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An analysis of discontinuous Galerkin methods for the compressible Euler equations: entropy and \(L_2\) stability

  • David M. WilliamsEmail author
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Abstract

The objective of this article is to characterize the entropy and \(L_2\) stability of several representative discontinuous Galerkin (DG) methods for solving the compressible Euler equations. Towards this end, three DG methods are constructed: one DG method with entropy variables as its unknowns, and two DG methods with conservative variables as their unknowns. These methods are employed in order to discretize the compressible Euler equations in space. Thereafter, the resulting semi-discrete formulations are analyzed, and the entropy and \(L_2\) stability characteristics are evaluated. It is shown that the semi-discrete formulation of the DG method with entropy variables is entropy and \(L_2\) stable. Furthermore, it is shown that the semi-discrete formulations of the DG methods with conservative variables are only guaranteed to be entropy and \(L_2\) stable under the following assumptions: the entropy projection errors vanish, or the terms containing the entropy projection errors are non-positive. Thereafter, the semi-discrete formulation with entropy variables, and one of the semi-discrete formulations with conservative variables, are discretized in time with an ‘algebraically stable’ Runge–Kutta (RK) scheme. The resulting formulations are fully-discrete and can be immediately applied to practical problems. In this article, they are employed to simulate a vortex propagating for long distances. It is shown that temporal stability is maintained by the DG method with entropy variables, but the DG method with conservative variables exhibits instability.

Mathematics Subject Classification

65M12 65M60 76N99 

Notes

Acknowledgements

The author would like to thank Dr. Siddhartha Mishra (ETH Zürich), Dr. Sigal Gottlieb (University of Massachusetts, Dartmouth), Dr. Dmitry Kamenetskiy (Boeing Research & Technology), Dr. Krzysztof Fidkowski (University of Michigan), Dr. Jesse Chan (Rice University), and Dr. Jason Hicken (Rensselaer Polytechnic Institute) for their participation in conversations that helped shape this work. The author would also like to thank Dr. David Darmofal, Dr. Marshall Galbraith, and Dr. Steven Allmaras (Massachusetts Institute of Technology) for developing the Solution Adaptive Numerical Simulator (SANS) code [30, 31] that was utilized for the numerical experiments.

