Skip to main content
SpringerLink
Log in

Construction of Modern Robust Nodal Discontinuous Galerkin Spectral Element Methods for the Compressible Navier–Stokes Equations

  • Chapter
  • First Online:
Efficient High-Order Discretizations for Computational Fluid Dynamics

Part of the book series: CISM International Centre for Mechanical Sciences ((CISM,volume 602))

Abstract

Discontinuous Galerkin (DG) methods have a long history in computational physics and engineering to approximate solutions of partial differential equations due to their high-order accuracy and geometric flexibility. However, DG is not perfect and there remain some issues. Concerning robustness, DG has undergone an extensive transformation over the past seven years into its modern form that provides statements on solution boundedness for linear and nonlinear problems. This chapter takes a constructive approach to introduce a modern incarnation of the DG spectral element method for the compressible Navier–Stokes equations in a three-dimensional curvilinear context. The groundwork of the numerical scheme comes from classic principles of spectral methods including polynomial approximations and Gauss-type quadratures. We identify aliasing as one underlying cause of the robustness issues for classical DG spectral methods. Removing said aliasing errors requires a particular differentiation matrix and careful discretization of the advective flux terms in the governing equations.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

Notes

  1. 1.

    https://cordis.europa.eu/docs/results/30/30719/122807181-6_en.pdf.

  2. 2.

    https://www.dlr.de/as/desktopdefault.aspx/tabid-7027/11654_read-27492/.

  3. 3.

    https://www.hopr-project.org.

  4. 4.

    https://github.com/project-fluxo.

References

  • Abarbanel, S., & Gottlieb, D. (1981). Optimal time splitting for two-and three-dimensional Navier-Stokes equations with mixed derivatives. Journal of Computational Physics, 41(1), 1–33.

    Article  MathSciNet  Google Scholar 

  • Altmann, C., Beck, A. D., Hindenlang, F., Staudenmaier, M., Gassner, G. J., & Munz, C.-D. (2013). An efficient high performance parallelization of a discontinuous Galerkin spectral element method. In Keller, R., Kramer, D., & Weiss, J-P. (eds.), Facing the Multicore-Challenge III, Lecture Notes in Computer Science, (pp. 37–47, vo. 7686). Berlin: Springer. ISBN 978-3-642-35892-0.

    Google Scholar 

  • Baggag, A., Atkins, H., & Keyes, D. (2000). Parallel implementation of the discontinuous Galerkin method. In Parallel computational fluid dynamics: Towards teraflops, optimization, and novel formulations, (pp. 115–122).

    Google Scholar 

  • Barth, T. J. (1999). Numerical methods for gasdynamic systems on unstructured meshes. In Dietmar Kröner, Mario Ohlberger, & Christian Rohde (Eds.), An Introduction to Recent Developments in Theory and Numerics for Conservation Laws (Vol. 5, pp. 195–285)., Lecture Notes in Computational Science and Engineering Berlin Heidelberg: Springer.

    Google Scholar 

  • Bassi, F., & Rebay, S. (1997). A high order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. Journal of Computational Physics, 131, 267–279.

    Article  MathSciNet  Google Scholar 

  • B. Biswas and H. Kumar. Entropy stable discontinuous Galerkin approximation for the relativistic hydrodynamic equations. arXiv preprintarXiv:1911.07488, 2019.

  • Black, K. (1999). A conservative spectral element method for the approximation of compressible fluid flow. Kybernetika, 35(1), 133–146.

    MathSciNet  Google Scholar 

  • Black, K. (2000). Spectral element approximation of convection-diffusion type problems. Applied Numerical Mathematics, 33(1–4), 373–379.

    Article  MathSciNet  Google Scholar 

  • Bohm, M., Winters, A. R., Gassner, G. J., Derigs, D., Hindenlang, F., &  Saur, J. (2018). An entropy stable nodal discontinuous Galerkin method for the resistive MHD equations. Part I: Theory and numerical verification. Journal of Computational Physics. https://doi.org/10.1016/j.jcp.2018.06.027.

