An extended range of energy stable flux reconstruction schemes, developed using a summation-by-parts approach, is presented on quadrilateral elements for various sets of polynomial bases. For the maximal order bases, a new set of correction functions which result in stable schemes is found. However, for a range of orders it is shown that only a single correction function can be cast as a tensor-product. Subsequently, correction functions are identified using a generalised analytic framework that results in stable schemes for total order and approximate Euclidean order polynomial bases on quadrilaterals -- which have not previously been explored in the context of flux reconstruction. It is shown that the approximate Euclidean order basis can provide similar numerical accuracy as the maximal order basis but with fewer points per element, and thus lower cost.
翻译:利用逐个总和的方法开发的能源稳定通量重建计划范围更广,是按多种多元基数的四边元素排列的。对于最大秩序基础,可以找到一套新的纠正功能,从而形成稳定的计划。然而,对于一系列命令,可以证明只有单一的纠正功能可以作为一种发压产品。随后,利用一个一般分析框架来确定了纠正功能,从而形成了四边基数上总秩序和接近欧几里德秩序的近似多面基数的稳定计划,而这种计划以前在通量重建中尚未探讨过,这表明大约欧几里德秩序基础可以提供与最大秩序基础相似的数字精确度,但每个要素的点数较少,因此成本较低。