References

  1. 1.
    Alexander, R.: Diagonally implicit Runge-Kutta methods for stiff ODE’s. SIAM J. Numer. Anal. 14(6), 1006–1021 (1977)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Balay, S., Abhyankar, S., Adams, M., Brown, J., Brune, P., Buschelman, K., Dalcin, L.D., Eijkhout, V., Gropp, W., Kaushik, D.: PETSc users manual revision 3.8. Tech. rep., Argonne National Lab.(ANL), Argonne, IL (United States) (2017)Google Scholar
  3. 3.
    Balsara, D.S., Shu, C.W.: Monotonicity preserving weighted essentially non-oscillatory schemes with increasingly high order of accuracy. J. Comput. Phys. 160(2), 405–452 (2000)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Barth, T., Charrier, P., Mansour, N.N.: Energy stable flux formulas for the discontinuous Galerkin discretization of first order nonlinear conservation laws. Tech. rep, NASA (2001)Google Scholar
  5. 5.
    Barth, T.J.: Numerical methods for gasdynamic systems on unstructured meshes. In: An Introduction to Recent Developments in Theory and Numerics for Conservation Laws, pp. 195–285. Springer, Berlin Heidelberg (1999)Google Scholar
  6. 6.
    Barth, T.J.: On discontinuous Galerkin approximations of Boltzmann moment systems with Levermore closure. Comput. Methods Appl. Mech. Eng. 195(25), 3311–3330 (2006)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)zbMATHGoogle Scholar
  8. 8.
    Burrage, K.: Efficiently implementable algebraically stable Runge-Kutta methods. SIAM J. Numer. Anal. 19(2), 245–258 (1982)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Castonguay, P., Vincent, P., Jameson, A.: Application of high-order energy stable flux reconstruction schemes to the Euler equations. In: 49th AIAA Aerospace Sciences Meeting including the New Horizons Forum and Aerospace Exposition (2011)Google Scholar
  10. 10.
    Chan, J., Demkowicz, L., Moser, R.: A DPG method for steady viscous compressible flow. Comput. Fluids 98, 69–90 (2014)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Chan, J., Demkowicz, L., Moser, R., Roberts, N.: A class of discontinuous Petrov–Galerkin methods. Part V: Solution of 1D Burgers and Navier–Stokes equations. ICES Report 29 (2010)Google Scholar
  12. 12.
    Ciarlet, P.G., Raviart, P.A.: General Lagrange and Hermite interpolation in Rn with applications to finite element methods. Arch. Ration. Mech. Anal. 46(3), 177–199 (1972)zbMATHGoogle Scholar
  13. 13.
    Cockburn, B.: Discontinuous Galerkin methods. ZAMM—J. Appl. Math. Mech. 83(11), 731–754 (2003)Google Scholar
  14. 14.
    Cockburn, B., Hou, S., Shu, C.W.: The Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case. Math. Comput. 54(190), 545–581 (1990)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Cockburn, B., Karniadakis, G.E., Shu, C.W.: The development of discontinuous Galerkin methods. In: Discontinuous Galerkin Methods, pp. 3–50. Springer (2000)Google Scholar
  16. 16.
    Cockburn, B., Lin, S.Y., Shu, C.W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems. J. Comput. Phys. 84(1), 90–113 (1989)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Cockburn, B., Shu, C.W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws II. General framework. Math. Comput. 52(186), 411–435 (1989)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Cockburn, B., Shu, C.W.: The Runge–Kutta local projection \(p^{1}\)-discontinuous Galerkin finite element method for scalar conservation laws. RAIRO-Modélisation mathématique et analyse numérique 25(3), 337–361 (1991)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Cockburn, B., Shu, C.W.: The local discontinuous Galerkin method for time-dependent convection-diffusion systems. SIAM J. Numer. Anal. 35(6), 2440–2463 (1998)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Cockburn, B., Shu, C.W.: The Runge–Kutta discontinuous Galerkin method for conservation laws V: multidimensional systems. J. Comput. Phys. 141(2), 199–224 (1998)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Dafermos, C.: Hyperbolic Conservation Laws in Continuum Physics. Springer, Berlin (2005)zbMATHGoogle Scholar
  22. 22.
    Demkowicz, L., Gopalakrishnan, J.: A class of discontinuous Petrov–Galerkin methods. Part I: the transport equation. Comput. Methods Appl. Mech. Eng. 199(23), 1558–1572 (2010)Google Scholar
  23. 23.