  • Bonev, B., Hesthaven, J. S., Giraldo, F. X., & Kopera, M. A. (2018). Discontinuous Galerkin scheme for the spherical shallow water equations with applications to tsunami modeling and prediction. Journal of Computational Physics, 362, 425–448.

    Article  MathSciNet  Google Scholar 

  • Canuto, C., & Quarteroni, A. (1982). Approximation results for orthogonal polynomials in Sobolev spaces. Mathematics of Computation, 38(157), 67–86.

    Article  MathSciNet  Google Scholar 

  • Canuto, C., Hussaini, M., Quarteroni, A., & Zang, T. (2007). Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics. Berlin: Springer.

    Book  Google Scholar 

  • Carlson, B. C. (1972). The logarithmic mean. The American Mathematical Monthly, 79(6), 615–618.

    Article  MathSciNet  Google Scholar 

  • Carpenter, M., Fisher, T., Nielsen, E., & Frankel, S. (2014). Entropy stable spectral collocation schemes for the Navier-Stokes equations: Discontinuous interfaces. SIAM Journal on Scientific Computing, 36(5), B835–B867.

    Article  MathSciNet  Google Scholar 

  • Chan, J. (2018). On discretely entropy conservative and entropy stable discontinuous Galerkin methods. Journal of Computational Physics, 362, 346–374.

    Article  MathSciNet  Google Scholar 

  • Chan, J., Hewett, R. J., & Warburton, T. (2017). Weight-adjusted discontinuous Galerkin methods: wave propogation in heterogeneous media. SIAM Journal on Scientific Computing, 39(6), A2935–A2961.

    Article  Google Scholar 

  • Chan, J., Del Rey Fernández, D. C., & Carpenter, M. H. (2019). Efficient entropy stable Gauss collocation methods. SIAM Journal on Scientific Computing, 41(5), A2938–A2966.

    Article  MathSciNet  Google Scholar 

  • Chandrashekar, P. (2013). Kinetic energy preserving and entropy stable finite volume schemes for compressible Euler and Navier-Stokes equations. Communications in Computational Physics, 14(1252–1286), 11.

    MathSciNet  Google Scholar 

  • Chen, T., & Shu, C.-W. (2017). Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws. Journal of Computational Physics, 345, 427–461.

    Article  MathSciNet  Google Scholar 

  • Cockburn, B., & Shu, C. W. (1991). The Runge-Kutta local projection \({P}^1\)-discontinuous Galerkin method for scalar conservation laws. Rairo-Matehmatical Modelling and Numerical Analysis-Modelisation Mathematique et Analyse Numerique, 25, 337–361.

    Article  MathSciNet  Google Scholar 

  • Cockburn, B., & Shu, C.-W. (1998a). The Runge-Kutta discontinuous Galerkin method for conservation laws V: Multidimensional systems. Journal of Computational Physics, 141(2), 199–224.

    Article  MathSciNet  Google Scholar 

  • Cockburn, B., & Shu, C.-W. (1998b). The local discontinuous Galerkin method for time-dependent convection diffusion systems. SIAM Journal on Numerical Analysis, 35, 2440–2463.

    Article  MathSciNet  Google Scholar 

  • Cockburn, B., Hou, S., & Shu, C.-W. (1990). The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV: The multidimensional case. Mathematics of Computation, 54(190), 545–581.

    Google Scholar 

  • Cockburn, B., Karniadakis, G. E., Shu, C-W. (2000). The development of discontinuous Galerkin methods. In Cockburn, B., Karniadakis, G., & Shu, C.-W. (eds.), Proceedings of the international symposium on discontinuous galerkin methods, (pp. 3–50), New York: Springer.

    Google Scholar 

  • Crean, J., Hicken, J. E., Del Rey Fernández, D. C., Zingg, D. W., & Carpenter, M. H. (2018). Entropy-stable summation-by-parts discretization of the Euler equations on general curved elements. Journal of Computational Physics, 356, 410–438.

    Article  MathSciNet  Google Scholar 

  • Dalcin, L., Rojas, D., Zampini, S., Del Rey Fernández, D. C., Carpenter, M. H., & Parsani, M. (2019). Conservative and entropy stable solid wall boundary conditions for the compressible Navier-Stokes equations: Adiabatic wall and heat entropy transfer. Journal of Computational Physics, 397, 108775.

