Multioperators Technique for Constructing Arbitrary High-Order Approximations and Schemes: Main Ideas and Applications to Fluid Dynamics Equations

  • Andrei I. TolstykhEmail author
Conference paper
Part of the Smart Innovation, Systems and Technologies book series (SIST, volume 133)


The main ideas of multioperators as tools for constructing arbitrary high-order approximations and schemes are outlined. Examples of multioperators-based numerical analysis formulas are presented. Schemes for fluid dynamics with multioperators approximations up to 36th-order are described. Numerical results of high fidelity long term calculations for instability and sound radiation problems are presented. The extension to strongly discontinuous solutions is briefly outlined.


Multioperators Multioperators-based formulas Schemes for fluid dynamics Euler and Navier-Stokes equations Instability and sound radiation Discontinuous solutions 


  1. 1.
    Tolstykh, A.I.: Multioperator high-order compact upwind methods for CFD parallel calculations. In: Emerson, D., Fox, P., Satofuka, N. (eds.) Parallel Computational Fluid Dynamics: Recent Developments and Advances Using Parallel Computing, Elsevier, pp. 383–390 (1998)Google Scholar
  2. 2.
    Tolstykh, A.I.: Development of arbitrary-order multioperators-based schemes for parallel calculations. Part 1. Higher-than-fifth order approximations to convection terms. J. Comput. Phys. 225(2), 2333–2353 (2007)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Tolstykh, A.I.: On the use of multioperators in the construction of high-order grid approximations. Comput. Math. Math. Phys. 56(6), 932–946 (2016)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Tolstykh, A.I.: High accuracy compact and multioperators approximations for partial differential equations. Nauka, Moscow (2015). (in Russian)Google Scholar
  5. 5.
    Tolstykh, A.I.: High accuracy non-centered compact difference schemes for fluid dynamics applications. World Scientific, Singapore (1994)CrossRefGoogle Scholar
  6. 6.
    Adams, N.A., Shariff, K.: A high resolution compact-ENO schemes for shock turbulence interaction problems. J. Comput. Phys. 127(1), 2–51 (1996)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Shu, C.-W.: Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws: NASA/CR-97-206253, ICASE Rept No. 97-65. Hampton: Langley Res. Center (1997)Google Scholar
  8. 8.
    Tam, C.K.W.: Problem 1-aliasing. 4th Computational Aeroacoustics (CAA) Workshop on benchmark problems, NASA/CP-2004–2159 (2004)Google Scholar
  9. 9.
    Tolstykh, A.I., Lipavskii, M.V.: Instability and acoustic fields of Rankine vortex as seen from long-term calculations with the tenth-order multioperators-based scheme. Math. Comput. Simul. 147, 301–320 (2018)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Tolstykh, A.I., Shirobokov, D.A.: Fast calculations of screech using highly accurate multioperators-based schemes. J. Appl. Acoustics. 74(1), 102–109 (2013)CrossRefGoogle Scholar
  11. 11.
    Tolstykh, A.I.: On 16th and 32th order multioperators-based schemes for smooth and discontinuous fluid dynamics solutions. Commun. in Comput. Phys. 22(2), 572–598 (2017)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Woodward, P., Colella, P.: The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54(1), 115–173 (1984)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Federal Research Center “Computer Science and Control” of the RASMoscowRussian Federation

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