Skip to main content
SpringerLink
Log in

Monotonization of a Family of Implicit Schemes for the Burgers Equation

  • Conference paper
  • First Online:
Modeling, Simulation and Optimization of Complex Processes HPSC 2018

  • 361 Accesses

Abstract

We study numerical methods for convection-dominated fluid dynamics problems. In particular, we consider initial-boundary value problems for the Burgers equation with small diffusion coefficients. Our goal is to investigate several strategies, which can be used to monotonize numerical methods and to ensure non-oscillatory and positivity-preserving properties of the computed solutions. We focus on fully implicit finite-element methods constructed using the backward Euler time discretization combined with high-order spatial approximations. We experimentally study the following three monotonization approaches: mesh refinement, increasing the time-step size and utilizing higher-order finite-element approximations. Feasibility of these three strategies is demonstrated on a number of numerical examples for both one- and two-dimensional Burgers equations.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

  1. Alnæs, M., Blechta, J., Hake, J., Johansson, A., Kehlet, B., Logg, A., Richardson, C., Ring, J., Rognes, M.E., Wells, G.N.: The FEniCS project version 1.5. Arch. Numer. Softw. 3(100), 9–23 (2015)

    Google Scholar 

  2. Anderson, J.D.: Computational Fluid Dynamics: The Basics with Applications. Mechanical Engineering Series. McGraw-Hill, New York (1995)

    Google Scholar 

  3. Ascher, U.M.: Numerical Methods for Evolutionary Differential Equations. Computational Science & Engineering, vol. 5. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2008)

    Google Scholar 

  4. Cai, Q., Kollmannsberger, S., Sala-Lardies, E., Huerta, A., Rank, E.: On the natural stabilization of convection dominated problems using high order Bubnov-Galerkin finite elements. Comput. Math. Appl. 66(12), 2545–2558 (2014)

    Article  MathSciNet  Google Scholar 

  5. Donea, J., Huerta, A.: Finite Element Methods for Flow Problems. Wiley, Hoboken (2003)

    Book  Google Scholar 

  6. Ern, A., Guermond, J.L.: Theory and Practice of Finite Elements. Applied Mathematical Sciences, vol. 159. Springer, New York (2004)

    Google Scholar 

  7. Godlewski, E., Raviart, P.A.: Numerical Approximation of Hyperbolic Systems of Conservation Laws. Applied Mathematical Sciences, vol. 118. Springer, New York (1996)

    Google Scholar 

  8. Guermond, J.L., Pasquetti, R., Popov, B.: Entropy viscosity method for nonlinear conservation laws. J. Comput. Phys. 230(11), 4248–4267 (2011)

    Article  MathSciNet  Google Scholar 

  9. Hartmann, R., Houston, P.: Adaptive discontinuous Galerkin finite element methods for nonlinear hyperbolic conservation laws. SIAM J. Sci. Comput. 24(3), 979–1004 (2002)

    Article  MathSciNet  Google Scholar 

  10. Hesthaven, J.S.: Numerical Methods for Conservation Laws: From Analysis to Algorithms. Computational Science & Engineering, vol. 18. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2018)

    Google Scholar 

  11. van der Houwen, P.J., Sommeijer, B.P., Kok, J.: The iterative solution of fully implicit discretizations of three-dimensional transport models. Appl. Numer. Math. 25(2–3), 243–256 (1997). Special issue on time integration (Amsterdam, 1996)

    Google Scholar 

  12. Hundsdorfer, W., Verwer, J.: Numerical Solution of Time-dependent Advection-Diffusion-Reaction Equations. Springer Series in Computational Mathematics, vol. 33. Springer, Berlin (2003)

    Google Scholar 

  13. Kröner, D.: Numerical Schemes for Conservation Laws. Wiley-Teubner Series Advances in Numerical Mathematics. Wiley, Chichester (1997)

    MATH  Google Scholar 

  14. Kulikovskii, A.G., Pogorelov, N.V., Semenov, A.Y.: Mathematical Aspects of Numerical Solution of Hyperbolic Systems. Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, vol. 118. Chapman & Hall/CRC, Boca Raton (2001)

    Google Scholar 

  15. Kurganov, A., Liu, Y.: New adaptive artificial viscosity method for hyperbolic systems of conservation laws. J. Comput. Phys. 231, 8114–8132 (2012)

    Article  MathSciNet  Google Scholar 

  16. Kurokawa, M., Uchiyama, Y., Nagai, S.: Tribological properties and gear performance of polyoxymethylene composites. J. Tribol. 122, 809–814 (2000)

    Article  Google Scholar 

  17. Kuzmin, D.: A Guide to Numerical Methods for Transport Equations. University Erlangen-Nuremberg (2010)

    Google Scholar 

  18. Larson, M.G., Bengzon, F.: The Finite Element Method: Theory, Implementation, and Applications. Texts in Computational Science and Engineering, vol. 10. Springer, Heidelberg (2013)