    Article  MathSciNet  Google Scholar 

  • Deng, S. (2007). Numerical simulation of optical coupling and light propagation in coupled optical resonators with size disorder. Applied Numerical Mathematics, 57(5–7), 475–485. ISSN 0168-9274.

    Google Scholar 

  • Deng, S.Z., Cai, W., & Astratov, V.N. (2004) Numerical study of light propagation via whispering gallery modes in microcylinder coupled resonator optical waveguides. Optics Express, 12(26), 6468–6480, DEC 27 2004. ISSN 1094-4087.

    Google Scholar 

  • Ducros, F., Laporte, F., Soulères, T., Guinot, V., Moinat, P., & Caruelle, B. (2000). High-order fluxes for conservative skew-symmetric-like schemes in structured meshes: Application to compressible flows. Journal of Computational Physics, 161, 114–139.

    Google Scholar 

  • Dutt, P. (1988). Stable boundary conditions and difference schemes for Navier-Stokes equations. SIAM Journal on Numerical Analysis, 25, 245–267.

    Google Scholar 

  • Evans, L. C. (2012). Partial differential equations. American Mathematical Society.

    Google Scholar 

  • Fagherazzi, S., Furbish, D. J., Rasetarinera, P., & Hussaini, M. Y. (2004a). Application of the discontinuous spectral Galerkin method to groundwater flow. Advances in Water Resourses, 27, 129–140.

    Article  Google Scholar 

  • Fagherazzi, S., Rasetarinera, P., Hussaini, M. Y., & Furbish, D. J. (2004b). Numerical solution of the dam-break problem with a discontinuous Galerkin method. Journal of Hydraulic Engineering, 130(6), 532–539. June.

    Google Scholar 

  • Farrashkhalvat, M., & Miles, J. P. (2003). Basic Structured Grid Generation: With an introduction to unstructured grid generation. Butterworth-Heinemann.

    Google Scholar 

  • Fisher, T., Carpenter, M. H., Nordström, J., Yamaleev, N. K., & Swanson, C. (2013). Discretely conservative finite-difference formulations for nonlinear conservation laws in split form: Theory and boundary conditions. Journal of Computational Physics, 234, 353–375.

    Article  MathSciNet  Google Scholar 

  • Fisher, T. C. (2012). High-order\(L_2\)stable multi-domain finite difference method for compressible flows. Ph.D. thesis, Purdue University.

    Google Scholar 

  • Fisher, T. C., & Carpenter, M. H. (2013). High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains. Journal of Computational Physics, 252, 518–557.

    Article  MathSciNet  Google Scholar 

  • Fjordholm, U. S., Mishra, S., & Tadmor, E. (2011). Well-balanced and energy stable schemes for the shallow water equations with discontiuous topography. Journal of Computational Physics, 230(14), 5587–5609.

    Article  MathSciNet  Google Scholar 

  • Flad, D., & Gassner, G. (2017). On the use of kinetic energy preservaing DG-scheme for large eddy simulation. Journal of Computational Physics, 350, 782–795.

    Article  MathSciNet  Google Scholar 

  • Flad, D., Beck, A., & Guthke, P. (2020). A large eddy simulation method for DGSEM using non-linearly optimized relaxation filters. Journal of Computational Physics, 408, 109303.

    Article  MathSciNet  Google Scholar 

  • Friedrich, L., Winters, A. R., Del Rey Fernández, G. J., Gassner, D. C., Parsani, M., & Carpenter, M. H. (2018). An entropy stable \(h/p\) non-conforming discontinuous Galerkin method with the summation-by-parts property. Journal of Scientific Computing, 77(2), 689–725.

    Article  MathSciNet  Google Scholar 

  • Friedrich, L., Schnücke, G., Winters, A. R., Del Rey Fernández, D. C., Gassner, G. J., & Carpenter, M. H. (2019). Entropy stable space-time discontinuous Galerkin schemes with summation-by-parts property for hyperbolic conservation laws. Journal of Scientific Computing, 80(1), 175–222.