    Demkowicz, L., Gopalakrishnan, J.: A class of discontinuous Petrov–Galerkin methods. Part II: optimal test functions. Numer. Methods Partial Differ. Equ. 27(1), 70–105 (2011)Google Scholar
  24. 24.
    Demkowicz, L., Gopalakrishnan, J., Niemi, A.H.: A class of discontinuous Petrov–Galerkin methods. Part III: adaptivity. Appl. Numer. Math. 62(4), 396–427 (2012)zbMATHGoogle Scholar
  25. 25.
    Di Pietro, D.A., Ern, A.: Mathematical Aspects of Discontinuous Galerkin Methods, vol. 69. Springer, Heidelberg (2011)zbMATHGoogle Scholar
  26. 26.
    Dutt, P.: Stable boundary conditions and difference schemes for Navier–Stokes equations. SIAM J. Numer. Anal. 25(2), 245–267 (1988)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Ellis, T., Demkowicz, L., Chan, J.: Locally conservative discontinuous Petrov–Galerkin finite elements for fluid problems. Comput. Math. Appl. 68(11), 1530–1549 (2014)MathSciNetzbMATHGoogle Scholar
  28. 28.
    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)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Fjordholm, U.S., Mishra, S., Tadmor, E.: ENO reconstruction and ENO interpolation are stable. Found. Comput. Math. 13(2), 139–159 (2013)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Galbraith, M.C., Allmaras, S., Darmofal, D.L.: A verification driven process for rapid development of CFD software. In: 53rd AIAA Aerospace Sciences Meeting (2015)Google Scholar
  31. 31.
    Galbraith, M.C., Allmaras, S.R., Darmofal, D.L.: SANS RANS solutions for 3D benchmark configurations. In: 2018 AIAA Aerospace Sciences Meeting, Kissimmee, Florida (2018)Google Scholar
  32. 32.
    Godunov, S.K.: An interesting class of quasilinear systems. In: Dokl. Akad. Nauk SSSR, pp. 521–523 (1961)Google Scholar
  33. 33.
    Harten, A.: On the symmetric form of systems of conservation laws with entropy. J. Comput. Phys. 49(1), 151–164 (1983)MathSciNetzbMATHGoogle Scholar
  34. 34.
    Harten, A.: ENO schemes with subcell resolution. J. Comput. Phys. 83(1), 148–184 (1989)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Harten, A., Engquist, B., Osher, S., Chakravarthy, S.R.: Uniformly high order accurate essentially non-oscillatory schemes: III. In: Upwind and High-Resolution Schemes, pp. 218–290. Springer (1987)Google Scholar
  36. 36.
    Harten, A., Osher, S.: Uniformly high-order accurate nonoscillatory schemes: I. SIAM J. Numer. Anal. 24(2), 279–309 (1987)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Harten, A., Osher, S., Engquist, B., Chakravarthy, S.R.: Some results on uniformly high-order accurate essentially nonoscillatory schemes. Appl. Numer. Math. 2(3–5), 347–377 (1986)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer, New York (2007)zbMATHGoogle Scholar
  39. 39.
    Hiltebrand, A., Mishra, S.: Entropy stable shock capturing space–time discontinuous Galerkin schemes for systems of conservation laws. Numerische Mathematik 126(1), 103–151 (2014)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Hou, S., Liu, X.D.: Solutions of multi-dimensional hyperbolic systems of conservation laws by square entropy condition satisfying discontinuous Galerkin method. J. Sci. Comput. 31(1–2), 127–151 (2007)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Hu, C., Shu, C.W.: Weighted essentially non-oscillatory schemes on triangular meshes. J. Comput. Phys. 150(1), 97–127 (1999)MathSciNetzbMATHGoogle Scholar
  42. 42.
    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)Google Scholar
  43. 43.
    Hughes, T.J.R., Mallet, M.: A new finite element formulation for computational fluid dynamics: III. The generalized streamline operator for multidimensional advective-diffusive systems. Comput. Methods Appl. Mech. Eng. 58(3), 305–328 (1986)Google Scholar
  44. 44.
    Hughes, T.J.R., Mallet, M.: A new finite element formulation for computational fluid dynamics: IV. A discontinuity-capturing operator for multidimensional advective-diffusive systems. Comput. Methods Appl. Mech. Eng. 58(3), 329–336 (1986)Google Scholar
  45. 45.
    Hughes, T.J.R., Mallet, M., Akira, M.: A new finite element formulation for computational fluid dynamics: II. Beyond SUPG. Comput. Methods Appl. Mech. Eng. 54(3), 341–355 (1986)Google Scholar
  46. 46.
