Solutions of Computational Acoustic Problems Using DRP Schemes



Computational acoustics is an important and active research area [12, 13, 45, 47, 49]. Acoustic signals propagate in the form of longitudinal waves in air. Pressure fluctuations associated with acoustic signals are usually very small compared to the large background pressure field. The atmospheric pressure is around \(10^{5}\) Pa, while the amplitude of the smallest recognizable acoustic disturbance for a human being is around \(10^{-5}\) Pa.


  1. 1.
    G. Ashcroft, X. Zhang, Optimized prefactored compact schemes. J. Comput. Phys. 190(2), 459–477 (2003)zbMATHCrossRefGoogle Scholar
  2. 2.
    C. Bailly, D. Juve, Numerical solution of acoustic propagation problems using linearized euler equations. AIAA J. 38(1), 22–29 (2000)CrossRefGoogle Scholar
  3. 3.
    M. Bernardini, S. Pirozzoli, A general strategy for the optimization of runge-kutta schemes for wave propagation phenomena. J. Comput. Phys. 228(11), 4182–4199 (2009)zbMATHCrossRefGoogle Scholar
  4. 4.
    Y.G. Bhumkar, High Performance Computing of Bypass Transition. Ph.D. Thesis, Department of Aerospace Engineering, Indian Institute of Technology, Kanpur (2012)Google Scholar
  5. 5.
    Y.G. Bhumkar, T.K. Sengupta, Adaptive multi-dimensional filters. Comput. Fluids 49 (2011)Google Scholar
  6. 6.
    Y.G. Bhumkar, T.W.H. Sheu, T.K. Sengupta, A dispersion relation preserving optimized upwind compact difference scheme for high accuracy flow simulations. J. Comput. Phys. 278, 378–399 (2014)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    B.J. Boersma, A staggered compact finite difference formulation for the compressible Navier-Stokes equations. J. Comput. Phys. 208(2), 675–690 (2005)zbMATHCrossRefGoogle Scholar
  8. 8.
    C. Bogey, C. Bailly, D. Juv\(\acute{e}\), Noise investigation of a high subsonic, moderate reynolds number jet using a compressible LES. Theor. Comput. Fluid Dyn. 16(4), 273–297 (2003)Google Scholar
  9. 9.
    V. Borue, S.A. Orszag, Local energy flux and subgrid-scale statistics in three-dimensional turbulence. J. Fluid Mech. 366, 1–31 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    P.H. Chiu, T.W.H. Sheu, On the development of a dispersion-relation-preserving dual-compact upwind scheme for convection-diffusion equation. J. Comput. Phys. 228, 3640–3655 (2009)Google Scholar
  11. 11.
    P.C. Chu, C. Fan, A three-point combined compact difference scheme. J. Comput. Phys. 140, 370–399 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    T. Colonius, S.K. Lele, Computational aeroacoustics: progress on nonlinear problems of sound generation. Prog. Aero. Sci. 40, 345–416 (2004)CrossRefGoogle Scholar
  13. 13.
    H.A.V. der Vorst, Bi-CGSTAB: A fast and smoothly converging variant of Bi-CG for the solution of non-symmetric linear systems. SIAM J. Sci. Stat. Comput. 13, 631–644 (1992)zbMATHCrossRefGoogle Scholar
  14. 14.
    K.Y. Fung, R.S.O. Man, S. Davis, A compact solution to computational acoustics, in ICASE/LaRC Workshop on Benchmark Problems in Computational Aeroacoustics (CAA) (1995), pp. 59–72Google Scholar
  15. 15.
    D.V. Gaitonde, J.S. Shang, J.L. Young, Practical aspects of higher-order numerical schemes for wave propagation phenomena. Int. J. Numer. Meth. Eng. 45(12), 1849–1869 (1999)zbMATHCrossRefGoogle Scholar
  16. 16.
    S.I. Green, Fluid Vortices: Fluid Mechanics and Its Applications (Springer, Berlin, 1995)Google Scholar
  17. 17.
    N. Haugen, A. Brandenburg, Inertial range scaling in numerical turbulence with hyperviscosity. Phys. Rev. E 70, 026405 (2004)Google Scholar
  18. 18.
    A.E. Honein, P. Moin, Higher entropy conservation and numerical stability of compressible turbulence simulations. J. Comput. Phys. 201(2), 531–545 (2004)zbMATHCrossRefGoogle Scholar
  19. 19.
