Skip to main content
SpringerLink
Log in
Book cover

Innovative Algorithms and Analysis pp 111–140Cite as

A Numerical Glimpse at Some Non-standard Solutions to Compressible Euler Equations

  • Chapter
  • First Online:

  • 1059 Accesses

Part of the book series: Springer INdAM Series ((SINDAMS,volume 16))

Abstract

In this note we review and recast some recent results on the existence of non-standard solutions to the compressible Euler equations as to make possible a preliminary numerical investigation. In particular, we are interested in studying numerically the forward in time evolution of some Lipschitz initial data which allow for non-standard solutions (“colliding data”). Numerical results indicate appearance of oscillations after the first break-up time along with a qualitative behavior seemingly compatible with relevant properties of non-standard solutions.

This is a preview of subscription content, 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   89.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   119.99
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

Learn about institutional subscriptions

Notes

  1. 1.

    https://www.symbolab.com/solver/polynomial-equation-calculator/.

References

  1. Acharya, A., Chen, G.Q., Slemrod, M., Wang, D.: Fluids, Elasticity, Geometry, and the Existence of Wrinkled Solutions (2016). arXiv: 1605.03058

    Google Scholar 

  2. Amadori, D., Gosse, L.: Error Estimates for Well-Balanced Schemes on Simple Balance Laws: One-Dimensional Position-Dependent Models. BCAM SpringerBriefs in Mathematics. Springer, New York (2015).

    Book  MATH  Google Scholar 

  3. Aregba-Driollet, D., Natalini, R.: Discrete kinetic schemes for multi-dimensional systems of conservation laws. SIAM J. Numer. Anal. 37, 1973–2004 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bianchini, S., Bressan, A.: Vanishing viscosity solutions of nonlinear hyperbolic systems. Ann. Math. 161, 223–342 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  5. Blondin, J.M., Marks, B.S.: Evolution of cold shock-bounded slabs. New Astronomy 1, 235–244 (1996)

    Article  Google Scholar 

  6. Brenier, Y., Corrias, L.: A kinetic formulation for multi-branch entropy solutions of scalar conservation laws. Ann. Inst. Henri Poincare Nonlinear Anal. 15, 169–190 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Chiodaroli, E.: A counterexample to well-posedness of entropy solutions to the compressible Euler system. J. Hyperbolic Differ. Equ. 11, 493–519 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  8. Chiodaroli, E., Kreml, O.: On the energy dissipation rate of solutions to the compressible isentropic Euler system. Arch. Ration. Mech. Anal. 214, 1019–1049 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chiodaroli, E., De Lellis, C., Kreml, O.: Global ill-posedness of the isentropic system of gas dynamics. Commun. Pure Appl. Math. 68, 1157–1190 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chiodaroli, E., Feireisl, E., Kreml, O.: On the weak solutions to the equations of a compressible heat conducting gas. Ann. Inst. H. Poincaré Anal. Non Linéaire 32, 225–243 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chiodaroli, E., Feireisl, E., Kreml, O., Wiedemann, E.: \(\mathcal{A}\)-free rigidity and applications to the compressible Euler system (2015). arXiv:1511.03114. https://arxiv.org/abs/1511.03114

  12. Colella, P.: Glimm’s method for gas dynamics. SIAM J. Sci. Stat. Comput. 3, 76–110 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  13. Crandall, M., Majda, A.: The method of fractional steps for conservation laws. Numer. Math. 34, 285 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  14. Dafermos, C.M.: Hyperbolic conservation laws in continuum physics. Grundleheren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 325, 3rd edn. Springer, Berlin (2010)

    Google Scholar 

  15. Davis, S.: A rotationally biased upwind difference scheme for the Euler equations. J. Comput. Phys. 56, 65–92 (1984)

    Article  MATH  Google Scholar 

  16. De Lellis, C., Székelyhidi, L.J.: The Euler equations as a differential inclusion. Ann. Math. 170, 1417–1436 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. De Lellis, C., Székelyhidi, L.J.: On admissibility criteria for weak solutions of the Euler equations. Arch. Ration. Mech. Anal. 195, 225–260 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  18. DiPerna, R.: Global solutions to a class of nonlinear hyperbolic systems of equations. Commun. Pure Appl. Math. 26, 1–28 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  19. Elling, V.: A possible counterexample to well posedness of entropy solutions and to Godunov scheme convergence. Math. Comput. 75, 1721–1733 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  20. Elling, V.: The carbuncle phenomenon is incurable. Acta Math. Sci. 29B (6), 1647–1656 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Elling, V.: Relative entropy and compressible potential flow. Acta Math. Sci. 35B (4), 763–776 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  22. Engquist, B., Sjögreen, B.: The convergence rate of finite difference schemes in the presence of shocks. SIAM J. Numer. Anal. 35, 2464–2485 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  23. Fjordholm, U.S., Mishra, S.: Vorticity preserving finite volume schemes for the shallow water equations. SIAM J. Sci. Comput. 33, 588–611 (2011).