    Article  MathSciNet  Google Scholar 

  • Gassner, G. (2013). A skew-symmetric discontinuous Galerkin spectral element discretization and its relation to SBP-SAT finite difference methods. SIAM Journal on Scientific Computing, 35(3), A1233–A1253.

    Article  MathSciNet  Google Scholar 

  • Gassner, G., & Kopriva, D. A. (2010). A comparison of the dispersion and dissipation errors of Gauss and Gauss-Lobatto discontinuous Galerkin spectral element methods. SIAM Journal on Scientific Computing, 33(5), 2560–2579.

    Article  MathSciNet  Google Scholar 

  • Gassner, G. J., & Winters, A. R. (2019). A novel robust strategy for discontinuous Galerkin methods in computational physics: Why? When? What? Where? Submitted to Frontiers in Physics.

    Google Scholar 

  • Gassner, G. J., Winters, A. R., & Kopriva, D. A. (2016a). A well balanced and entropy conservative discontinuous Galerkin spectral element method for the shallow water equations. Applied Mathematics and Computation, 272. Part, 2, 291–308.

    Google Scholar 

  • Gassner, G. J., Winters, A. R., & Kopriva, D. A. (2016b). Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations. Journal of Computational Physics, 327, 39–66.

    Article  MathSciNet  Google Scholar 

  • Gassner, G. J., Winters, A. R., Hindenlang, F. J., & Kopriva, D. A. (2018). The BR1 scheme is stable for the compressible Navier-Stokes equations. Journal of Scientific Computing, 77(1), 154–200.

    Article  MathSciNet  Google Scholar 

  • Geuzaine, C., & Remacle, J.-F. (2009). Gmsh: A 3-D finite element mesh generator with built-in pre-and post-processing facilities. International Journal for Numerical Methods in Engineering, 79(11), 1309–1331.

    Article  MathSciNet  Google Scholar 

  • Giraldo, F. X., & Restelli, M. (2008). A study of spectral element and discontinuous Galerkin methods for the Navier-Stokes equations in nonhydrostatic mesoscale atmospheric modeling: Equation sets and test cases. Journal of Computational Physics, 227(8), 3849–3877.

    Article  MathSciNet  Google Scholar 

  • Giraldo, F. X., Hesthaven, J. S., & Warburton, T. (2002). Nodal high-order discontinuous Galerkin methods for the spherical shallow water equations. Journal of Computational Physics, 181(2), 499–525.

    Article  MathSciNet  Google Scholar 

  • Gordon, W. J., & Hall, C. A. (1973). Construction of curvilinear co-ordinate systems and their applications to mesh generation. International Journal for Numerical Methods in Engineering Engineering, 7, 461–477.

    Article  MathSciNet  Google Scholar 

  • Gottlieb, D., & Orszag, S.A. (1977). Numerical Analysis of Spectral Methods: Theory and Applications. SIAM-CMBS.

    Google Scholar 

  • Harten, A. (1983). On the symmetric form of systems of conservation laws with entropy. Journal of Computational Physics, 49, 151–164.

    Google Scholar 

  • Hennemann, S., & Gassner, G. J. (2020). A provably entropy stable subcell shock capturing approach for high order split form DG. Submitted to Journal of Computational Physics.

    Google Scholar 

  • Hesthaven, J. S., & Warburton, T. (2008). Nodal discontinuous galerkin methods. Springer.

    Google Scholar 

  • Hicken, J. E., Fernández, D. C. D. R., & Zingg, D. W. (2016). Multidimensional summation-by-parts operators: General theory and application to simplex elements. SIAM Journal on Scientific Computing, 38(4), A1935–A1958.

    Google Scholar 

  • Hindenlang, F. (2014). Mesh curving techniques for high order parallel simulations on unstructured meshes. Ph.D. thesis, University of Stuttgart.

    Google Scholar 

  • Hindenlang, F., Gassner, G. J., Altmann, C., Beck, A., Staudenmaier, M., & Munz, C.-D. (2012). Explicit discontinuous Galerkin methods for unsteady problems. Computers and Fluids, 61, 86–93.

