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On the Filtering of Spurious Oscillations in the Numerical Simulations of Convection Dominated Problems

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Abstract

This paper presents the numerical filtering as a valuable tool used for the removal of spurious oscillations in the numerical solution of some specific fluids flows. The qualitative results based on the application of two distinct representative filters are discussed. The compact low-pass filters and the digital Total Variation Diminishing (TVD) filters are described as examples of the filtering approach. Their use is shown on selected one-dimensional advection problems, demonstrating the essential behavior of the filtered solution. Finally an example of a more realistic biomedical problem solved using finite-volume scheme is discussed including the effect of the digital filter. The conclusions sum up some of the experience from the application of both filtering techniques.

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Notes

  1. Sometimes called spectral function or spectral response (see e.g. [21] or [16] for more details)

References

  1. Bodnár, T.: On the use of non-linear TVD filters in finite-volume simulations. In: Algoritmy 2012 Proceedings of Contributed Papers and Posters, pp. 190–199. Slovak University of Technology, Faculty of Civil Engineering, Bratislava (2012)

  2. Bodnár, T., Beneš, L.: On some high resolution schemes for stably stratified fluid flows. In: Fořt, J. et al. (eds.) Finite Volumes for Complex Applications VI Problems & Perspectives. Springer Proceedings in Mathematics, vol. 4, pp. 145–153. Springer, Berlin (2011)

  3. Bodnár, T., Beneš, L., Fraunié, Ph., Kozel, K.: Application of compact finite-difference schemes to simulations of stably stratified fluid flows. Appl. Math. Comput. 219, 3336–3353 (2012)

    MathSciNet  MATH  Google Scholar 

  4. Bodnár, T., Fasano, A., Sequeira, A.: Mathematical models for blood coagulation. In: Bodnár, T., Galdi, G. P., Nečasová, Š. (eds.) Fluid-Structure Interaction and Biomedical Applications. Advances in Mathematical Fluid, pp. 483–569. Birkhäuser, Basel (2014)

    Google Scholar 

  5. Bodnár, T., Fraunié, Ph.: Numerical simulation of the wake structure behind three-dimensional hill under stable stratification. In: Šimurda, D., Bodnár, T (eds.) Proceedings Topical Problems of Fluid Mechanics 2016, pp. 17–22. Institute of Thermomechanics CAS, Prague (2016)

  6. Bodnár, T., Fraunié, Ph., Řezníček, H.: Numerical tests of far-field boundary conditions for stably stratified stratified flows. In: Šimurda, D., Bodnár, T (eds.) Proceedings Topical Problems of Fluid Mechanics 2019, pp. 17–24. Institute of Thermomechanics CAS, Prague (2019)

  7. Bodnár, T., Příhoda, J.: Numerical simulation of turbulent free-surface flow in curved channel. Flow. Turbul. Combust. 76, 429–442 (2006)

    Article  Google Scholar 

  8. Bodnár, T., Sequeira, A.: Numerical simulation of the coagulation dynamics of blood. Comput. Math. Methods Med. 9, 83–104 (2008)

    Article  MathSciNet  Google Scholar 

  9. Bürgel, A., Sonar, T.: Discrete filtering of numerical solutions to hyperbolic conservation laws. Int. J. Numer. Methods Fluids 40, 263–271 (2002)

    Article  MathSciNet  Google Scholar 

  10. Engquist, B., Lötstedt, P., Sjögreen, B.: Nonlinear filters for efficient shock computation. Math. Comput. 52, 509–537 (1989)

    Article  MathSciNet  Google Scholar 

  11. Gaitonde, D.V., Shang, J.S., Young, J.L.: Practical aspects of higher-order numerical schemes for wave propagation phenomena. Int. J. Numer. Methods Eng. 45, 1849–1869 (1999)

    Article  Google Scholar 

  12. Lafon, F., Osher, S.: High order filtering methods for approximating hyperbolic systems of conservation laws. Tech. Rep. ICASE Report No. 90-25, Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Hampton, Virinia, 23665-5225 (1990)

