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.
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References
Abe, Y., Haga, T., Nonomura, T., Fujii, K.: On the freestream preservation of high-order conservative flux-reconstruction schemes. J. Comput. Phys. 281, 28–54 (2015)
Adjerid, S., Devine, K.D., Flaherty, J.E., Krivodonova, L.: A posteriori error estimation for discontinuous Galerkin solutions of hyperbolic problems. Comput. Methods Appl. Mech. Eng. 191, 1097–1112 (2002)
Ainsworth, M.: Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods. J. Comput. Phys. 198(1), 106–130 (2004)
Alhawwary, M., Wang, Z.J.: Fourier analysis and evaluation of DG, FD and compact difference methods for conservation laws. J. Comput. Phys. 373, 835–862 (2015)
Bassi, F., Rebay, S.: A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equations. J. Comput. Phys. 131(1997), 267–279 (1997)
Bassi, F., Rebay, S.: High-order accurate discontinuous finite element solution for the 2D Euler equations. J. Comput. Phys. 138, 251–285 (1997)
Bassi, F., Rebay, S.: A high order discontinuous Galerkin method for compressible turbulent flows. In: Cockburn, B., Karniadakis, G., Shu, C.-W. (eds.) Discontinuous Galerkin Methods: Theory, Computation, and Application. Lecture Notes in Computational Science and Engineering, pp. 77–88. Springer, Berlin (2000)
Black, K.: A conservative spectral element method for the approximation of compressible fluid flow. Kybernetika 35(1), 133–146 (1999)
Black, K.: Spectral element approximation of convection-diffusion type problems. Appl. Numer. Math. 33(1–4), 373–379 (2000)
Cockburn, B., Karniadakis, G., Shu, C.-W. (eds.): Discontinuous Galerkin Methods: Theory, Computation, and Application. Springer, Berlin (2000)
Cockburn, B., Shu, C.-W.: The local discontinuous Galerkin methods for time-dependent convection diffusion systems. SIAM J. Numer. Anal. 35, 2440–2463 (1998)
Cockburn, B., Shu, C.-W.: Foreword for the special issue on discontinuous Galerkin method. J. Sci. Comput. 22–23, 1–3 (2005)
Cockburn, B., Shu, C.-W.: Foreword for the special issue on discontinuous Galerkin method. J. Sci. Comput. 40, 1–3 (2009)
Gassner, G., Kopriva, D.A.: A comparison of the dispersion and dissipation errors of Gauss and Gauss–Lobatto discontinuous Galerkin spectral element methods. SIAM J. Sci. Comput. 33(5), 2560–2579 (2011)
Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods. Springer, Berlin (2008)
Hildebrand, F.B.: Introduction to Numerical Analysis, 2nd edn. Dover Books on Advanced Mathematics, New York (1987)
Hu, F.Q., Hussaini, M.Y., Rasetarinera, P.: An analysis of the discontinuous Galerkin method for wave propagation problems. J. Comput. Phys. 151, 921–946 (1999)
Hu, F., Atkins, H.: Eigensolution analysis of the discontinuous Galerkin method with nonuniform grids: I. One space dimension. J. Comput. Phys. 182(2), 516–545 (2002)
Huynh, H.T.: A Flux Reconstruction Approach to High-Order Schemes Including Discontinuous Galerkin Methods. AIAA Paper 2007-4079 (2007)
Huynh, H.T.: A Reconstruction Approach to High-Order Schemes Including Discontinuous Galerkin for Diffusion. AIAA Paper 2009-403 (2009)
Huynh, H.T., Wang, Z.J., Vincent, P.E.: High-order methods for computational fluid dynamics: a brief review of compact differential formulations on unstructured grids. Comput. Fluids 98, 209–220 (2014)
Johnson, P.E., Johnsen, E., Huynh, H.T.: A novel flux reconstruction method for diffusion problems. In: AIAA Paper, AIAA Aviation Conference (2019, to appear)