    Article  MathSciNet  Google Scholar 

  • Hindenlang, F. J., Gassner, G. J., Kopriva, D. A. (2019). Stability of wall boundary condition procedures for discontinuous Galerkin spectral element approximations of the compressible Euler equations. arXiv:1901.04924.

  • Hu, F. Q., Hussaini, M. Y., & Rasetarinera, P. (1999). An analysis of the discontinuous Galerkin method for wave propagation problems. Journal of Computational Physics, 151(2), 921–946.

    Google Scholar 

  • Ismail, F., & Roe, P. L. (2009). Affordable, entropy-consistent Euler flux functions II: Entropy production at shocks. Journal of Computational Physics, 228(15), 5410–5436.

    Google Scholar 

  • Karniadakis, G. E., & Sherwin, S. J. (2005). Spectral/hp element methods for computational fluid dynamics. Oxford University Press.

    Google Scholar 

  • Kennedy, C. A., & Gruber, A. (2008). Reduced aliasing formulations of the convective terms within the Navier-Stokes equations for a compressible fluid. Journal of Computational Physics, 227, 1676–1700.

    Article  MathSciNet  Google Scholar 

  • Knupp, P. M., Steinberg, S. (1993). Fundamentals of grid generation. CRC-Press.

    Google Scholar 

  • Kopriva, D. A. (2006). Metric identities and the discontinuous spectral element method on curvilinear meshes. Journal of Scientific Computing, 26(3), 301–327.

    Article  MathSciNet  Google Scholar 

  • Kopriva, D. A. (2009). Implementing spectral methods for partial differential equations. Scientific computation.

    Google Scholar 

  • Kopriva, D. A., & Gassner, G. J. (2014). An energy stable discontinuous Galerkin spectral element discretization for variable coefficient advection problems. SIAM Journal on Scientific Computing, 34(4), A2076–A2099.

    Article  MathSciNet  Google Scholar 

  • Kopriva, D. A., Woodruff, S. L., & Hussaini, M. Y. (2000). 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 (pp. 355–361), New York: Springer.

    Google Scholar 

  • Kopriva, D. A., Woodruff, S. L., & Hussaini, M. Y. (2002). Computation of electromagnetic scattering with a non-conforming discontinuous spectral element method. International Journal for Numerical Methods in Engineering, 53(1), 105–122.

    Article  Google Scholar 

  • Kopriva, D. A., Winters, A. R., Bohm, M., & Gassner, G. J. (2016). A provably stable discontinuous Galerkin spectral element approximation for moving hexahedral meshes. Computers & Fluids, 139, 148–160.

    Article  MathSciNet  Google Scholar 

  • Kopriva, D. A., Hindenlang, F. J., Bolemann, T., & Gassner, G. J. (2019). Free-stream preservation for curved geometrically non-conforming discontinuous Galerkin spectral elements. Journal of Scientific Computing, 79(3), 1389–1408.

    Article  MathSciNet  Google Scholar 

  • Krais, N.  Schnücke, G.,  Bolemann, T., & Gassner, G. (2020). Split form ALE discontinuous Galerkin methods with applications to under-resolved turbulent low-Mach number flows. arXiv:2003.02296.

  • Kreiss, H.-O., Oliger, J. (1973). Methods for the approximate solution of time-dependent problems. World Meteorological Organization, Geneva, 1973. GARP Rept. No.10.

    Google Scholar 

  • Kreiss, H.-O., & Olliger, J. (1972). Comparison of accurate methods for the integration of hyperbolic equations. Tellus, 24, 199–215.

    Article  MathSciNet  Google Scholar 

  • Kreiss, H.-O., Scherer, G. (1974). Finite element and finite difference methods for hyperbolic partial differential equations. In Mathematical aspects of finite elements in partial differential equations (pp. 195–212). Elsevier.

    Google Scholar 

  • Kreiss, H.-O., Scherer, G. (1977). On the existence of energy estimates for difference approximations for hyperbolic systems. Technical report, Deptpartment of Scientific Computing, Uppsala University.

    Google Scholar 

  • Lax, P. D. (1954). Weak solutions of nonlinear hyperbolic conservation equations and their numerical computation. Communications on Pure and Applied Mathematics, 7(1), 159–193.