    Google Scholar 

  19. LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  20. LeVeque, R.J.: Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2007)

    Book  Google Scholar 

  21. Logg, A., Mardal, K.A., Wells, G.: Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book, vol. 84. Springer Science & Business Media, Berlin (2012)

    Book  Google Scholar 

  22. Mao, K., Greenwood, D., Ramakrishnan, R., Goodship, V., Shrouti, C., Chetwynd, D., Langlois, P.: The wear resistance improvement of fibre reinforced polymer composite gears. Wear 426–427, 1033–1039 (2019)

    Article  Google Scholar 

  23. Morton, K.W.: Numerical Solution of Convection-Diffusion Problems. Applied Mathematics and Mathematical Computation, vol. 12. Chapman & Hall, London (1996)

    Google Scholar 

  24. Ortega, J.M., Rheinboldt, W.C.: Iterative Solution of Nonlinear Equations in Several Variables. Classics in Applied Mathematics, vol. 30. Society for Industrial and Applied Mathematics (SIAM), Philadelphia (2000). Reprint of the 1970 original

    Google Scholar 

  25. Samarskii, A.A.: The Theory of Difference Schemes. Monographs and Textbooks in Pure and Applied Mathematics, vol. 240. Marcel Dekker Inc, New York (2001)

    Google Scholar 

  26. Samarskii, A.A., Gulin, A.V.: Ustoĭchivost’ Raznostnykh Skhem (Russian) [Stability of Difference Schemes], 2nd edn. Editorial URSS, Moscow (2004)

    Google Scholar 

  27. Samarskii, A.A., Vabishchevich, P.N.: Numerical Methods for Solving Convection-Diffusion. URSS, Moscow (1999)

    Google Scholar 

  28. Senthilvelan, S., Gnanamoorthy, R.: Effect of gear tooth fillet radius on the performance of injection molded nylon 6/6 gears. Mater. Des. 27, 632–639 (2006)

    Article  Google Scholar 

  29. Thomée, V.: Galerkin Finite Element Methods for Parabolic Problems. Springer Series in Computational Mathematics, vol. 25, 2nd edn. Springer, Berlin (2006)

    Google Scholar 

  30. Vabishchevich, P.N.: Decoupling schemes for predicting compressible fluid flows. Comput. Fluids 171, 94–102 (2018)

    Article  MathSciNet  Google Scholar 

  31. Von Neumann, J., Richtmyer, R.D.: A method for the numerical calculation of hydrodynamic shocks. J. Appl. Phys. 21, 232–237 (1950)

    Article  MathSciNet  Google Scholar 

  32. Wesseling, P.: Principles of Computational Fluid Dynamics. Springer Series in Computational Mathematics, vol. 29. Springer, Berlin (2001)

    Google Scholar 

  33. Zienkiewicz, O.C., Taylor, R.L., Nithiarasu, P.: The Finite Element Method for Fluid Dynamics, 7th edn. Elsevier/Butterworth Heinemann, Amsterdam (2014)

    MATH  Google Scholar 

Download references

Acknowledgements

The work of A. Kurganov was supported in part by NSFC grant 11771201. The work of P. N. Vabishchevich was supported in part by Russian Federation Government mega-grant 14.Y26.31.0013. The authors would like to thank anonymous reviewers for their valuable suggestions and corrections.

Author information

Authors and Affiliations

  1. Department of Mathematics, Southern University of Science and Technology, Shenzhen, 518055, China

    Alexander Kurganov

  2. Nuclear Safety Institute of the Russian Academy of Sciences, 115191, Moscow, Russia

    Petr N. Vabishchevich

  3. North-Eastern Federal University, 677000, Yakutsk, Russia

    Petr N. Vabishchevich

Authors
  1. Alexander Kurganov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Petr N. Vabishchevich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Kurganov .

Editor information

Editors and Affiliations

  1. Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, Heidelberg, Germany

    Hans Georg Bock

  2. Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, Heidelberg, Germany

    Willi Jäger

  3. Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, Heidelberg, Germany

    Ekaterina Kostina

  4. Institute of Mathematics, Vietnam Academy of Science and Technology, Hanoi, Vietnam

    Hoang Xuan Phu

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Kurganov, A., Vabishchevich, P.N. (2021). Monotonization of a Family of Implicit Schemes for the Burgers Equation. In: Bock, H.G., Jäger, W., Kostina, E., Phu, H.X. (eds) Modeling, Simulation and Optimization of Complex Processes HPSC 2018. Springer, Cham. https://doi.org/10.1007/978-3-030-55240-4_12

Download citation

Publish with us

Policies and ethics

Search

Navigation