A unified meshfree pseudospectral method for solving both classical and fractional PDEs
A unified meshfree pseudospectral method for solving both classical and fractional PDEs
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DOI：
10.1137/20m1335959
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发表时间：
2020-09
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影响因子：
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通讯作者：
J. Burkardt;Yixuan Wu;Yanzhi Zhang
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文献类型：
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作者：
J. Burkardt;Yixuan Wu;Yanzhi Zhang
在本文中，我们提出了一种基于高斯径向基函数（RBF）的无网格方法来解决经典和分数偏微分方程。该方法利用高斯函数的解析拉普拉斯算子，将经典拉普拉斯算子和分数阶拉普拉斯算子的离散化统一在一个框架内，避免了分数阶导数数值计算的巨大计算开销.这些重要的优点使它区别于其他分数阶偏微分方程的数值方法。此外，我们的方法是简单的，易于处理复杂的几何和局部细化，其计算机程序实现保持相同的任何尺寸$d \ge 1$。大量的数值实验研究我们的方法在逼近Dirichlet拉普拉斯算子和解决偏微分方程问题的性能。与最近提出的Wendland径向基函数方法相比，我们的方法准确地将狄利克雷边界条件融入到方案中，并且没有文献中观察到的吉布斯现象。我们的研究表明，为了获得良好的精度，形状参数不能太小或太大，最佳形状参数可能取决于径向基函数的中心点和解决方案的性质。
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.