High-resolution, high-order, finite-difference algorithms for computational aeroacoustics

  • John W. Goodrich
Algorithms Euler
Part of the Lecture Notes in Physics book series (LNP, volume 490)


Computational Aeroacoustics requires accurate wave propagation over long distances for a wide range of frequencies ([7],[12]). This requirement places a severe demand on numerical algorithms, creating a need for methods with high accuracy and high resolution ([11],[13]). This paper briefly presents high order finite difference methods for the Linearized Euler Equations in both one and two space dimensions. The methods may all be represented as single step explicit algorithms on central stencils. These methods all correctly incorporate the proper wave dynamics of linear constant coefficient hyperbolic systems in one or multiple space dimensions, including propagation along characteristics surfaces for nondiagonalizable systems in two space dimensions. These methods all have the same order of accuracy in time as in space, and they all are stable if the time to space step size ratio is less than one over one plus the maximum mean convection velocity in any single coordinate direction. Two classes of methods will be presented and compared: the first has dispersive truncation error with up to eighth order accuracy in both space and time [3]; the second has diffusive error with up to eleventh order accuracy in both space and time [4].


Maximum Absolute Error Artificial Boundary Condition Point Stencil Linearize Euler Equation High Resolution Method 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    L. Collatz, The Numerical Treatment of Differential Equations, (Springer-Verlag, New York, 1966).Google Scholar
  2. [2]
    P. Garabedian, Partial Differential Equations, (Wiley, New York, 1964).zbMATHGoogle Scholar
  3. [3]
    J. W. Goodrich, “An Approach to the Development of Numerical Algorithms for first Order Linear Hyperbolic Systems in Multiple Space Dimensions: The Constant Coefficient Case,” NASA TM 106928 (September, 1995), and submitted.Google Scholar
  4. [4]
    J. W. Goodrich, in preparation.Google Scholar
  5. [5]
    J. W. Goodrich, and T. Hagstrom, “Accurate Algorithms and Radiation Boundary Conditions for Linearized Euler Equations,” AIAA Paper 96-1660, (1996).Google Scholar
  6. [6]
    T. Hagstrom, “On High Order Radiation Boundary Conditions,” to appear in the proceedings of The IMA Workshop on Computational Wave Propagation, (Minneapolis, Minn., September, 1994).Google Scholar
  7. [7]
    J. Hardin, “Numerical Considerations For Computational Aeroacoustics,” to appear in “Computational Aeroacoustics,” edited by J. Hardin and J. Hussaini, (Springer Verlag, New York, 1993).Google Scholar
  8. [8]
    A. Harten, B. Engquist, S. Osher and S. R. Chakravarthy, “Uniformly High Order Accurate Essentially Non-oscillatory Schemes, III,” J. Comput. Phys., 71, 231 (1987).zbMATHCrossRefMathSciNetADSGoogle Scholar
  9. [9]
    H. O. Kreiss and J. Oliger, “Comparison of accurate methods for the integration of hyperbolic equations,” Tellus, 24, 199 (1972).ADSMathSciNetCrossRefGoogle Scholar
  10. [10]
    P. D. Lax and B. Wendroff, “Systems of Conservation Laws,” Comm. Pure Appl. Math, 13, 217 (1960).zbMATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    S. K. Lele, “Compact Finite Difference Schemes with Spectral like Resolution”, J. Comput. Phys. 103, pp16–42 (1992).zbMATHCrossRefMathSciNetADSGoogle Scholar
  12. [12]
    J. Lighthill, “Report On The Final Panel Discussion On Computational Aeroacoustics,” NASA Contractor Report 189718, ICASE Report No. 92-53 (October 1992).Google Scholar
  13. [13]
    C. K. W. Tam and J. C. Webb, “Dispersion Relation Preserving Finite Difference Schemes For Computational Acoustics,” J. Comput. Phys. 107, pp262–281 (1993).zbMATHCrossRefADSMathSciNetGoogle Scholar
  14. [14]
    T. Yabe, Y. Kadota, F. Ikeda, “A Multidimensional cubic interpolated pseudoparticle (CIP) method without time splitting technique for hyperbolic equations,” J. Phys. Soc. Japan, 59, 2301 (1990).CrossRefMathSciNetADSGoogle Scholar

Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • John W. Goodrich
    • 1
  1. 1.NASA Lewis Research CenterClevelandUSA

Personalised recommendations