Kopriva, D.A.: Metric identities and the discontinuous spectral element method on curvilinear meshes. J. Sci. Comput. 26, 301–326 (2006)
Kopriva, D.A., Woodruff, S.L., Hussaini, M.Y.: Discontinuous spectral element approximation of Maxwell’s equations. In: Cockburn, B., Karniadakis, G., Shu, C.-W. (eds.) Proceedings of the International Symposium on Discontinuous Galerkin Methods. Springer, New York (2000)
Kopriva, D.A., Woodruff, S.L., Hussaini, M.Y.: Computation of electromagnetic scattering with a non-conforming discontinuous spectral element method. Int. J. Numer. Meth. Eng. 53, 105–122 (2002)
LaSaint, P., Raviart, P.A.: On a finite element method for solving the neutron transport equation. In: de Boor, C. (ed.) Mathematical Aspects of Finite Elements in Partial Differential Equations, pp. 89–145. Academic Press, Cambridge (1974)
Mengaldo, G., De Grazia, D., Moura, R.C., Sherwin, S.J.: Spatial eigensolution analysis of energy-stable flux reconstruction schemes and influence of the numerical flux on accuracy and robustness. J. Comput. Phys. 358, 1–20 (2018)
Moura, R.C., Sherwin, S.J., Peiró, J.: Linear dispersion-diffusion analysis and its application to under-resolved turbulence simulations using discontinuous Galerkin spectral/hp methods. J. Comput. Phys. 298, 695–710 (2015)
Reed, W.H., Hill, T.R.: Triangular Mesh Methods for the Neutron Transport Equation. Los Alamos Scientific Laboratory Report, LA-UR-73-479 (1973)
Romero, J., Asthana, K., Jameson, A.: A simplified formulation of the flux reconstruction method. J. Sci. Comput. 67(1), 351–374 (2016)
Romero, J., Witherden, F.D., Jameson, A.: A direct flux reconstruction scheme for advection–diffusion problems on triangular grids. J. Sci. Comput. 73(2–3), 1115–1144 (2017)
Shu, C.-W.: Discontinuous Galerkin method for time dependent problems: survey and recent developments. In: Feng, X., et al. (eds.) Recent Developments in Discontinuous Galerkin Finite Element Methods for Partial Differential Equations, The IMA Volumes in Mathematics and its Applications 157. Springer, Berlin (2012)
Van den Abeele, K.: Development of High-Order Accurate Schemes for Unstructured Grids. Ph.D. thesis, Vrije Universiteit Brussel (2009)
Vincent, P.E., Castonguay, P., Jameson, A.: Insights from von Neumann analysis of high-order flux reconstruction schemes. J. Comput. Phys. 230(22), 8134–8154 (2011)
Wang, Z.J., et al.: High-order CFD methods: current status and perspective. Int. J. Numer. Meth. Fluids 72(8), 811–845 (2013)
Wang, Z.J., Huynh, H.T.: A review of flux reconstruction or correction procedure via reconstruction method for the Navier–Stokes equations. Mech. Eng. Rev. 3(1), 1–16 (2016)
Wang, L., Yu, M.: Compact direct flux reconstruction for conservation laws. J. Sci. Comput. 75, 253–275 (2018)
Witherden, F.D., Vincent, P.E., Jameson, A.: High-order flux reconstruction schemes. In: Abgrall, R., Shu, C.-W. (eds.) Handbook of Numerical Analysis, Chapter 10, vol. 17, pp. 227–263. Elsevier, Amsterdam (2016)
Yang, H., Li, F., Qiu, J.: Dispersion and dissipation errors of two fully discrete discontinuous Galerkin methods. J. Sci. Comput. 55(3), 552–574 (2013)
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The author was supported by the Transformational Tools and Technologies Project of NASA.
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Huynh, H.T. Discontinuous Galerkin via Interpolation: The Direct Flux Reconstruction Method. J Sci Comput 82, 75 (2020). https://doi.org/10.1007/s10915-020-01175-3
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DOI: https://doi.org/10.1007/s10915-020-01175-3