    G.S. Karamanos, G.E. Karniadakis, A spectral vanishing viscosity method for large-eddy simulations. J. Comput. Phys. 163(1), 22–50 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    T. Kawai, Sound diffraction by a many-sided barrier or pillar. J. Sound Vib. 79(2), 229–242 (1981)zbMATHCrossRefGoogle Scholar
  21. 21.
    L.E. Kinsler, A.R. Frey, A.B. Coppens, J.V. Sanders, Fundamentals of Acoustics (Wiley, New York, 2000)Google Scholar
  22. 22.
    E. Lamballais, V. Fortun\(\acute{e}\), S.L. Aizet, Straightforward high-order numerical dissipation via the viscous term for direct and large eddy simulation. J. Comput. Phys. 230(9), 3270–3275 (2011)Google Scholar
  23. 23.
    A.G. Lamorgese, D.A. Caughey, S.B. Pope, Direct numerical simulation of homogeneous turbulence with hyperviscosity. Phys. Fluids 17 (2005)Google Scholar
  24. 24.
    S.K. Lele, Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103, 16–42 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    M.J. Lighthill, On sound generated aerodynamically. I. general theory. Proc. Roy. Soc. Lond. A: Math. Phys. Eng. Sci. 211, 564–587 (1952)Google Scholar
  26. 26.
    K. Mahesh, A family of high-order finite difference schemes with good spectral resolution. J. Comput. Phys. 145, 332–358 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    S. Nagarajan, S.K. Lele, J.H. Ferziger, A robust high-order compact method for large eddy simulation. J. Comput. Phys. 191(2), 392–419 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    R. Pasquetti, Spectral vanishing viscosity method for large-eddy simulation of turbulent flows. J. Sci. Comput. 27, 365–375 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    S. Pirozzoli, Performance analysis and optimization of finite-difference schemes for wave propagation problems. J. Comput. Phys. 222(2), 809–831 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    T. Poinsot, D. Veynante, Theoretical and Numerical Combustion. R. T. Edwards Inc., (2005)Google Scholar
  31. 31.
    J. Pradhan, S. Jindal, B. Mahato, Y.G. Bhumkar, Joint optimization of the spatial and the temporal discretization scheme for accurate computation of acoustic problems. Commun. Comput. Phys. 24(2), 408–434 (2018)Google Scholar
  32. 32.
    J. Pradhan, B. Mahato, S.D. Dhandole, Y.G. Bhumkar, Construction, analysis and application of coupled compact difference scheme in computational acoustics and fluid flow problems. Commun. Comput. Phys. 18(4), 957–984 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  33. 33.
    M.K. Rajpoot, T.K. Sengupta, P.K. Dutt, Optimal time advancing dispersion relation preserving schemes. J. Comput. Phys. 229, 3623–3651 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    T. Rylander, P. Ingelström, A. Bondeson, Computational Electromagnetics (Springer, Berlin, 2007)Google Scholar
  35. 35.
    N.D. Sandham, Q. Li, H.C. Yee, Entropy splitting for high-order numerical simulation of compressible turbulence. J. Comput. Phys. 178(2), 307–322 (2002)zbMATHCrossRefGoogle Scholar
  36. 36.
    T.K. Sengupta, S. Bhaumik, Y.G. Bhumkar, Direct numerical simulation of two-dimensional wall-bounded turbulent flows from receptivity stage. Phys. Rev. E 85(2), 026308 (2012)Google Scholar
  37. 37.
    T.K. Sengupta, High Accuracy Computing Methods: Fluid Flows and Wave Phenomena (Cambridge University Press, USA, 2013)zbMATHCrossRefGoogle Scholar
  38. 38.
    T.K. Sengupta, Y.G. Bhumkar, New explicit two-dimensional higher order filters. Comput. Fluids 39, 1848–1863 (2010)zbMATHCrossRefGoogle Scholar
  39. 39.
    T.K. Sengupta, G. Ganeriwal, S. De, Analysis of central and upwind compact schemes. J. Comput. Phys. 192, 677–694 (2003)zbMATHCrossRefGoogle Scholar
  40. 40.
    T.K. Sengupta, S.K. Sircar, A. Dipankar, High accuracy schemes for DNS and acoustics. J. Sci. Comput. 26, 151–193 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    T.K. Sengupta, A. Dipankar, P. Sagaut, Error dynamics: Beyond von neumann analysis. J. Comput. Phys. 226, 1211–1218 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  42. 42.
    T.K. Sengupta, Y.G. Bhumkar, V. Lakshmanan, Design and analysis of a new filter for LES and DES. Comput. Struct. 87, 735–750 (2009)CrossRefGoogle Scholar
  43. 43.
