm based on local radial basis functions (RBFs) combined with the differential quadrature (DQ) method to provide numerical approximations of the solutions of time-dependent, nonlinear and spatially one-dimensional reaction-diffusion systems and to capture their evolving patterns. The combination of local RBFs and the DQ method is applied to discretize the system in space; implicit multistep methods are subsequently used to discretize in time.

Design/methodology/approach

In a method of lines setting, a meshless method for their discretization in space is proposed. This discretization is based on a DQ approach, and RBFs are used as test functions. A local approach is followed where only selected RBFs feature in the computation of a particular DQ weight.

Findings

The proposed method is applied on four reaction-diffusion models: Huxley’s equation, a linear reaction-diffusion system, the Gray–Scott model and the two-dimensional Brusselator model. The method captured the various patterns of the models similar to available in literature. The method shows second order of convergence in space variables and works reliably and efficiently for the problems.

Originality/value

The originality lies in the following facts: A meshless method is proposed for reaction-diffusion models based on local RBFs; the proposed scheme is able to capture patterns of the models for big time T; the scheme has second order of convergence in both time and space variables and Nuemann boundary conditions are easy to implement in this scheme.

中文翻译：

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用于模拟瞬态反应扩散问题的局部径向基函数微分正交半离散化技术

目的

本文旨在开发一种基于局部径向基函数 (RBF) 并结合微分正交 (DQ) 方法的无网格算法，以提供瞬态、非线性和空间一维反应扩散系统解的数值近似，并捕捉他们不断发展的模式。将局部RBFs与DQ方法相结合，对系统进行空间离散化；随后使用隐式多步方法进行时间离散。

设计/方法/方法

在线条设置方法中，提出了一种空间离散化的无网格方法。这种离散化基于 DQ 方法，并且 RBF 用作测试函数。遵循本地方法，其中仅选定的 RBF 具有计算特定 DQ 权重的特征。

发现

所提出的方法应用于四种反应扩散模型：赫胥黎方程、线性反应扩散系统、Gray-Scott 模型和二维 Brusselator 模型。该方法捕获了类似于文献中可用的模型的各种模式。该方法显示了空间变量中的二阶收敛性，并且可以可靠有效地解决问题。

原创性/价值

其独创性在于：提出了一种基于局部RBFs的反应扩散模型的无网格方法；所提出的方案能够捕获大时间T的模型模式；该方案在时间和空间变量上均具有二阶收敛性，并且在该方案中易于实现 Nuemann 边界条件。