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Local Wavelet Adaptation of Cartesian Grids in Computational Fluid Dynamics

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Abstract

A method for dynamic local adaptation of graded Cartesian trees for the numerical solution of fluid dynamics problems is presented. Local wavelet analysis of a gas-dynamic field based on nonuniform B-splines is applied independently to each cell of the computational grid and makes it possible to identify nonsmooth or significantly nonlinear sections of the solution (or, vice versa, sufficiently smooth and linear ones) and modify the grid to calculate the next time step so that nonuniformly scaled flow features had adequate grid resolution. In combination with other computational fluid dynamics methods, such as the free boundary method, the presented technique allows one to effectively solve nonstationary problems involving flow around moving bodies. The operation of the proposed version of wavelet adaptation is demonstrated using a number of such problems.

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REFERENCES

  1. S. K. Godunov et al., Numerical Solution of Multidimensional Fluid Dynamics Problems (Nauka, Moscow, 1976) [in Russian].

    Google Scholar 

  2. Theoretical Foundation and Designing Computational Algorithms Ed. by K. I. Babenko (Nauka, Moscow, 1978) [in Russian].

    Google Scholar 

  3. D. P. Young, R. G. Melvin, M. B. Bieterman, F. T. Johnson, S. S. Samant, and J. E. Bussoletti, “A locally refined rectangular grid finite element method: Application to computational fluid dynamics and computational physics,” J. Comput. Phys. 92 (1), 1 (1991).

    Article  MathSciNet  Google Scholar 

  4. A. M. Khokhlov, “Fully threaded tree algorithms for adaptive refinement fluid dynamics simulations,” J. Comput. Phys. 143, 519–543 (1998).

    Article  MathSciNet  Google Scholar 

  5. M. J. Aftosmis, M. J. Berger, and J. E. Melton, “Adaptive Cartesian mesh generation,” in Handbook of Grid Generation, Chapter 22, Ed. by J. Thompson, B. Soni, and N. Weatherill (CRC, 1999).

    Google Scholar 

  6. R. B. Pember, J. B. Bell, P. Colella, W. Y. Crutchfield, and M. L. Welcome, “An adaptive Cartesian grid method for unsteady compressible flow in complex geometries,” AIAA Paper 93-3385-CP (1993).

  7. P. V. Breslavskii and V. I. Mazhukin, “A dynamic adaptation method in fluid dynamics problems,” Mat. Model. 7 (12), 48–78 (1995).

    MathSciNet  Google Scholar 

  8. P. V. Breslavskii and V. I. Mazhukin, “Dynamically adapted grids for interacting discontinuous solutions,” Comput. Math. Mat. Phys. 47 (4), 687–706 (2007).

    Article  MathSciNet  Google Scholar 

  9. A. L. Afendikov, A. T. Lutskii, and A. V. Plenkin, “Wavelet analysis of localized structures in ideal and viscid models,” Mat. Model. 23 (1), 41–50 (2011).

    Google Scholar 

  10. A. L. Afendikov et al., “Localization of discontinuites if fields of gas-dynamic functions using wavelet analysis,” Mat. Model., No. 7, 65–84 (2008).

  11. A. L. Afendikov, A. T. Lutskii, and A. V. Plenkin, “Localization of singularities of gas-dynamic fields and adaptation of computational grid to the location of discontinuities,” Mat. Model. 24 (12), 49–54 (2012).

    Google Scholar 

  12. A. Harten, “Multiresolution algorithms for the numerical solution of hyperbolic conservation laws,” Comm. Pure Appl. Math. 48 (12), 1305–1342 (1995).

    Article  MathSciNet  Google Scholar 

  13. A. Harten, “Multiresolution representation of data: A general framework,” SIAM J. Numer. Anal. 33 (3), 1205–1256 (1996).

    Article  MathSciNet  Google Scholar 

  14. G. Zumbusch, “Parallel multilevel methods. Adaptive mesh refinement and loadbalancing,” in Advanced Numerical Mathematics (Teubner, Wiesbaden, 2003).

    Google Scholar 

  15. F. Bramkamp, P. Lamby, and S. Mueller, “An adaptive multiscale finite volume solver for unsteady and steady state flow computations,” J. Comput. Phys. 197, 460–490 (2004).

    Article  MathSciNet  Google Scholar 

  16. D. Hartmann, M. Meinke, and W. Schroder, “A strictly conservative Cartesian cut-cell method for compressible viscous flows on adaptive grids,” Comput. Meth. Appl. Mech. Eng. 200, 1038–1052 (2011).

    Article  MathSciNet  Google Scholar 

  17. K. I. Babenko, Foundations of Numerical Analysis (Nauka, Moscow, 1998) [in Russian].

    Google Scholar 

  18. M. Sh. Birman and M. Z. Solomyak, “Piecewise linear approximations of functions of the classes \(W_{p}^{\alpha }\),” Mat. Sb. 73, No. 115, 331–355 (1967).

    MathSciNet  Google Scholar 

  19. Yu. V. Vasilevskii et al., Automated Techologies for Constructing Unstructured Computationl Meshes (Fizmatlit, Moscow, 2016) [in Russian].

    Google Scholar 

  20. I. S. Menshov and M. A. Kornev, “Free-boundary method for the numerical solution of gas-dynamic equations in domains with varying geometry, Math. Models Comput. Simul. 6 (6), 612–621 (2014).