    Article  MathSciNet  Google Scholar 

  • Lax, P. D. (1967). Hyperbolic difference equations: A review of the Courant-Friedrichs-Lewy paper in the light of recent developments. IBM Journal of Reseach and Development, 11(2), 235–238.

    Article  MathSciNet  Google Scholar 

  • LeFloch, P., & Rohde, C. (2000). High-order schemes, entropy inequalities, and nonclassical shocks. SIAM Journal on Numerical Analysis, 37(6), 2023–2060.

    Article  MathSciNet  Google Scholar 

  • LeVeque, R. J. (2020) Finite Volume Methods for Hyperbolic Problems. Cambridge University Press.

    Google Scholar 

  • Liu, Y., Shu, C.-W., & Zhang, M. (2018). Entropy stable high order discontinuous Galerkin methods for ideal compressible MHD on structured meshes. Journal of Computational Physics, 354, 163–178.

    Article  MathSciNet  Google Scholar 

  • Manzanero, J., Ferrer, E., Rubio, G., & Valero, E. (2020a) Design of a Smagorinsky spectral vanishing viscosity turbulence model for discontinuous Galerkin methods. Computers & Fluids, 104440.

    Google Scholar 

  • Manzanero, J., Rubio, G., Kopriva, D. A., Ferrer, E., & Valero, E. (2020b). An entropy-stable discontinuous Galerkin approximation for the incompressible Navier-Stokes equations with variable density and artificial compressibility. Journal of Computational Physics, 408, 109241.

    Article  MathSciNet  Google Scholar 

  • Manzanero, J., Rubio, G., Kopriva, D. A., Ferrer, E., & Valero, E. (2020c) Entropy–stable discontinuous Galerkin approximation with summation–by–parts property for the incompressible Navier–Stokes/Cahn–Hilliard system. Journal of Computational Physics, 109363.

    Google Scholar 

  • Manzanero, J., Rubio, G., Kopriva, D. A., Ferrer, E., & Valero, E. (2020d). A free-energy stable nodal discontinuous Galerkin approximation with summation-by-parts property for the Cahn-Hilliard equation. Journal of Computational Physics, 403, 109072.

    Article  MathSciNet  Google Scholar 

  • Merriam, M. L. (1987). Smoothing and the second law. Computer Methods in Applied Mechanics and Engineering, 64(1–3), 177–193.

    Article  MathSciNet  Google Scholar 

  • Merriam, M. L. (1989). An entropy-based approach to nonlinear stability. NASA Technical Memorandum, 101086(64), 1–154.

    MathSciNet  Google Scholar 

  • Nitsche, J. A. (1971). Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg, 36, 9–15.

    Article  MathSciNet  Google Scholar 

  • Nordström, J., & Svärd, M. (2005). Well-posed boundary conditions for the Navier-Stokes equations. SIAM Journal on Numerical Analysis, 43(3), 1231–1255.

    Article  MathSciNet  Google Scholar 

  • Parsani, M., Carpenter, M. H., & Nielsen, E. J. (2015). Entropy stable wall boundary conditions for the three-dimensional compressible Navier-Stokes equations. Journal of Computational Physics, 292, 88–113.

    Article  MathSciNet  Google Scholar 

  • Pazner, W., & Persson, P.-O. (2019). Analysis and entropy stability of the line-based discontinuous Galerkin method. Journal of Scientific Computing, 80(1), 376–402.

    Article  MathSciNet  Google Scholar 

  • Pirozzoli, S. (2010). Generalized conservative approximations of split convective derivative operators. Journal of Computational Physics, 229(19), 7180–7190.

    Article  MathSciNet  Google Scholar 

  • Ranocha, H. (2018). Comparison of some entropy conservative numerical fluxes for the Euler equations. Journal of Scientific Computing, 76(1), 216–242.

    Article  MathSciNet  Google Scholar 

  • Ranocha, H., Sayyari, M., Dalcin, L., Parsani, M., & Ketcheson, D. I. (2020). Relaxation Runge-Kutta methods: Fully discrete explicit entropy-stable schemes for the compressible Euler and Navier-Stokes equations. SIAM Journal on Scientific Computing, 42(2), A612–A638.

