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.
翻译:Flus 重建提供了解决部分差异方程式的框架,其中函数在元素中不连续地相近。 通常, 这样做的方法是使用多元基体来测试多种基体形状, 而在极小的数值下, 将光基函数作为基本功能近似的一种方法, 使用分析和数字方法进行探讨 。 在某些网状密度下, RBF 流量重建发现优于多球通量重建, 并且随着 RBF 间插器的宽度增加, 网状密度范围变得更小。 一种避免平板RBF 条件差的方法被用来测试多种基体形状, 在极小的数值下, 恢复多球行为 。 改变解决方案点的位置的效果与多球状调的效果相似, 高子- legendre 点效果最为有效 。 改变功能中心的位置被认为对性能的影响很小。 对非线状布尔格斯 等式也确定了类似的行为 。