Skip to main content
SpringerLink
Log in

Skew-Symmetric Entropy Stable Modal Discontinuous Galerkin Formulations

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

High order entropy stable discontinuous Galerkin (DG) methods for nonlinear conservation laws satisfy an inherent discrete entropy inequality. The construction of such schemes has relied on the use of carefully chosen nodal points (Gassner in SIAM J Sci Comput 35(3):A1233–A1253, 2013; Fisher and Carpenter in J Comput Phys 252:518–557, 2013; Carpenter et al. in SIAM J Sci Comput 36(5):B835–B867, 2014; Crean et al. in J Comput Phys 356:410–438, 2018; Chan et al. in Efficient entropy stable Gauss collocation methods, 2018. arXiv:1809.01178) or volume and surface quadrature rules (Chan in J Comput Phys 362:346–374, 2018; Chan and Wilcox in J Comput Phys 378:366–393, 2019) to produce operators which satisfy a summation-by-parts (SBP) property. In this work, we show how to construct “modal” DG formulations which are entropy stable for volume and surface quadratures under which the SBP property in Chan (2018) does not hold. These formulations rely on an alternative skew-symmetric construction of operators which automatically satisfy the SBP property. Entropy stability then follows for choices of volume and surface quadrature which satisfy sufficient accuracy conditions. The accuracy of these new SBP operators depends on a separate set of conditions on quadrature accuracy, with design order accuracy recovered under the usual assumptions of degree \(2N-1\) volume quadratures and degree 2N surface quadratures. We conclude with numerical experiments verifying the accuracy and stability of the proposed formulations, and discuss an application of these formulations for entropy stable DG schemes on mixed quadrilateral-triangle meshes.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Notes

  1. The term “hybridized” SBP operator was introduced in the review paper [32]. These operators were originally referred to as “decoupled” SBP operators in [15]).

  2. This interpolation step must be performed using interpolation points with an appropriate number of nodes on each boundary [16]. These include, for example, GLL nodes on tensor product elements, and optimized interpolation nodes on non-tensor product elements [42,43,44].

  3. It is possible to construct the geometric terms for \(N_{\mathrm{geo}} = N\) using a local \(H_\mathrm{div}\) basis where

    $$\begin{aligned} \varvec{r}^i \in Q^{N-1,N,N} \times Q^{N,N-1,N} \times Q^{N,N,N-1}. \end{aligned}$$

    Then, the geometric terms satisfy \(\nabla \times \varvec{r}^i \in Q^{N,N-1,N-1}\times Q^{N-1,N,N-1} \times Q^{N-1,N-1,N}\) with traces in \(Q^{N-1}\), and Assumption 1 holds under degree \((2N-1)\) volume and surface quadrature. This approach will be investigated in more detail in future work.

References

  1. Wang, Z.J., Fidkowski, K., Abgrall, R., Bassi, F., Caraeni, D., Cary, A., Deconinck, H., Hartmann, R., Hillewaert, K., Huynh, H.T., et al.: High-order CFD methods: current status and perspective. Int. J. Numer. Methods Fluids 72(8), 811–845 (2013)

    Article  MathSciNet  Google Scholar 

  2. Ainsworth, M.: Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods. J. Comput. Phys. 198(1), 106–130 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Visbal, M.R., Gaitonde, D.V.: High-order-accurate methods for complex unsteady subsonic flows. AIAA J. 37(10), 1231–1239 (1999)

    Article  Google Scholar 

  4. Tadmor, E.: The numerical viscosity of entropy stable schemes for systems of conservation laws. I. Math. Comput. 49(179), 91–103 (1987)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Fjordholm, U.S., Mishra, S., Tadmor, E.: Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws. SIAM J. Numer. Anal. 50(2), 544–573 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chandrashekar, P.: Kinetic energy preserving and entropy stable finite volume schemes for compressible Euler and Navier–Stokes equations. Commun. Comput. Phys. 14(5), 1252–1286 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Tadmor, E.: Entropy stable schemes. Handbook Numer. Anal. 17, 467–493 (2016)

