Application of direct meshless local Petrov–Galerkin (DMLPG) method for some Turing-type models

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• 作者：Mohammad Ilati ; Mehdi Dehghan
• 关键词：Meshless local weak form methods ; Generalized moving least squares (GMLS) approximation ; MLPG methods ; Direct MLPG (DMLPG) methods ; Schnakenberg model ; Gierer–Meinhardt model ; FitzHugh–Nagumo model ; Gray–Scott model
• 刊名：Engineering with Computers
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：33
• 期：1
• 页码：107-124
• 全文大小：
• 刊物类别：Computer Science
• 刊物主题：Computer-Aided Engineering (CAD, CAE) and Design; Math. Applications in Chemistry; Systems Theory, Control; Calculus of Variations and Optimal Control; Optimization; Classical Mechanics; Appl.Mathemat
• 出版者：Springer London
• ISSN：1435-5663
• 卷排序：33

文摘

Mathematical modeling of pattern formation in developmental biology leads to non-linear reaction–diffusion systems which are usually highly stiff in both diffusion and reaction terms. In this paper, the Direct Meshless Local Petrov–Galerkin (DMLPG) procedure is applied to find the numerical solution of some non-linear time-dependent reaction–diffusion systems such as Schnakenberg model, Gierer–Meinhardt model, FitzHugh–Nagumo model and Gray–Scott model. As far as we know, it is the first time that DMLPG method is applied for solving non-linear partial differential equations (PDEs) and systems of PDEs. Computational efficiency is the most significant advantage of the DMLPG method in comparison with the classic Meshless Local Petrov–Galerkin (MLPG) method. This is due to the fact that DMLPG shifts the numerical integrations over low-degree polynomials instead of over complicated moving least squares (MLS) shape functions and this reduces the computational costs, significantly. The main aim of this paper is to show that the DMLPG method is also suitable for solving the non-linear time-dependent systems, especially reaction–diffusion systems. Numerical results support the good efficiency of the proposed method for solving non-linear reaction–diffusion systems. Also it is shown that DMLPG provides considerable savings in computational time in comparison with the classical MLPG method.