The aim of this paper is to investigate the application of radial basis function-generated finite difference (RBF-FD) methods for convection–diffusion partial differential equations (PDEs) on a sphere. In the application of RBF-FD method, choosing a reasonable value of shape parameter is important to the computation of PDEs. The work is devoted to the numerical study of the range of near optimal shape parameters for the convection–diffusion equations. Because the RBF-FD Direct method often leads to ill-conditioned problems for small shape parameters, the RBF-QR method is applied locally to overcome the ill-conditioning in the context of RBF-FD mode. Additionally, for convection-dominated problems, it can be found that the results of using central-type stencil present spurious oscillations. Therefore, we propose an upwind RBF-FD (URBF-FD) scheme to overcome the problems, which is well adapted to the problems on the sphere and easy to be implemented. Further numerical results show that the proposed URBF-FD method is stable and effective for convection-dominated PDEs on the sphere.

中文翻译：

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使用 RBF-FD 和 RBF-QR 方法对球体上的对流扩散偏微分方程进行数值模拟

本文的目的是研究径向基函数生成的有限差分 (RBF-FD) 方法在球体上对流-扩散偏微分方程 (PDE) 的应用。在RBF-FD方法的应用中，选择合理的形状参数取值对偏微分方程的计算很重要。这项工作致力于对流-扩散方程的近似最优形状参数范围的数值研究。由于 RBF-FD Direct 方法经常导致小形状参数的病态问题，因此在 RBF-FD 模式下局部应用 RBF-QR 方法来克服病态问题。此外，对于以对流为主的问题，可以发现使用中心型模板的结果存在虚假振荡。所以，我们提出了一种迎风RBF-FD（URBF-FD）方案来克服这些问题，该方案很好地适应了球体上的问题并且易于实现。进一步的数值结果表明，所提出的 URBF-FD 方法对于球体上以对流为主的偏微分方程是稳定有效的。