    Article  MathSciNet  Google Scholar 

  21. I. S. Menshov and P. V. Pavlukhin, “Efficient parallel shock-capturing method for aerodynamics simulations on body-unfitted cartesian grids,” Comput. Math. Math. Phys. 56 (9), 1651–1664 (2016).

    Article  MathSciNet  Google Scholar 

  22. A. L. Afendikov et al., Adaptive wavelet algorithms on Cartesian grids, Preprint IPM RAN (Keldysh Inst. of Applied Mathematics, Russian Academy of Sciences, 2016).

  23. C. de Boor, “Splines as linear combinations of B-splines. A Survey,” in Approximation Theory, II Ed. by G. G. Lorentz, C. K. Chui, and L. L. Schumaker (Academic New York, 1976), pp. 1–47 (emended version, 1986).

  24. W. Sweldens, “The Lifting Scheme: A custom-design construction of biorthogonal wavelets,” Appl. Comput. Harmonic Anal. 3, 186–200 (1996).

    Article  MathSciNet  Google Scholar 

  25. J. M. Ford, I. V. Oseledets, and E. E. Tyrtyshnikov, “Matrix approximations and solvers using tensor products and non-standard wavelet transforms related to irregular grids,” Russ. J. Numer. Anal. Math. Model. 19 (2), 185–204 (2004).

    Article  MathSciNet  Google Scholar 

  26. Yu. k. Dem’yanovich, “Calibration relation for B-splines on a nonuniform grid,” Mat. Model. 13 (9), 98–100 (2001).

    MathSciNet  Google Scholar 

  27. A. L. Afendikov, A. E. Lutskii, I. S. Menshov, V. S. Nikitin, and Ya. V. Khasaeva, Numerical simulation of pellet ejection from a blunt body, Preprint IPM RAN (Keldysh Inst. of Applied Mathematics, Russian Academy of Sciences, 2017).

  28. A. L. Afendikov and V. S. Nikitin, “Numerical simulation of free motion of a system of bodies in a supersonic gas flow on adaptive grids,” Math. Models Comput. Simul. 13 (4), 667–673 (2021).

    Article  MathSciNet  Google Scholar 

  29. J. D. Regele and O. V. Vasilyev, “An adaptive wavelet-collocation method for shock computations,” Int. J. Comput. Fluid Dynam. 23 (7), 503–518 (2009).

    Article  Google Scholar 

  30. O. V. Vasilyev and C. Bowman, “Second generation wavelet collocation method for the solution of partial differential equations,” J. Comput. Phys. 165, 660–693 (2000).

    Article  MathSciNet  Google Scholar 

  31. O. V. Vasilyev, N. K. R. Kevlahan, D. E. Goldstein, A. V. Vezolainen, J. Regele, A. Nejadmalayeri, S. Reckinger, E. Brown Dymkoski, and E. Rogoz, “Adaptive wavelet environment for in silico universal multiscale modeling,” AWESUMM, 2019.

    Google Scholar 

  32. E. Brown Dymkoski, N. Kasimov, and O. V. Vasilyev, “A characteristic based volume penalization method for general evolution problems applied to compressible viscous flows,” J. Comput. Phys. 262, 344–357 (2014).

    Article  MathSciNet  Google Scholar 

  33. E. Brown Dymkoski, N. Kasimov, and O. V. Vasilyev, “Characteristic based volume penalization method for arbitrary Mach flows around solid obstacles,” in Direct and Large Eddy Simulation IX, Ed. by J. Frohlich, H. Kuerten, B. Geurts, and V. Armenio (Springer, 2015), pp. 10 –115.

    Google Scholar 

  34. O. V. Vasilyev and N. K. R. Kevlahan, “Hybrid wavelet collocation Brinkman penalization method for complex geometry flows,” Int. J. Numer. Meth. Fluid. 40, 531–538 (2002).

    Article  MathSciNet  Google Scholar 

  35. C. K. Chui, An Introduction to Wavelets (Elsevier, 2014).

    Google Scholar 

  36. Chi-Wang Shu, “High order ENO and WENO schemes for computational fluid dynamics,” in High-Order Methods for Computational Physics (Springer, Heidelberg, 1999), pp. 438–480.

    Google Scholar 

  37. V. F. Tishkin, E. E. Peskova, R. V. Zhalnin, and V. A. Goryunov, “On the Construction of WENO schemes for hyperbolic sysyems of equations on unstructured meshes,” Izv. Vyssh. Uchebn. Zaved., Ser. Fiz.-Mat. Nauki, No. 29 (2014).

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Funding

This work was supported by the Moscow Center of Fundamental and Applied Mathematics, Agreement with the Ministry of Science and Higher Education no. 075-15-2019-1623.

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  1. Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, 125047, Moscow, Russia

    A. L. Afendikov & V. S. Nikitin

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  1. A. L. Afendikov
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  2. V. S. Nikitin
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Translated by A. Klimontovich

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Afendikov, A.L., Nikitin, V.S. Local Wavelet Adaptation of Cartesian Grids in Computational Fluid Dynamics. Comput. Math. and Math. Phys. 64, 300–313 (2024). https://doi.org/10.1134/S0965542524020027

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