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Local Discontinuous Galerkin Methods for the Khokhlov–Zabolotskaya–Kuznetzov Equation

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Abstract

Khokhlov–Zabolotskaya–Kuznetzov (KZK) equation is a model that describes the propagation of the ultrasound beams in the thermoviscous fluid. It contains a nonlocal diffraction term, an absorption term and a nonlinear term. Accurate numerical methods to simulate the KZK equation are important to its broad applications in medical ultrasound simulations. In this paper, we propose a local discontinuous Galerkin method to solve the KZK equation. We prove the \(L^2\) stability of our scheme and conduct a series of numerical experiments including the focused circular short tone burst excitation and the propagation of unfocused sound beams, which show that our scheme leads to accurate solutions and performs better than the benchmark solutions in the literature.

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References

  1. Aanonsen, S.I., Barkve, T., Tjotta, J.N., Tjotta, S.: Distortion and harmonic generation in the nearfield of a finite amplitude sound beam. J. Acoust. Soc. Am. 75, 749–768 (1984)

    Article  MATH  Google Scholar 

  2. Baker, A.C., Berg, A., Sahin, A., Tjotta, J.N.: The nonlinear pressure field of plane rectangular apertures: experimental and theoretical results. J. Acoust. Soc. Am 97, 3510–3517 (1995)

    Article  Google Scholar 

  3. Bakhvalov, N.S., Zhileikin, Y.M., Zabolotskaya, S.A.: Nonlinear Theory of Sound Beams. American Institute of Physics, New York (1987)

    Google Scholar 

  4. Blackstock, D.T.: Connection between the Fay and Fubini solutions for plane sound waves of finite amplitude. J. Acoust. Soc. Am. 39, 1019–1026 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  5. Cockburn, B., Hou, S., Shu, C.-W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case. J. Comput. Phys. 141, 199–224 (1998)

    Article  MathSciNet  Google Scholar 

  6. Cockburn, B., Karniadakis, G., Shu, C.-W.: The development of discontinuous Galerkin methods. In: Discontinuous Galerkin Methods, pp. 3–50. Springer, Berlin, Heidelberg (2000)

  7. 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, 90–113 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cockburn, B., Shu, C.-W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws II: general framework. Math. Comput. 52, 411–435 (1989)

    MATH  Google Scholar 

  9. Cockburn, B., Shu, C.-W.: TVB Runge–Kutta local projection discontinuous Galerkin finite element method for scalar conservation laws V: multidimensional systems. Math. Comput. 52, 411–435 (1989)

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  11. Froysa, K.-E., Tjotta, J.N., Tjotta, S.: Linear propagation of a pulsed sound beam from a plane or focusing source. J. Acoust. Soc. Am. 93, 80–92 (1993)

    Article  Google Scholar 

  12. Hajihasani, M., Farjami, Y., Gharibzadeh, S., Tavakkoli, J.: A novel numerical solution to the diffraction term in the KZK nonlinear wave equation. In: 2009 38th Annual Symposium of the Ultrasonic Industry Association (2009)

  13. Hamilton, M.F.: Comparison of three transient solutions for the axial pressure in a focused sound beam. J. Acoust. Soc. Am. 92, 527–532 (1992)

    Article  Google Scholar 

  14. Lee, Y.S.: Numerical solution of the KZK equation for pulsed finite amplitude sound beams in thermoviscous fluid. Ph.D. thesis, University of Texas (1993)

  15. Lee, Y.S., Hamilton, M.F.: Time-domain modeling of pulsed finite-amplitude sound beams. J. Acoust. Soc. Am. 97, 906–917 (1995)

    Article  Google Scholar 

  16. Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Tech. report LA-UR-73-479, Los Alamos Scientific Laboratory, Los Alamos, NM (1973)

  17. Shu, C.-W.: TVB uniformly high-order schemes for conservation laws. Math. Comput. 49, 105–121 (1987)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  19. Xu, Y., Shu, C.-W.: Local discontinuous Galerkin methods for two classes of two-dimensional nonlinear wave equations. Phys. D 208, 21–58 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  20. Xu, Y., Shu, C.-W.: Local discontinuous Galerkin methods for high-order time-dependent partial differential equations. Commun. Comput. Phys. 7, 1–46 (2010)

    MATH  MathSciNet  Google Scholar 

  21. Yan, J., Shu, C.-W.: A local discontinuous galerkin method for KdV-type equations. SIAM J. Numer. Anal. 40, 769–791 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  22. Yan, J., Shu, C.-W.: Local discontinuous Galerkin methods for partial differential equations with higher order derivatives. J. Sci. Comput. 17, 27–47 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  23. Yang, X., Cleveland, R.O.: Time domain simulation of nonlinear acoustic beams generated by rectangular pistons with application to harmonic imaging. J. Acoust. Soc. Am 117, 113–123 (2005)

    Article  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, The Ohio State University, Columbus, OH, 43210, USA

    Ching-Shan Chou, Weizhou Sun & He Yang

  2. Department of Mathematics, University of California Riverside, Riverside, CA, 92521, USA

    Yulong Xing

Authors
  1. Ching-Shan Chou
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  2. Weizhou Sun
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  3. Yulong Xing
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  4. He Yang
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Corresponding author

Correspondence to Ching-Shan Chou.

Additional information

In honor of Professor Shu’s 60th birthday.

CSC is partially supported by NSF Grant DMS 1253481. YX is partially supported by NSF Grant DMS-1621111 and ONR Grant N00014-16-1-2714.

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Chou, CS., Sun, W., Xing, Y. et al. Local Discontinuous Galerkin Methods for the Khokhlov–Zabolotskaya–Kuznetzov Equation. J Sci Comput 73, 593–616 (2017). https://doi.org/10.1007/s10915-017-0502-z

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  • DOI: https://doi.org/10.1007/s10915-017-0502-z

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