This work presents a numerical algorithm for using radial basis function-generated finite differences (RBF-FD) to solve partial differential equations (PDEs) on ${\mathbb{S}}^2$ using polyharmonic splines with added polynomials defined in a 2D plane (PHS+Poly). We introduce a novel method for calculating RBF-FD PHS+Poly differentiation weights on ${\mathbb{S}}^2$ using first a Householder reflection and then a projection onto the tangent plane. The new PHS+Poly RBF-FD method is implemented on two standard test cases: 1) solid body rotation on ${\mathbb{S}}^2$ and 2) 3D tracer transport within Earth's atmosphere. Compared to existing methods (including those in the RBF literature) at similar resolutions, this approach requires fewer degrees of freedom and is algorithmically much simpler. A MATLAB code to implement the method is included in the Appendix.

这项工作提出了一种使用径向基函数生成的有限差分（RBF-FD）求解偏微分方程（PDE）的数值算法。 ${\mathbb{小号}}^2$使用在2D平面（PHS + Poly）中定义的具有添加多项式的多谐波样条曲线。我们介绍了一种新的方法来计算RBF-FD PHS +多态性权重${\mathbb{小号}}^2$首先使用Householder反射，然后使用切线平面上的投影。新的PHS + Poly RBF-FD方法在两个标准测试用例上实现：1）实体旋转${\mathbb{小号}}^2$2）3D示踪剂在地球大气层中的运输。与具有类似分辨率的现有方法（包括RBF文献中的方法）相比，此方法需要较少的自由度，并且在算法上更简单。附录中包含实现该方法的MATLAB代码。