Discontinuous Galerkin via interpolation: the direct flux reconstruction method. (English) Zbl 1445.76062
Summary: The discontinuous Galerkin (DG) method is based on the idea of projection using integration. The recent direct flux reconstruction (DFR) method by J. Romero et al. [J. Sci. Comput. 67, No. 1, 351–374 (2016; Zbl 1339.65180)] 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.
MSC:
| 76M99 | Basic methods in fluid mechanics |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
high-order method; stability; Fourier analysis; von Neumann analysis; Gauss quadrature; Lobatto polynomialCitations:
Zbl 1339.65180References:
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