This article presents a local mesh-less procedure for simulating the Galilei invariant fractional advection–diffusion (GI-FAD) equations in one, two, and three-dimensional spaces. The proposed method combines a second-order Crank–Nicolson scheme for time discretization and the second-order weighted and shifted Grünwald difference (WSGD) formula. This time discretization scheme ensures unconditional stability and convergence with an order of . In the spatial domain, a mesh-less weak form is employed based on the direct mesh-less local Petrov–Galerkin (DMLPG) method. The DMLPG method employs the generalized moving least-square (GMLS) approximation in conjunction with the local weak form of the equation. By utilizing simple polynomials as shape functions in the GMLS approximation, the necessity for complex shape function construction in the MLS approximation is eliminated. To validate and demonstrate the effectiveness of the proposed algorithm, a variety of problems in one, two, and three dimensions are investigated on both regular and irregular computational domains. The numerical results obtained from these investigations confirm the accuracy and reliability of the developed approach in simulating GI-FAD equations.

中文翻译：

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使用直接无网格局部 Petrov-Galerkin 方法对多维 Galilei 不变分数平流扩散方程进行数值研究


本文提出了一种局部无网格程序，用于模拟一维、二维和三维空间中的伽利略不变分数平流扩散 (GI-FAD) 方程。所提出的方法结合了用于时间离散化的二阶 Crank-Nicolson 方案和二阶加权移位 Grünwald 差分（WSGD）公式。这种时间离散化方案确保了无条件稳定性和收敛性，其量级为 。在空间域中，采用基于直接无网格局部 Petrov-Galerkin (DMLPG) 方法的无网格弱形式。 DMLPG 方法采用广义移动最小二乘 (GMLS) 近似以及方程的局部弱形式。通过利用简单多项式作为 GMLS 近似中的形状函数，消除了 MLS 近似中复杂形状函数构造的必要性。为了验证和证明所提出算法的有效性，在规则和不规则计算域上研究了一维、二维和三维的各种问题。这些研究获得的数值结果证实了所开发的 GI-FAD 方程模拟方法的准确性和可靠性。