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Discontinuous Galerkin via Interpolation: The Direct Flux Reconstruction Method
Journal of Scientific Computing ( IF 2.5 ) Pub Date : 2020-03-07 , DOI: 10.1007/s10915-020-01175-3
H. T. Huynh

Abstract

The discontinuous Galerkin (DG) method is based on the idea of projection using integration. The recent direct flux reconstruction (DFR) method by Romero et al. (J Sci Comput 67(1):351–374, 2016) is derived via interpolation and results in a scheme identical to DG (on hexahedral meshes). The DFR method is further studied and developed here. Two proofs for its equivalence with the DG scheme considerably simpler than the original proof are presented. The first proof employs the \( 2K - 1 \) degree of precision by a \( K \)-point Gauss quadrature. The second shows the equivalence of DG, FR, and DFR by using the property that the derivative of the degree \( K + 1 \) Lobatto polynomial vanishes at the \( K \) Gauss points. Fourier analysis for these schemes are presented using an approach more geometric compared with existing analytic approaches. The effects of nonuniform mesh and those of high-order mesh transformation (a precursor for curved meshes in two and three spatial dimensions) on stability and accuracy are examined. These nonstandard analyses are obtained via an in-depth study of the behavior of eigenvalues and eigenvectors.



中文翻译:

通过插值的不连续Galerkin:直接通量重建方法

摘要

不连续Galerkin(DG)方法基于使用集成进行投影的思想。Romero等人最近的直接通量重建(DFR)方法。(J Sci Comput 67(1):351–374,2016)是通过插值得出的,其结果与DG(在六面体网格上)相同。DFR方法在这里进一步研究和开发。给出了与DG方案等效的两个证明,比原始证明要简单得多。第一个证明采用\(K \)点高斯正交积分的\(2K-1 \)精度。第二个通过使用以下性质来显示DG,FR和DFR的等价性:\(K + 1 \) Lobatto多项式的导数在\(K \)处消失高斯点。与现有的分析方法相比,使用更几何的方法对这些方案进行了傅里叶分析。检验了非均匀网格和高阶网格转换(二维和三个空间维曲面网格的先驱)对稳定性和准确性的影响。这些非标准分析是通过对特征值和特征向量的行为进行深入研究而获得的。

更新日期:2020-03-20
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