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Regularization of Singularities in the Weighted Summation of Dirac-Delta Functions for the Spectral Solution of Hyperbolic Conservation Laws

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Abstract

Singular source terms expressed as weighted summations of Dirac-delta functions are regularized through approximation theory with convolution operators. We consider the numerical solution of scalar and one-dimensional hyperbolic conservation laws with the singular source by spectral Chebyshev collocation methods. The regularization is obtained by convolution with a high-order compactly supported Dirac-delta approximation whose overall accuracy is controlled by the number of vanishing moments, degree of smoothness and length of the support (scaling parameter). An optimal scaling parameter that leads to a high-order accurate representation of the singular source at smooth parts and full convergence order away from the singularities in the spectral solution is derived. The accuracy of the regularization and the spectral solution is assessed by solving an advection and Burgers equation with smooth initial data. Numerical results illustrate the enhanced accuracy of the spectral method through the proposed regularization.

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References

  1. Abe, H., Natsuhiko, S., Itatani, R.: High-order spline interpolations in the particle simulation. J. Comput. Phys. 63, 247–267 (1986)

    Article  MATH  Google Scholar 

  2. Castro, M., Costa, B., Don, W.S.: High order weighted essentially non-oscillatory WENO-Z schemes for hyperbolic conservation laws. J. Comput. Phys. 230(5), 1766–1792 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Costa, B., Don, W.S.: Multi-domain hybrid spectral-WENO methods for hyperbolic conservation laws. J. Comput. Phys. 224(2), 970–991 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  4. Crowe, C.T., Sharma, M.P., Stock, D.E.: The particle-source in cell (PSI-Cell) model for gas-droplet flows. J. Fluids Eng. 99(2), 325–332 (1977)

    Article  Google Scholar 

  5. Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration, 2nd edn. Academic Press, Inc., Cambridge (1984)

    MATH  Google Scholar 

  6. Folland, G.B.: Real Analysis: Modern Techniques and Their Applications. Wiley, New York (1984)

    MATH  Google Scholar 

  7. Gelb, A., Cates, D.: Segmentation of images from fourier spectral data. Commun. Comput. Phys. 5, 326–349 (2009)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Hesthaven, J.S., Gottlieb, S., Gottlieb, D.: Spectral Methods for Time-Dependent Problems, vol. 21. Cambridge University Press, Cambridge (2007)

    Book  MATH  Google Scholar 

  10. Hockney, R.W., Eastwood, J.W.: Computer Simulation Using Particles. Taylor & Francis Group, New York (1988)

    Book  MATH  Google Scholar 

  11. Jacobs, G., Don, W.S.: A high-order WENO-Z finite difference scheme based particle-source-in-cell method for computation of particle-laden flows with shocks. J. Comput. Phys. 228, 1365–1379 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kaufmann, A., Moreau, M., Simonin, O., Helie, J.: Comparison between lagrangian and mesoscopic eulerian modelling approaches for inertial particles suspended in decaying isotropic turbulence. J. Comput. Phys. 227(13), 6448–6472 (2008)

    Article  MATH  Google Scholar 

  13. Malgouyres, F., Guichard, F.: Edge direction preserving image zooming: a mathematical and numerical analysis. SIAM J. Numer. Anal. 39(1), 1–37 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  14. Meijering, E.H.W., Niessen, W.J., Viergever, M.A.: Quantitative evaluation of convolution based methods for medical image interpolation. Med. Image Anal. 5(2), 111–126 (2001)

    Article  Google Scholar 

  15. Monaghan, J.J.: Shock simulation by the particle method SPH. J. Comput. Phys. 52, 374–389 (1983)

    Article  MATH  Google Scholar 

  16. Monaghan, J.J.: Extrapolating B splines for interpolation. J. Comput. Phys. 60, 253–262 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  17. Monaghan, J.J.: Smoothed particle hydrodynamics. Rep. Prog. Phys. 68, 1703–1759 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  18. Panda, R., Chatterji, B.N.: Least squares generalized B-spline signal and image processing. Signal Process. 81, 2005–2017 (2001)

    Article  Google Scholar 

  19. Ryan, J., Shu, C.W.: On a one-sided post-processing technique for the discontinuous Galerkin methods. Methods Appl. Anal. 10(2), 295–308 (2003)

    MathSciNet  MATH  Google Scholar 

  20. Schaum, A.: Theory and design of local interpolators. CVGIP: Gr. Model. Image. Process 55(6), 464–481 (1993)

    Google Scholar 

  21. Shu, C.W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws. Lect. Notes Math. 1697, 325–432 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  22. Suarez, J.P., Jacobs, G., 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(4), A1831–A1849 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Thévenaz, P., Blu, T., Unser, M.: Interpolation revisited. IEEE Trans. Med. Imaging 19(7), 739–758 (2000)

    Article  Google Scholar 

  24. Tornberg, A.K.: Multi-dimensional quadrature of singular and discontinuous functions. BIT 42(3), 644–669 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  25. Yang, Y., Shu, C.W.: Discontinuous Galerkin method for hyperbolic equations involving \(\delta \)-singularities: negative-order norm error estimates and applications. Numer. Math. 124, 753–781 (2013)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Computational Science Research Center, San Diego State University, San Diego, CA, 92182-1245, USA

    Jean-Piero Suarez

  2. Department of Aerospace Engineering and Engineering Mechanics, San Diego State University, San Diego, CA, 92182, USA

    Gustaaf B. Jacobs

Authors
  1. Jean-Piero Suarez
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  2. Gustaaf B. Jacobs
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Correspondence to Gustaaf B. Jacobs.

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This research was supported by the Air Force Office of Scientific Research (AFOSR-F9550-09-1-0097), National Science Foundation (NSF-DMS-1115705) and the Computational Science Reseach Center (CSRC) at San Diego State University.

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Suarez, JP., 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 72, 1080–1092 (2017). https://doi.org/10.1007/s10915-017-0389-8

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  • DOI: https://doi.org/10.1007/s10915-017-0389-8

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