论文标题
一种用于解决经典和分数PDE的统一的无网状伪谱法
A unified meshfree pseudospectral method for solving both classical and fractional PDEs
论文作者
论文摘要
在本文中，我们提出了一种基于高斯径向基函数（RBF）的网格方法，以解决经典和分数PDE。所提出的方法利用了高斯函数的分析拉普拉斯，以适应单个框架中经典和分数laplacian的离散化，并避免使用较大的计算成本，以用于分数衍生物的数值评估。这些重要的优点将其与分数PDE的其他数值方法区分开。此外，我们的方法简单易处理复杂的几何形状和本地改进，其计算机程序实现对于任何维度$ d \ ge 1 $保持不变。提供了广泛的数值实验，以研究我们方法在近似Dirichlet Laplace操作员和解决PDE问题的过程中的性能。与最近提出的Wendland RBF方法相比，我们的方法将Dirichlet边界条件完全融合到了该方案中，并且不含文献中观察到的Gibbs现象。我们的研究表明，要获得良好的精度，形状参数不能太小或太大，最佳形状参数可能取决于RBF中心点和溶液属性。
In this paper, we propose a meshfree method based on the Gaussian radial basis function (RBF) to solve both classical and fractional PDEs. The proposed method takes advantage of the analytical Laplacian of Gaussian functions so as to accommodate the discretization of the classical and fractional Laplacian in a single framework and avoid the large computational cost for numerical evaluation of the fractional derivatives. These important merits distinguish it from other numerical methods for fractional PDEs. Moreover, our method is simple and easy to handle complex geometry and local refinement, and its computer program implementation remains the same for any dimension $d \ge 1$. Extensive numerical experiments are provided to study the performance of our method in both approximating the Dirichlet Laplace operators and solving PDE problems. Compared to the recently proposed Wendland RBF method, our method exactly incorporates the Dirichlet boundary conditions into the scheme and is free of the Gibbs phenomenon as observed in the literature. Our studies suggest that to obtain good accuracy the shape parameter cannot be too small or too big, and the optimal shape parameter might depend on the RBF center points and the solution properties.