    MathSciNet  Google Scholar 

  9. Ray, D., Chandrashekar, P., Fjordholm, U.S., Mishra, S.: Entropy stable scheme on two-dimensional unstructured grids for Euler equations. Commun. Comput. Phys. 19(5), 1111–1140 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  10. Fisher, T.C., Carpenter, M.H.: High-order entropy stable finite difference schemes for nonlinear conservation laws: finite domains. J. Comput. Phys. 252, 518–557 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  11. Carpenter, M.H., Fisher, T.C., Nielsen, E.J., Frankel, S.H.: Entropy stable spectral collocation schemes for the Navier–Stokes equations: discontinuous interfaces. SIAM J. Sci. Comput. 36(5), B835–B867 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gassner, G.J., Winters, A.R., Kopriva, D.A.: Split form nodal discontinuous Galerkin schemes with summation-by-parts property for the compressible Euler equations. J. Comput. Phys. 327, 39–66 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Gassner, G.J., Winters, A.R., Hindenlang, F.J., Kopriva, D.A.: The BR1 scheme is stable for the compressible Navier-Stokes equations. J. Sci. Comput. 77, 1–47 (2017)

    MathSciNet  MATH  Google Scholar 

  14. Crean, J., Hicken, J.E., Fernández, D.C.D.R., Zingg, D.W., Carpenter, M.H.: Entropy-stable summation-by-parts discretization of the Euler equations on general curved elements. J. Comput. Phys. 356, 410–438 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chan, J.: On discretely entropy conservative and entropy stable discontinuous Galerkin methods. J. Comput. Phys. 362, 346–374 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  16. Chan, J., Wilcox, L.C.: On discretely entropy stable weight-adjusted discontinuous Galerkin methods: curvilinear meshes. J. Comput. Phys. 378, 366–393 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  17. Crean, J., Hicken, J.E., Fernández, D.C.D.R., Zingg, D.W., Carpenter, M.H: High-order, entropy-stable discretizations of the Euler equations for complex geometries. In: 23rd AIAA Computational Fluid Dynamics Conference. American Institute of Aeronautics and Astronautics (2017)

  18. Chen, T., Shu, C.-W.: Entropy stable high order discontinuous Galerkin methods with suitable quadrature rules for hyperbolic conservation laws. J. Comput. Phys. 345, 427–461 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Parsani, M., Carpenter, M.H., Fisher, T.C., Nielsen, E.J.: Entropy stable staggered grid discontinuous spectral collocation methods of any order for the compressible Navier–Stokes equations. SIAM J. Sci. Comput. 38(5), A3129–A3162 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  20. Del Rey Fernández, D.C., Crean, J., Carpenter, M.H., Hicken, J.E.: Staggered-grid entropy-stable multidimensional summation-by-parts discretizations on curvilinear coordinates. J. Comput. Phys. 392, 161–186 (2019)

    Article  MathSciNet  Google Scholar 

  21. Chan, J., Fernandez, D.C., Carpenter, M.H.: Efficient entropy stable Gauss collocation methods. arXiv preprint arXiv:1809.01178. (2018)

  22. Friedrich, L., Winters, A.R., Fernández, D.C., Gassner, G.J., Parsani, M., Carpenter, M.H.: An Entropy Stable h/p Non-conforming Discontinuous Galerkin Method with the Summation-by-Parts Property. arXiv preprint arXiv:1712.10234. (2017)

  23. Hicken, J.E., Del Rey Fernández, D.C., Zingg, D.W.: Multidimensional summation-by-parts operators: General theory and application to simplex elements. SIAM J. Sci. Comput. 38(4), A1935–A1958 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hughes, T.J.R., Franca, L.P., Mallet, M.: A new finite element formulation for computational fluid dynamics: I. Symmetric forms of the compressible Euler and Navier-Stokes equations and the second law of thermodynamics. Comput. Methods Appl. Mech. Eng. 54(2), 223–234 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  25. Mock, M.S.: Systems of conservation laws of mixed type. J. Differ. Equ. 37(1), 70–88 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  26. Harten, A.: On the symmetric form of systems of conservation laws with entropy. J. Comput. Phys. 49(1), 151–164 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  27. Dafermos, C.M.: Hyperbolic Conservation Laws in Continuum Physics. Springer, Berlin (2005)

