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首页> 外文期刊>Journal of Computational Physics >On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphere
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On choosing a radial basis function and a shape parameter when solving a convective PDE on a sphere

机译：在求解球体上时选择径向基函数和形状参数

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摘要

Radial basis function (RBF) approximations have been used for some time to interpolate data on a sphere (as well as on many other types of domains). Their ability to solve, to spectral accuracy, convection-type PDEs over a sphere has been demonstrated only very recently. In such applications, there are two main choices that have to be made: (i) which type of radial function to use, and (ii) what value to choose for their shape parameter (denoted by e, and with flat basis functions stretched out in the radial direction - corresponding to epsilon = 0). The recent RBF-QR algorithm has made it practical to compute stably also for small values of epsilon. Results from solving a convective-type PDE on a sphere are compared here for many choices of radial functions over the complete range of epsilon-values (from very large down to the limit of epsilon -> 0). The results are analyzed with a methodology that has similarities to the customary Fourier analysis in equispaced 1-D periodic settings. In particular, we find that high accuracy can be maintained also over very long time integrations. We furthermore gain insights into why RBFs sometimes offer higher accuracy than spherical harmonics (since the latter arise as an often non-optimal special case of the former). Anticipated future application areas for RBF-based methods in spherical geometries include weather and climate modeling. (c) 2007 Elsevier Inc. All rights reserved.
机译：径向基函数（RBF）近似已经使用了一段时间来插入球体上的数据（以及许多其他类型的域）。它们的解决能力，以光谱准确性，在球体上的对流式PDE已经已经展示。在这种应用中，有两个主要选择必须进行：（i）使用哪种类型的径向功能，并且（ii）选择它们的形状参数的值是什么（由e表示，并且延伸的平坦函数）在径向方向 - 对应于epsilon = 0）。最近的RBF-QR算法使得稳定计算较小的epsilon。在这里，在球体上求解对流型PDE的结果在此处比较了在完整范围内的胃泌素值（从非常大到epsilon - > 0的极限）的径向功能的许多选择。通过一种方法分析结果，该方法具有与常规的1-D定期设置中的常规傅立叶分析相似。特别是，我们发现可以在很长的时间集成上保持高精度。我们进一步深入了解为什么RBF有时提供比球面谐波更高的精度（因为后者由于前者的通常是非最佳特殊情况）。预期的未来适用于球形几何形状中的基于RBF的方法包括天气和气候建模。 （c）2007年elestvier Inc.保留所有权利。

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