    Article  MathSciNet  MATH  Google Scholar 

  24. Fjordholm, U.S., Mishra S., Tadmor, E.: On the computation of measure-valued solutions. Acta Numer. 25 567–679 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  25. Gosse, L.: Using K-branch entropy solutions for multivalued geometric optics computations. J. Comput. Phys. 180, 155–182 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  26. Gosse, L.: Computing Qualitatively Correct Approximations of Balance Laws. Exponential-Fit, Well-Balanced and Asymptotic-Preserving. SIMAI Springer Series. Springer, Milan (2013). ISBN: 978-88-470-2891-3

    Book  MATH  Google Scholar 

  27. Holden, H., Karlsen, K.H., Lie, K.-A., Risebro, N.H.: Splitting Methods for Partial Differential Equations with Rough Solutions. Analysis and MATLAB Programs. Series of Lectures in Mathematics. European Mathematical Society (EMS), Zürich (2010)

    Book  MATH  Google Scholar 

  28. Jeltsch, R., Torrilhon, M.: On curl-preserving finite volume discretizations for shallow water equations. BIT Numer. Math. 46, S35–S53 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  29. Larroutourou, B.: How to preserve the mass fractions positivity when computing multi-component flows. J. Comput. Phys. 95, 59–84 (1991)

    Article  MathSciNet  Google Scholar 

  30. LeVeque, R.J.: Nonlinear conservation laws and finite-volume methods. In: Computational Methods for Astrophysical Fluid Flow. The series Saas-Fee Advanced Courses, vol. 27, pp. 1–159. Springer (1998)

    Google Scholar 

  31. Morton, K.W., Roe, P.L.: Vorticity-preserving lax-Wendroff type schemes for the system wave equation. SIAM J. Sci. Comput. 23, 170–192 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  32. Roe, P.L.: Discrete models for the numerical analysis of time- dependent multidimensional gas dynamics. J. Comput. Phys. 63, 458–476 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  33. Roe, P.L.: Discontinuous solutions to hyperbolic systems under operator splitting. Numer. Methods Partial Differ. Equ. 7, 277–297 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  34. Roe, P.L.: Beyond Riemann problem. In: Algorithmic Trends in Computational Fluid Dynamics. Part of the Series ICASE/NASA LaRC Series, pp. 341–367. Springer, New York (1993)

    Google Scholar 

  35. Wang, Y., Li, J.: Numerical defects of the HLL scheme and dissipation matrices for the Euler equations. SIAM J. Numer. Anal. 52, 207–219 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  36. Warming, R.F., Hyett, B.J.: The modified equation approach to the stability and accuracy analysis of finite difference methods. J. Comput. Phys. 14, 159–179 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  37. Xu, K., Hu, J.: Projection dynamics in Godunov-type schemes. J. Comput. Phys. 42, 412–427 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  38. Zheng, Y.: Systems of Conservation Laws: Two-Dimensional Riemann Problems. Progress in Nonlinear Differential Equations and Their Applications, vol. 38. Birkhauser, Basel (2001)

    Google Scholar 

Download references

Acknowledgements

Both the authors are happy to thank Denise Aregba-Driollet (Bordeaux) and Roger Käppeli (Zürich) who gently accepted to perform fine-grid computations, partly reported in §6, by using (formally) second-order extensions of 2D BGK and HLL schemes. Eleuterio Toro (Trento) suggested to set up the averaging process reported in Remark 1.

Author information

Authors and Affiliations

  1. EPFL Lausanne, Station 8, 1015, Lausanne, Switzerland

    Elisabetta Chiodaroli

  2. IAC, CNR, via dei Taurini 19, 00185, Roma, Italy

    Laurent Gosse

Authors
  1. Elisabetta Chiodaroli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Laurent Gosse
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Elisabetta Chiodaroli .

Editor information

Editors and Affiliations

  1. Istituto per le Applicazioni del Calcolo CNR, Rome, Italy

    Laurent Gosse

  2. Istituto per le Applicazioni del Calcolo CNR, Rome, Italy

    Roberto Natalini

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Chiodaroli, E., Gosse, L. (2017). A Numerical Glimpse at Some Non-standard Solutions to Compressible Euler Equations. In: Gosse, L., Natalini, R. (eds) Innovative Algorithms and Analysis. Springer INdAM Series, vol 16. Springer, Cham. https://doi.org/10.1007/978-3-319-49262-9_4

Download citation

Publish with us

Policies and ethics

Search

Navigation