    Book  MATH  Google Scholar 

  28. Chan, J., Wang, Z., Modave, A., Remacle, J.-F., Warburton, T.: GPU-accelerated discontinuous Galerkin methods on hybrid meshes. J. Comput. Phys. 318, 142–168 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  29. Gassner, G.J.: A skew-symmetric discontinuous Galerkin spectral element discretization and its relation to SBP-SAT finite difference methods. SIAM J. Sci. Comput. 35(3), A1233–A1253 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  30. Del Rey Fernández, D.C., Boom, P.D., Zingg, D.W.: A generalized framework for nodal first derivative summation-by-parts operators. J. Comput. Phys. 266, 214–239 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  31. Ranocha, H.: Generalised summation-by-parts operators and variable coefficients. J. Comput. Phys. 362, 20–48 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  32. Chen, T., Shu, C.-W.: Review of entropy stable discontinuous Galerkin methods for systems of conservation laws on unstructured simplex meshes. https://www.brown.edu/research/projects/scientificcomputing/sites/brown.edu.research.projects.scientific-computing/files/uploads/Review%20of%20entropy%20stable%20discontinuous%20Galerkin%20methods.pdf

  33. Xiao, H., Gimbutas, Z.: A numerical algorithm for the construction of efficient quadrature rules in two and higher dimensions. Comput. Math. Appl. 59, 663–676 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  34. Shi, C., Shu, C.-W.: On local conservation of numerical methods for conservation laws. Computers and Fluids (2017)

  35. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Springer, Berlin (2007)

    Book  MATH  Google Scholar 

  36. Kopriva, D.A., Gassner, G.J.: Geometry effects in nodal discontinuous Galerkin methods on curved elements that are provably stable. Appl. Math. Comput. 272, 274–290 (2016)

    MathSciNet  MATH  Google Scholar 

  37. Kopriva, D.A., Hindenlang, F.J., Bolemann, T., Gassner, G.J.: Free-Stream Preservation for Curved Geometrically Non-conforming Discontinuous Galerkin Spectral Elements. J. Sci. Comput. 79(3), 1389–1408 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  38. Kopriva, D.A.: Metric identities and the discontinuous spectral element method on curvilinear meshes. J. Sci. Comput. 26(3), 301–327 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  39. Thomas, P.D., Lombard, C.K.: Geometric conservation law and its application to flow computations on moving grids. AIAA J. 17(10), 1030–1037 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  40. Visbal, M.R., Gaitonde, D.V.: On the use of higher-order finite-difference schemes on curvilinear and deforming meshes. J. Comput. Phys. 181(1), 155–185 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  42. Hesthaven, J.S.: From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex. SIAM J. Numer. Anal. 35(2), 655–676 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  43. Warburton, T.: An explicit construction of interpolation nodes on the simplex. J. Eng. Math. 56(3), 247–262 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  44. Chan, J., Warburton, T.: A comparison of high order interpolation nodes for the pyramid. SIAM J. Sci. Comput. 37(5), A2151–A2170 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  45. Winters, A.R., Derigs, D., Gassner, G.J., Walch, S.: A uniquely defined entropy stable matrix dissipation operator for high Mach number ideal MHD and compressible Euler simulations. J. Comput. Phys. 332, 274–289 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  46. Ismail, F., Roe, P.L.: Affordable, entropy-consistent Euler flux functions II: entropy production at shocks. J. Comput. Phys. 228(15), 5410–5436 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  47. Ranocha, H.: Comparison of some entropy conservative numerical fluxes for the Euler equations. J. Sci. Comput. 76(1), 216–242 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  48. Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. In: Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, pp. 325–432. Springer, Berlin (1998)

  49. Carpenter, M.H., Kennedy, C.A.: Fourth-order \(2n\)-storage Runge–Kutta schemes. Technical Report NASA-TM-109112, NAS 1.15:109112, NASA Langley Research Center (1994)

  50. Chan, J., Evans, J.A.: Multi-patch discontinuous Galerkin isogeometric analysis for wave propagation: Explicit time-stepping and efficient mass matrix inversion. Comput. Methods Appl. Mech. Eng. 333, 22–54 (2018)

    Article  MathSciNet  Google Scholar 

  51. Hindenlang, F.J., Gassner, G.J.: On the order reduction of entropy stable DGSEM for the compressible Euler equations. arXiv preprint arXiv:1901.05812 (2019)

  52. 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)

    Article  MathSciNet  MATH  Google Scholar 

  53. Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer, Berlin (1994)

    Book  MATH  Google Scholar 

  54. Krivodonova, L., Qin, R.: An analysis of the spectrum of the discontinuous Galerkin method. Appl. Numer. Math. 64, 1–18 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The author thanks David C. Del Rey Fernandez for helpful discussions, as well as the two anonymous reviewers whose comments significantly improved the readabilty of this manuscript. Jesse Chan is supported by the National Science Foundation under awards DMS-1719818 and DMS-1712639.

