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Shock Regularization with Smoothness-Increasing Accuracy-Conserving Dirac-Delta Polynomial Kernels

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Abstract

A smoothness-increasing accuracy conserving filtering approach to the regularization of discontinuities is presented for single domain spectral collocation approximations of hyperbolic conservation laws. The filter is based on convolution of a polynomial kernel that approximates a delta-sequence. The kernel combines a kth order smoothness with an arbitrary number of m zero moments. The zero moments ensure a mth order accurate approximation of the delta-sequence to the delta function. Through exact quadrature the projection error of the polynomial kernel on the spectral basis is ensured to be less than the moment error. A number of test cases on the advection equation, Burger’s equation and Euler equations in 1D and 2D show that the filter regularizes discontinuities while preserving high-order resolution away from a discontinuity.

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References

  1. Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 77, 439–471 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cockburn, B.: Devising discontinuous galerkin methods for non-linear hyperbolic conservation laws. J. Comput. Appl. Math. 128, 187–204 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cockburn, B., Shu, C.W.: Runge–Kutta discontinuous galerkin methods for convection-dominated problems. J. Sci. Comput. 16, 173–261 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dumbser, M., Zanotti, O., Loubére, R., Diot, S.: A posteriori subcell limiting of the discontinuous Galerkin finite element method for hyperbolic conservation laws. J. Comput. Phys. 278, 47–75 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Chaudhuri, A., Jacobs, G.B., Don, W.S., Abassi, H., Mashayek, F.: Explicit discontinuous spectral element method with entropy generation based artificial viscosity for shocked viscous flows. J. Comput. Phys. 332, 99–117 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hughes, T.J., Franca, L., 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, 223–234 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  7. Guermond, J.L., Pasquetti, R., Popov, B.: Entropy viscosity method for nonlinear conservation laws. J. Comput. Phys. 230, 4248–4267 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Persson, P.O., Peraire, J.: Sub-cell shock capturing for discontinuous Galerkin methods. Proceedings of the 44th AIAA Aerospace Sciences Meeting and Exhibit, January 2006. AIAA-2006-112 (2006)

  9. Don, W.S.: Numerical Study of pseudospectral methods in shock wave applications. J. Comput. Phys. 110, 103–111 (1994)

    Article  MATH  Google Scholar 

  10. Vandeven, H.: Family of spectral filters for discontinuous problems. J. Sci. Comput. 6(2), 159–192 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bramble, J.H., Schatz, A.H.: Higher order local accuracy by averaging in the finite element method. Math. Comput. 31, 94–111 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  12. Cockburn, B., Luskin, M., Shu, C.W., Sli, E.: Enhanced accuracy by post-processing for finite element methods for hyperbolic equations. Math. Comput. 72, 577–606 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  13. Mock, M.S., Lax, P.D.: The computation of discontinuous solutions of linear hyperbolic equations. Commun. Pure Appl. Math. 18, 423–430 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  14. Suarez, J.P., Jacobs, G.B., Don, W.S.: A high-order Dirac-delta regularization with optimal scaling in the spectral solution of one-dimensional singular hyperbolic conservation laws. SIAM J. Sci. Comput. 36, 1831–1849 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Tornberg, A.K.: Multi-dimensional quadrature of singular and discontinuous functions. BIT 42, 644–669 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  16. Suarez, J.P., Jacobs, G.B.: Regularization of singularities in the weighted summation of Dirac-delta functions for the spectral solution of hyperbolic conservation laws. J. Sci. Comput. arXiv:1611.05510

  17. Gottlieb, S., Hesthaven, J., Gottlieb, D.: Spectral Methods for Time-Dependent Problems. Cambridge University Press, Cambridge (2007)

  18. Gottlieb, S., Shu, C.W.: Total variation diminishing Runge-Kutta schemes. Math. Comput. 67(221), 73–85 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ryan, J.K., Shu, C.W., Atkins, H.L.: Extension of a post-processing technique for discontinuous Galerkin methods for hyperbolic equations with application to an aeroacoustic problem. SIAM J. Sci. Comput. 26, 821–843 (2004)

    Article  MATH  Google Scholar 

  20. King, J., Mirzaee, H., Ryan, J.K., Kirby, R.M.: Smoothness-increasing accuracy-conserving (SIAC) filtering for discontinuous Galerkin solutions: Improved errors versus higher-order accuracy. J. Sci. Comput. 53, 129–149 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Curtis, S., Kirby, R.M., Ryan, J.K., Shu, C.W.: Post-processing for the discontinuous Galerkin method over non-uniform meshes. SIAM J. Sci. Comput. 30, 272–289 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ji, L., van Slingerland, P., Ryan, J.K., Vuik, C.: Superconvergent error estimates for a position-dependent smoothness-increasing accuracy-conserving filter for DG solutions. Math. Comput. 83, 2239–2262 (2014)

    Article  MATH  Google Scholar 

  23. Mirzaee, H., Ji, L., Ryan, J.K., Kirby, R.M.: Smoothness-increasing accuracy-conserving (SIAC) post-processing for discontinuous Galerkin solutions over structured triangular meshes. SIAM J. Numer. Anal. 49, 1899–1920 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ryan, J.K., Shu, C.W.: On a one-sided post-processing technique for the discontinuous Galerkin methods. Methods Appl. Anal. 10, 295–307 (2003)

    MathSciNet  MATH  Google Scholar 

  25. van Slingerland, P., Ryan, J.K., Vuik, C.: Position-dependent smoothness-increasing accuracy-conserving (SIAC) filtering for accuracy for improving discontinuous Galerkin solutions. SIAM J. Sci. Comput. 33, 802–825 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Mirzaee, H., Ryan, J.K., Kirby, R.M.: Quantification of errors introduced in the numerical approximation and implementation of smoothness-increasing accuracy-conserving (SIAC) filtering of discontinuous Galerkin (DG) fields. J. Sci. Comput. 45, 447–470 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  27. Mirzaee, H., Ryan, J.K., Kirby, R.M.: Efficient implementation of smoothness-increasing accuracy-conserving (SIAC) filters for discontinuous Galerkin solutions. J. Sci. Comput. 52, 85–112 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  28. Steffan, M., Curtis, S., Kirby, R.M., Ryan, J.K.: Investigation of smoothness-increasing accuracy-conserving filters for improving streamline integration through discontinous fields. IEEE-TVCG 14, 680–692 (2008)

    Google Scholar 

  29. Walfisch, D., Ryan, J.K., Kirby, R.M., Haimes, R.: One-sided smoothness-increasing accuracy-conserving filtering for enhanced streamline integration through discontinuous fields. J. Sci. Comput. 38, 164–184 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  30. Toro, E.F.: Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Aerospace Engineering, San Diego State University, San Diego, CA, USA

    B. W. Wissink & G. B. Jacobs

  2. School of Mathematics, University of East Anglia, Norwich, UK

    J. K. Ryan

  3. School of Mathematical Sciences, Ocean University of China, Qingdao, China

    W. S. Don

  4. Faculty of Engineering Technology, University of Twente, Enschede, The Netherlands

    B. W. Wissink & E. T. A. van der Weide

Authors
  1. B. W. Wissink
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  2. G. B. Jacobs
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  3. J. K. Ryan
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  4. W. S. Don
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  5. E. T. A. van der Weide
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Correspondence to G. B. Jacobs.

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Wissink, B.W., Jacobs, G.B., Ryan, J.K. et al. Shock Regularization with Smoothness-Increasing Accuracy-Conserving Dirac-Delta Polynomial Kernels. J Sci Comput 77, 579–596 (2018). https://doi.org/10.1007/s10915-018-0719-5

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  • DOI: https://doi.org/10.1007/s10915-018-0719-5

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