论文标题
关于使用RBF插值进行通量重建
On the Use of RBF Interpolation for Flux Reconstruction
论文作者
论文摘要
通量重建提供了一个框架，用于求解部分微分方程，其中函数在元素中不连续近似。通常，这是通过使用多项式来完成的。在这里，使用分析方法和数值方法，在一个维度中探索了径向基函数作为基础功能近似方法的方法。在某些网格密度下，发现RBF通量重建超过了多项式通量重建，并且随着RBF interpolator的宽度的增加，这种网格密度变得更细。一种避免扁平RBF条件差的方法用于测试各种基础形状，并且在很小的值下，恢复了多项式行为。发现更改溶液点的位置的效果与多项式FR中的效果相似，而高斯 - legendre点是最有效的。发现更改功能中心的位置对性能的影响很小。针对非线性汉堡方程式确定类似的行为。
Flux reconstruction provides a framework for solving partial differential equations in which functions are discontinuously approximated within elements. Typically, this is done by using polynomials. Here, the use of radial basis functions as a methods for underlying functional approximation is explored in one dimension, using both analytical and numerical methods. At some mesh densities, RBF flux reconstruction is found to outperform polynomial flux reconstruction, and this range of mesh densities becomes finer as the width of the RBF interpolator is increased. A method which avoids the poor conditioning of flat RBFs is used to test a wide range of basis shapes, and at very small values, the polynomial behaviour is recovered. Changing the location of the solution points is found to have an effect similar to that in polynomial FR, with the Gauss--Legendre points being the most effective. Altering the location of the functional centres is found to have only a very small effect on performance. Similar behaviours are determined for the non-linear Burgers' equation.