  13. Lele, S.K.: Compact finite difference schemes with spectral-like resolution. J. Comput. Phys. 103, 16–42 (1992)

    Article  MathSciNet  Google Scholar 

  14. Lo, S.C., Blaisdell, G.A., Lyrintzis, A.S.: High-order shock capturing schemes for turbulence calculations. In: 45th AIAA Aerospace Sciences Meeting and Exhibit, January 8–11, 2007, Reno, Nevada, USA, AIAA Paper 2007-827. American Institute of Aeronautics and Astronautics (2007)

  15. Ortleb, S., Meister, A., Sonar, T.: Adaptive spectral filtering and digital total variation postprocessing for the DG method on triangular grids: Application to the Euler equations. In: Hesthaven, J.S., Rønquist, E.M. (eds.) Spectral and High Order Methods for Partial Differential Equations. Lecture Notes in Computational Science and Engineering, vol. 76, pp. 469–477. Springer, Berlin (2011)

    MATH  Google Scholar 

  16. Sengupta, T.K.: High Accuracy Computing Methods: Fluid Flows and Wave Phenomena. Cambridge University Press, Cambridge (2013)

    Book  Google Scholar 

  17. Sequeira, A., Bodnár, T.: Blood coagulation simulations using a viscoelastic model. Math. Model. Nat. Phenom. 9, 34–45 (2014)

    Article  MathSciNet  Google Scholar 

  18. Sequeira, A., Santos, R.F., Bodnár, T.: Blood coagulation dynamics: mathematical modeling and stability results. Math. Biosci. Eng. 8, 425–443 (2011)

    Article  MathSciNet  Google Scholar 

  19. Shyy, W., Chen, M.-H., Mittal, R., Udaykumar, H.S.: On the suppression of numerical oscillations using a non-linear filter. J. Comput. Phys. 102, 49–62 (1992)

    Article  Google Scholar 

  20. Tadmor, E.: Approximate solutions of nonlinear conservation laws. In: Cockburn, B. et al. (eds.) Advanced Numerical Approximation of Nonlinear Hyperbolic Equations. Lecture Notes in Mathematics, vol. 1697, pp. 1–149. Springer, Berlin Heidelberg (1998)

    Google Scholar 

  21. Visbal, M.R., Gaitonde, D.V.: On the use of higher-order finite-difference schemes on curvilinear and deforming meshes. J. Comput. Phys. 181, 155–185 (2002)

    Article  MathSciNet  Google Scholar 

  22. Yee, H.C., Sandham, N.D., Djomehri, M.J.: Low-dissipative high-order shock-capturing methods using characteristic-based filters. J. Comput. Phys. 150, 199–238 (1999)

    Article  MathSciNet  Google Scholar 

  23. Yost, S., Rao, P.: A non-linear filter for one- and two-dimensional open channel flows with shocks. Adv. Water Resour. 24, 187–193 (2000)

    Article  Google Scholar 

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Acknowledgements

The work of A. Sequeira has been partially supported by Fundação para a Ciência e a Tecnologia - FCT (Portugal) through the Projects UID/Multi/04621/2013 and UID/Multi/04621/2019 of CEMAT Center for Computational and Stochastic Mathematics (University of Lisbon). The financial support of T. Bodnár for the present work was partly provided by the Czech Science Foundation under the grant No. P201-16-03230S and partly by the European Regional Development Fund-Project “Center for Advanced Applied Science” No.CZ.02.1.01/0.0/0.0/16_019/0000778.

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Authors and Affiliations

  1. Department of Mathematics and Center for Computational and Stochastic Mathematics - CEMAT Instituto Superior Técnico, University of Lisbon, Av. Rovisco Pais, 1049-001, Lisboa, Portugal

    Adélia Sequeira

  2. Institute of Mathematics, Czech Academy of Sciences, žitná 25, 115 67, Prague 1, Czech Republic

    Tomáš Bodnár

  3. Department of Technical Mathematics, Faculty of Mechanical Engineering, Czech Technical University in Prague, Karlovo Náměstí 13, 121 35, Prague 2, Czech Republic

    Tomáš Bodnár

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  1. Adélia Sequeira
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  2. Tomáš Bodnár
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Correspondence to Adélia Sequeira.

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Sequeira, A., Bodnár, T. On the Filtering of Spurious Oscillations in the Numerical Simulations of Convection Dominated Problems. Vietnam J. Math. 47, 851–864 (2019). https://doi.org/10.1007/s10013-019-00369-z

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