    Article  MathSciNet  Google Scholar 

  • Rasetarinera, P., & Hussaini, M. Y. (2001). An efficient implicit discontinuous spectral Galerkin method. Journal of Computational Physics, 172, 718–738.

    Article  Google Scholar 

  • Rasetarinera, P., Kopriva, D. A., & Hussaini, M. Y. (2001). Discontinuous spectral element solution of acoustic radiation from thin airfoils. AIAA Journal, 39(11), 2070–2075.

    Article  Google Scholar 

  • Reed, W. H., & Hill, T. R. (1973) Triangular mesh methods for the neutron transport equation. Technical Report LA-UR-73-479, Los Alamos National Laboratory.

    Google Scholar 

  • Renac, F. (2019). Entropy stable DGSEM for nonlinear hyperbolic systems in nonconservative form with application to two-phase flows. Journal of Computational Physics, 382, 1–26.

    Article  MathSciNet  Google Scholar 

  • Restelli, M., & Giraldo, F. X. (2009). A conservative discontinuous Galerkin semi-implicit formulation for the Navier-Stokes equations in nonhydrostatic mesoscale modeling. SIAM Journal on Scientific Computing, 31(3), 2231–2257.

    Article  MathSciNet  Google Scholar 

  • Roe, P. L. (1997). Approximate Riemann solvers, parameter vectors, and difference schemes. Journal of Computational Physics, 135(2), 250–258.

    Article  MathSciNet  Google Scholar 

  • Schnücke, G., Krais, N., Bolemann, T., & Gassner, G. J. (2020). Entropy stable discontinuous Galerkin schemes on moving meshes for hyperbolic conservation laws. Journal of Scientific Computing, 82(3), 1–42.

    Article  MathSciNet  Google Scholar 

  • Sjögreen, B., Yee, H. C., & Kotov, D. (2017). Skew-symmetric splitting and stability of high order central schemes. Journal of Physics: Conference Series, 837(1), 012019.

    Google Scholar 

  • Stanescu, D., Farassat, F., & Hussaini, M. Y. (2002a) Aircraft engine noise scattering - parallel discontinuous Galerkin spectral element method. Paper 2002-0800, AIAA.

    Google Scholar 

  • Stanescu, D., Xu, J., Farassat, F., & Hussaini, M. Y. (2002b). Computation of engine noise propagation and scattering off an aircraft. Aeroacoustics, 1(4), 403–420.

    Article  Google Scholar 

  • Strand, B. (1994). Summation by parts for finite difference approximations for \(d/dx\). Journal of Computational Physics, 110,

    Google Scholar 

  • Svärd, M., & Nordström, J. (2014). Review of summation-by-parts schemes for initial-boundary-value problems. Journal of Computational Physics, 268, 17–38.

    Article  MathSciNet  Google Scholar 

  • Tadmor, E. (1984). Skew-selfadjoint form for systems of conservation laws. Journal of Mathematical Analysis and Applications, 103(2), 428–442.

    Article  MathSciNet  Google Scholar 

  • Tadmor, E. (1987). Entropy functions for symmetric systems of conservation laws. Journal of Mathematical Analysis and Applications, 122(2), 355–359.

    Article  MathSciNet  Google Scholar 

  • Tadmor, E. (2003) Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numerica, 12, 451–512, 5 (2003).

    Google Scholar 

  • Tadmor, E. (2016). Perfect derivatives, conservative differences and entropy stable computation of hyperbolic conservation laws. Discrete and Continuous Dynamical Systems-A, 36(8), 4579–4598.

    Article  MathSciNet  Google Scholar 

  • Tadmor, E., & Zhong, W. (2006). Entropy stable approximations of Navier-Stokes equations with no artificial numerical viscosity. Journal of Hyperbolic Differential Equations, 3(3), 529–559.

    Article  MathSciNet  Google Scholar 

  • Wang, Z. J., Fidkowski, K., Abgrall, R., Bassi, F., Caraeni, D., Cary, A., et al. (2013). High-order CFD methods: current status and perspective. International Journal for Numerical Methods in Fluids, 72(8), 811–845.