Author information

Authors and Affiliations

  1. Department of Computational and Applied Mathematics, Rice University, 6100 Main Street, Houston, TX, 77005, USA

    Jesse Chan

Authors
  1. Jesse Chan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jesse Chan.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

A Dependence of Inverse and Trace Constants on Quadrature

A Dependence of Inverse and Trace Constants on Quadrature

The maximum stable timestep under explicit time-stepping depends on specific choices of volume and surface quadrature. The dependence of timestep on quadrature has been documented for tensor product elements in [52], where they showed that for a high order Taylor method in time, the maximum stable timestep under \((N+1)\)-point GLL volume and surface quadratures is roughly twice as large as the maximum stable timestep when volume/surface integrals are approximated using \((N+1)\) point Gauss quadratures.

This discrepancy can be understood in terms of constants in finite element inverse and trace inequalities. It was shown in [28, 50] that, for linear problems, the maximum stable time-step scales inversely with the order-dependent constants \(C_I, C_T\), where

$$\begin{aligned} \int _{{\widehat{D}}} \left| \nabla u \right| ^2 \le C_I \int _{{\widehat{D}}} u^2, \qquad \int _{\partial {\widehat{D}}} u^2 \le C_T \int _{{\widehat{D}}} u^2, \qquad \forall u \in V^N . \end{aligned}$$
(31)

Here, the integrals over \({\widehat{D}}, \partial {\widehat{D}}\) are computed using the same volume and surface quadrature rules used for computations. These constants can be used to bound surface integrals which appear in DG formulations, which can in turn be used to construct bounds on the spectral radius of DG discretization matrices. The maximum stable timestep \(dt_{\max }\) is thus inversely proportional to the inverse and trace constants

$$\begin{aligned} dt_{\max } \propto C_T^{-1}, C_I^{-1} \end{aligned}$$

The constants \(C_I, C_T\) depend on the choices of volume and surface quadrature used to evaluate each of the integrals in (31). It is known that \(L^2\) norm computed using GLL quadrature is weaker than the full \(L^2\) norm [35, 53]. For the domain \({\widehat{D}} = [-1,1]^d\) in d dimensions, it can be shown that

$$\begin{aligned} \int _{{\widehat{D}}} u^2 \le \int _{{\widehat{D}}, \mathrm{GLL}} u^2 \le \left( 2+\frac{1}{N} \right) ^{d/2}\int _{{\widehat{D}}} u^2 \qquad \forall u \in Q^{N}, \end{aligned}$$
(32)

where the middle integral is under-integrated using GLL quadrature. In other words, the discrete \(L^2\) norm induced using GLL quadrature is weaker than the \(L^2\) norm induced by a more accurate quadrature rule, which will be reflected in the trace and inverse constants.

Table 2 Inverse and trace constants for triangular and quadrilateral elements with different quadrature configurations
Full size table

Table 2 shows trace and inverse constants for triangular and quadrilateral elements under several different configurations of quadrature. For quadrilateral elements at high orders, we observe that the degree N inverse constants \(C_I\) under Gauss quadrature are roughly as large as the degree \((N+1)\) inverse constants under (volume) GLL quadrature. The degree N trace constants \(C_T\) under Gauss quadrature are exactly equal to the degree \((N+1)\) trace constants under GLL quadrature, which was proven in [28]. Trace constants under GLL volume and Gauss surface quadrature are also identical to trace constants computed using GLL quadrature for both volume and surface integrals, which is a consequence of the lower bound in (32).

Several observations can be made based on the values of \(C_I, C_T\) presented in Table 2. On quadrilaterals, the maximum stable time-step for a degree N DG scheme using Gauss quadrature is expected to be smaller than that of a degree N scheme using GLL quadrature, which matches observations in [52]. Additionally, the maximum stable timestep under GLL volume quadrature and Gauss surface quadrature should be the same as the maximum stable timestep when GLL quadrature is used for both volume and surface integrals (e.g. DG-SEM). For triangles, the maximum stable timestep should be smaller under surface GLL quadrature compared to surface Gauss quadrature. However, we note that, while bounds on the maximum stable time-step can be derived based on the constants \(C_I, C_T\) [28, 50], these bounds are not tight for upwind or dissipative fluxes [54].

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chan, J. Skew-Symmetric Entropy Stable Modal Discontinuous Galerkin Formulations. J Sci Comput 81, 459–485 (2019). https://doi.org/10.1007/s10915-019-01026-w

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-019-01026-w

Keywords

Search

Navigation