    T.K. Sengupta, V. Lakshmanan, V.V.S.N. Vijay, A new combined stable and dispersion relation preserving compact scheme for non-periodic problems. J. Comput. Phys. 228, 3048–3071 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    T.K. Sengupta, V.V.S.N. Vijay, S. Bhaumik, Further improvement and analysis of CCD scheme: dissipation discretization and de-aliasing properties. J. Comput. Phys. 228(17), 6150–6168 (2009)zbMATHCrossRefGoogle Scholar
  45. 45.
    T.K. Sengupta, M.K. Rajpoot, S. Saurabh, V.V.S.N. Vijay, Analysis of anisotropy of numerical wave solutions by high accuracy finite difference methods. J. Comput. Phys. 230, 27–60 (2011)MathSciNetzbMATHCrossRefGoogle Scholar
  46. 46.
    T.K. Sengupta, Y.G. Bhumkar, M. Rajpoot, V.K. Suman, S. Saurabh, Spurious waves in discrete computation of wave phenomena and flow problems. Appl. Math. Comput. 218, 9035–9065 (2012)MathSciNetzbMATHGoogle Scholar
  47. 47.
    T.K. Sengupta, Y.G. Bhumkar, S. Sengupta, Dynamics and instability of a shielded vortex in close proximity of a wall. Comput. Fluids 70, 166–175 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  48. 48.
    P.L. Shah, J. Hardin, Second-order numerical solution of time-dependent, first-order hyperbolic equations, in ICASE/LaRC Workshop on Benchmark Problems in Computational Aeroacoustics (CAA) (1995), pp. 133–141Google Scholar
  49. 49.
    E. Tadmor, Convergence of spectral methods for nonlinear conservation laws. SIAM J. Num. Analysis 26(1), 30–44 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  50. 50.
    C.K.W. Tam, J.C. Hardin, Second computational aeroacoustics (caa) workshop on benchmark problems, in NASA Conference Publication (1997), p. 3352Google Scholar
  51. 51.
    C.K.W. Tam, K.A. Kurbatskii, J. Fang, Numerical boundary conditions for computational aeroacoustics benchmark problems, in NASA Conference Publication (1997), p. 3352Google Scholar
  52. 52.
    C.K.W. Tam, H. Shen, K.A. Kurbatskii, L. Auriault, Z. Dong, J.C. Webb, Solutions of the benchmark problems by the dispersion-relation-preserving scheme, in ICASE/LaRC Workshop on Benchmark Problems in Computational Aeroacoustics (CAA), vol. 1 (1995), pp. 149–172Google Scholar
  53. 53.
    C.K.W. Tam, Computational Aeroacoustics a Wave Number Approach (Cambridge University Press, New York, 2012)zbMATHCrossRefGoogle Scholar
  54. 54.
    C.K.W. Tam, Z. Dong, Radiation and outflow boundary conditions for direct computation of acoustic and flow disturbances in a nonuniform mean flow. J. Comput. Acoust. 4(2), 175–201 (1996)CrossRefGoogle Scholar
  55. 55.
    C.K.W. Tam, J.C. Webb, Dispersion-relation-preserving finite difference schemes for computational acoustics. J. Comput. Phys. 107, 262–281 (1993)MathSciNetzbMATHCrossRefGoogle Scholar
  56. 56.
    R. Vichnevetsky, J.B. Bowles, Fourier Analysis of Numerical Approximations of Hyperbolic Equations (SIAM Stud. Appl. Math, Philadelphia, 1982)zbMATHCrossRefGoogle Scholar
  57. 57.
    M.R. Visbal, D.V. Gaitonde, On the use of higher-order finite-difference schemes on curvilinear and deforming meshes. J. Comput. Phys. 181, 155–185 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    C.H. Yu, Y.G. Bhumkar, T.W.H. Sheu, Dispersion relation preserving combined compact difference schemes for flow problems. J. Sci. Comput. 62(2), 482–516 (2015)MathSciNetzbMATHCrossRefGoogle Scholar
  59. 59.
    Q. Zhou, Z. Yao, F. He, M.Y. Shen, A new family of high-order compact upwind difference schemes with good spectral resolution. J. Comput. Phys. 227, 1306–1339 (2007)MathSciNetzbMATHCrossRefGoogle Scholar

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© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.Department of Aerospace EngineeringIndian Institute of Technology KanpurKanpurIndia
  2. 2.School of Mechanical SciencesIndian Institute of Technology BhubaneswarBhubaneswarIndia

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