    Article  MathSciNet  Google Scholar 

  • Wilcox, L. C., Stadler, G., Burstedde, C., & Ghattas, O. (2010). A high-order discontinuous Galerkin method for wave propagation through coupled elastic-acoustic media. Journal of Computational Physics, 229(24), 9373–9396.

    Article  MathSciNet  Google Scholar 

  • Wintermeyer, N., Winters, A. R., Gassner, G. J., & Kopriva, D. A. (2017). An entropy stable nodal discontinuous Galerkin method for the two dimensional shallow water equations on unstructured curvilinear meshes with discontinuous bathymetry. Journal of Computational Physics, 340, 200–242.

    Article  MathSciNet  Google Scholar 

  • Wintermeyer, N., Winters, A. R., Gassner, G. J., & Warburton, T. (2018). An entropy stable discontinuous Galerkin method for the shallow water equations on curvilinear meshes with wet/dry fronts accelerated by GPUs. Journal of Computational Physics, 375, 447–480.

    Article  MathSciNet  Google Scholar 

  • Winters, A. R., & Gassner, G. J. (2016). Affordable, entropy conserving and entropy stable flux functions for the ideal MHD equations. Journal of Computational Physics, 301, 72–108.

    Article  MathSciNet  Google Scholar 

  • Winters, A. R., Derigs, D., Gassner, G. J., & Walch, S. (2017). A uniquely defined entropy stable matrix dissipation operator for high Mach number ideal MHD and compressible Euler simulations. Journal of Computational Physics, 332, 274–289.

    Article  MathSciNet  Google Scholar 

  • Winters, A. R., Moura, R. C., Mengaldo, G., Gassner, G. J., Walch, S., Peiro, J., et al. (2018). A comparative study on polynomial dealiasing and split form discontinuous Galerkin schemes for under-resolved turbulence computations. Journal of Computational Physics, 372, 1–21.

    Google Scholar 

  • Winters, A. R., Czernik, C., Schily, M. B., & Gassner, G. J. (2019). Entropy stable numerical approximations for the isothermal and polytropic Euler equations. BIT Numerical Mathematics, 1–34.

    Google Scholar 

  • Wu, K., & Shu, C.-W. (2019) Entropy symmetrization and high-order accurate entropy stable numerical schemes for relativistic MHD equations. arXiv:1907.07467.

  • Xie, Z., Wang, L.-L., & Zhao, X. (2013). On exponential convergence of gegenbauer interpolation and spectral differentiation. Mathematics of Computation, 82, 1017–1036.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Linköping University, Linköping, Sweden

    Andrew R. Winters

  2. The Florida State University, FL, USA

    David A. Kopriva

  3. San Diego State University, San Diego, CA, USA

    David A. Kopriva

  4. University of Cologne, Cologne, Germany

    Gregor J. Gassner

  5. Max Planck Institute for Plasma Physics, Garching, Germany

    Florian Hindenlang

Authors
  1. Andrew R. Winters
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David A. Kopriva
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Gregor J. Gassner
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Florian Hindenlang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gregor J. Gassner .

Editor information

Editors and Affiliations

  1. Institute for Computational Mechanics, Technical University Munich, Garching bei München, Germany

    Martin Kronbichler

  2. Department of Mathematics, University of California, Berkeley, Berkeley, CA, USA

    Per-Olof Persson

Rights and permissions

Reprints and permissions

Copyright information

© 2021 CISM International Centre for Mechanical Sciences, Udine

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Winters, A.R., Kopriva, D.A., Gassner, G.J., Hindenlang, F. (2021). Construction of Modern Robust Nodal Discontinuous Galerkin Spectral Element Methods for the Compressible Navier–Stokes Equations. In: Kronbichler, M., Persson, PO. (eds) Efficient High-Order Discretizations for Computational Fluid Dynamics. CISM International Centre for Mechanical Sciences, vol 602. Springer, Cham. https://doi.org/10.1007/978-3-030-60610-7_3

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-60610-7_3

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-60609-1

  • Online ISBN: 978-3-030-60610-7

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation