Top

Journal of Scientific Computing

Published in:

30-04-2018

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

Authors: B. W. Wissink, G. B. Jacobs, J. K. Ryan, W. S. Don, E. T. A. van der Weide

Published in: Journal of Scientific Computing | Issue 1/2018

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

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.

previous article Hybrid Optimized Low-Dissipation and Adaptive MUSCL Reconstruction Technique for Hyperbolic Conservation Laws

next article PCBC: Quasiconformality of Point Cloud Mappings

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

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

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

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

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)MathSciNetCrossRefMATH

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)MathSciNetCrossRefMATH

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)MathSciNetCrossRefMATH

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

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)

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

10.

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

11.

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

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)MathSciNetCrossRefMATH

13.

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

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)MathSciNetCrossRefMATH

15.

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

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)MathSciNetCrossRefMATH

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)CrossRefMATH

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)MathSciNetCrossRefMATH

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)MathSciNetCrossRefMATH

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)CrossRefMATH

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)MathSciNetCrossRefMATH

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)MathSciNetMATH

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)MathSciNetCrossRefMATH

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)MathSciNetCrossRefMATH

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)MathSciNetCrossRefMATH

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)

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)MathSciNetCrossRefMATH

30.

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

Title: Shock Regularization with Smoothness-Increasing Accuracy-Conserving Dirac-Delta Polynomial Kernels
Authors: B. W. Wissink
G. B. Jacobs
J. K. Ryan
W. S. Don
E. T. A. van der Weide
Publication date: 30-04-2018
Publisher: Springer US
Published in: Journal of Scientific Computing / Issue 1/2018
Print ISSN: 0885-7474
Electronic ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-018-0719-5

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Wirtschaft"

Springer Professional "Technik"

Other articles of this Issue 1/2018

PCBC: Quasiconformality of Point Cloud Mappings

A Sparse Grid Stochastic Collocation Upwind Finite Volume Element Method for the Constrained Optimal Control Problem Governed by Random Convection Diffusion Equations

High Spatial Order Energy Stable FDTD Methods for Maxwell’s Equations in Nonlinear Optical Media in One Dimension

A Stable Fast Time-Stepping Method for Fractional Integral and Derivative Operators

Optimization with Respect to Order in a Fractional Diffusion Model: Analysis, Approximation and Algorithmic Aspects

Dispersion Properties of Explicit Finite Element Methods for Wave Propagation Modelling on Tetrahedral Meshes

Premium Partner