    Jiang, G.S., Shu, C.W.: On a cell entropy inequality for discontinuous Galerkin methods. Math. Comput. 62(206), 531–538 (1994)MathSciNetzbMATHGoogle Scholar
  47. 47.
    Kirby, R.M., Karniadakis, G.E.: De-aliasing on non-uniform grids: algorithms and applications. J. Comput. Phys. 191(1), 249–264 (2003)zbMATHGoogle Scholar
  48. 48.
    Lax, P.D.: Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, vol. 11. SIAM, Philadelphia (1973)zbMATHGoogle Scholar
  49. 49.
    Lefloch, P.G., Mercier, J.M., Rohde, C.: Fully discrete, entropy conservative schemes of arbitrary order. SIAM J. Numer. Anal. 40(5), 1968–1992 (2002)MathSciNetzbMATHGoogle Scholar
  50. 50.
    Liu, X.D., Osher, S., Chan, T.: Weighted essentially non-oscillatory schemes. J. Comput. Phys. 115(1), 200–212 (1994)MathSciNetzbMATHGoogle Scholar
  51. 51.
    Mock, M.S.: Systems of conservation laws of mixed type. J. Differ. Equ. 37(1), 70–88 (1980)MathSciNetzbMATHGoogle Scholar
  52. 52.
    Nesterov, Y.: Introductory Lectures on Convex Optimization: A Basic Course. Springer, New York (2004)zbMATHGoogle Scholar
  53. 53.
    Pugh, C.C.: Real Mathematical Analysis. Springer, New York (2002)zbMATHGoogle Scholar
  54. 54.
    Quarteroni, A., Sacco, R., Saleri, F.: Numerical Mathematics, vol. 37. Springer, New York (2010)zbMATHGoogle Scholar
  55. 55.
    Saad, Y., Schultz, M.H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems. SIAM J. Sci. Stat. Comput. 7(3), 856–869 (1986)MathSciNetzbMATHGoogle Scholar
  56. 56.
    Shakib, F., Hughes, T.J.R., Johan, Z.: A new finite element formulation for computational fluid dynamics: X. The compressible Euler and Navier–Stokes equations. In: Computer Methods in Applied Mechanics and Engineering, pp. 141–219 (1991)Google Scholar
  57. 57.
    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 (1998)Google Scholar
  58. 58.
    Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77(2), 439–471 (1988)MathSciNetzbMATHGoogle Scholar
  59. 59.
    Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes: II. In: Upwind and High-Resolution Schemes, pp. 328–374. Springer (1989)Google Scholar
  60. 60.
    Spiegel, S.C., Huynh, H.T., DeBonis, J.R.: A survey of the isentropic Euler vortex problem using high-order methods. In: 22nd AIAA Computational Fluid Dynamics Conference (2015)Google Scholar
  61. 61.
    Strang, G.: Introduction to Linear Algebra. Wellesley-Cambridge Press Wellesley, Wellesley (2016)zbMATHGoogle Scholar
  62. 62.
    Svärd, M.: Weak solutions and convergent numerical schemes of modified compressible Navier-Stokes equations. J. Comput. Phys. 288, 19–51 (2015)MathSciNetzbMATHGoogle Scholar
  63. 63.
    Svärd, M., Özcan, H.: Entropy-stable schemes for the Euler equations with far-field and wall boundary conditions. J. Sci. Comput. 58(1), 61–89 (2014)MathSciNetzbMATHGoogle Scholar
  64. 64.
    Tadmor, E.: The numerical viscosity of entropy stable schemes for systems of conservation laws. I. Math. Comput. 49(179), 91–103 (1987)MathSciNetzbMATHGoogle Scholar
  65. 65.
    Tadmor, E.: Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numerica 12, 451–512 (2003)MathSciNetzbMATHGoogle Scholar
  66. 66.
    Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics: A Practical Introduction. Springer, Berlin (2013)Google Scholar
  67. 67.
    Wang, Z.J.: Adaptive High-Order Methods in Computational Fluid Dynamics, vol. 2. World Scientific, Singapore (2011)zbMATHGoogle Scholar
  68. 68.
    Wang, Z.J., Liu, Y., May, G., Jameson, A.: Spectral difference method for unstructured grids II: extension to the Euler equations. J. Sci. Comput. 32(1), 45–71 (2007)MathSciNetzbMATHGoogle Scholar
  69. 69.
    Williams, D.: An entropy stable, hybridizable discontinuous Galerkin method for the compressible Navier-Stokes equations. Math. Comput. 87(309), 95–121 (2018)MathSciNetzbMATHGoogle Scholar
  70. 70.
    Zitelli, J., Muga, I., Demkowicz, L., Gopalakrishnan, J., Pardo, D., Calo, V.M.: A class of discontinuous Petrov–Galerkin methods. Part IV: the optimal test norm and time-harmonic wave propagation in 1D. J. Comput. Phys. 230(7), 2406–2432 (2011)Google Scholar

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Authors and Affiliations

  1. 1.Department of Mechanical EngineeringThe Pennsylvania State UniversityUniversity ParkUSA

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