Skip to main content
Springer Nature Link
Log in

Entropy Production by Explicit Runge–Kutta Schemes

  • Published:
Journal of Scientific Computing Aims and scope Submit manuscript

Abstract

We propose and test a formula for the separate computation of the temporal and spatial entropy production of fully discrete, finite-volume, explicit Runge–Kutta discretizations of systems of conservation laws.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

Notes

  1. Physical entropy is a concave function of the state variables, but it is customary to set \(\eta \) equal to a negative function of the entropy, which causes the inequality to go in the opposite sense.

  2. SSP \(\equiv \) Strong Stability Preserving schemes [29] are RK or multistep schemes that can be written as convex combinations of Forward-Euler schemes.

  3. we are tacitly assuming here that we have a steady problem for which a terminal (steady) state is eventually attained, at which point \(U^{n+1}=U^{n}\) and, thus, \(\eta _i^{n+1} -\eta _i^n =0\) for all i.

  4. we assume that \(a_{21} a_{32} b_3 \ne 0\), which is automatically verified for \(3^{\mathrm{rd}}\)-order RK-3 schemes.

  5. For n real numbers, Jensen’s inequality yields

    $$\begin{aligned} (x_1 +\cdots +x_n )^{2}=\left( \frac{1}{n}nx_1 +\cdots +\frac{1}{n}nx_n \right) ^{2}\le \frac{1}{n}(nx_1 )^{2}+\cdots +\frac{1}{n}(nx_n )^{2}=n\left( {x_1 ^{2}+\cdots +x_n ^{2}} \right) \end{aligned}$$

    Hence, for the squared norm of the sum of \(n\,p\)-dimensional vectors we have

    $$\begin{aligned} \left( {\sum _{i=1}^n {\vec {v}_i } } \right) ^{2}= & {} \left( v_1^1 +\cdots +v_n^1\right) ^{2}+\cdots +(v_1^p +\cdots +v_n^p )^{2}\le n\left( {(v_1^1 )^{2}+\cdots +(v_n^1 )^{2}} \right) \\&+\cdots +n\left( {(v_1^p )^{2}+\cdots +(v_n^p )^{2}} \right) \\= & {} n\left( {(v_1^1 )^{2}+\cdots +(v_1^p )^{2}} \right) +\cdots +n\left( {(v_n^1 )^{2}+\cdots +(v_n^p )^{2}} \right) =n\sum _{i=1}^n {\vec {v}_i ^{2}}\end{aligned}$$

  6. An entropy conservative scheme is one for which the numerical flux verifies \(\Delta \hbox {v}_{i+\frac{1}{2}}^T \tilde{F}_{i+\frac{1}{2}} =\Delta \Theta _{i+\frac{1}{2}}\) and thus the entropy production \(\Pi _{i+\frac{1}{2}} =0\). In the scalar case, the entropy conservative flux is unique for each choice of entropy function and can be computed as \(\tilde{F}_{i+\frac{1}{2}} =\Delta \Theta _{i+\frac{1}{2}} /\Delta \hbox {v}_{i+\frac{1}{2}}\).

References

  1. Osher, S.: Riemann solvers, the entropy condition, and difference approximations. SIAM J. Numer. Anal. 21(2), 217–235 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  2. Harten, A.: On the symmetric form of systems of conservation laws with entropy. J. Comput. Phys. 49(1), 151–164 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  3. Tadmor, E.: The numerical viscosity of entropy stable schemes for systems of conservation laws. I. Math. Comput. 49(179), 91–103 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  4. Puppo, G., Semplice, M.: Entropy and the numerical integration of conservation laws. In: Proceedings of the 2nd Annual Meeting of the Lebanese Society for Mathematical Sciences (LSMS-2011) (2011)

  5. Puppo, G.: Numerical entropy production for central schemes. SIAM J. Sci. Comput. 25(4), 1382–1415 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  6. Puppo, G., Semplice, M.: Numerical entropy and adaptivity for finite volume schemes. Commun. Comput. Phys. 10(5), 1132–1160 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Golay, F.: Numerical entropy production and error indicator for compressible flows. C. R. Mecanique 337, 233–237 (2009)

    Article  MATH  Google Scholar 

  8. Altazin, T., Ersoy, M., Golay, F., Sous, D., Yushchenko, L.: Numerical investigation of BB-AMR scheme using entropy production as refinement criterion. Int. J. Comput. Fluid Dyn. 30(3), 256–271 (2016)

    Article  MathSciNet  Google Scholar 

  9. Jameson, A.: The construction of discretely conservative finite volume schemes that also globally conserve energy or entropy. J. Sci. Comput. 34, 152–187 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  10. Jameson, A.: Formulation of kinetic energy preserving conservative schemes for gas dynamics and direct numerical simulation of one-dimensional viscous compressible flow in a shock tube using entropy and kinetic energy preserving schemes. J. Sci. Comput. 34, 188–208 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ismail, F., Roe, P.L.: Affordable, entropy-consistent Euler flux functions II: entropy production at shocks. J. Comput. Phys. 228(15), 5410–5436 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lax, P.: Shock waves and entropy. In: Zarantonello, E. (ed.) Contributions to Nonlinear Functional Analysis, pp. 603–634. Academic Press, New York (1971)

    Chapter  Google Scholar 

  13. Harten, A., Hyman, J.M., Lax, P.D., Keyfitz, B.: On finite-difference approximations and entropy conditions for shocks. Commun. Pure Appl. Math. 29, 297–322 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  14. Tadmor, E.: Entropy stable schemes. In: Abgrall, R., Shu, C.W. (eds.) Handbook of Numerical Methods for Hyperbolic Problems, Chapter 18, vol. XVII, pp. 467–493. Elsevier, Amsterdam (2016)

    Google Scholar 

  15. Tadmor, E.: Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems. Acta Numer. 12, 451–512 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lefloch, P.G., Mercier, J.M., Rohde, C.: Fully discrete, entropy conservative schemes of arbitrary order. SIAM J. Numer. Anal. 40(5), 1968–1992 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  17. Fjordholm, U., Mishra, S., Tadmor, E.: Arbitrarily high-order accurate entropy stable essentially nonoscillatory schemes for systems of conservation laws. SIAM J. Numer. Anal. 50(2), 544–573 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  18. Jiang, G.-S., Shu, C.-W.: On a cell entropy inequality for discontinuous Galerkin method. Math. Comput. 62, 531–538 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  19. Bertoluzza, S., Fallette, S., Russo, G., Shu, C.-W.: Discontinuous Galerkin methods: general approach and stability. In: Numerical Solutions of Partial Differential Equations, Advanced Courses in Mathematics CRM Barcelona, pp. 149–201. Birkhauser, Basel (2009)

  20. Qiu, J., Zhang, Q.: Stability, error estimate and limiters of discontinuous Galerkin methods. In: Abgrall, R., Shu, C.W. (eds.) Handbook of Numerical Methods for Hyperbolic Problems. Chapter 7, vol. XVII, pp. 147–171. Elsevier, Amsterdam (2016)

    Google Scholar 

  21. Carpenter, M., Fisher, T., Nielsen, E., Parsan, M., Svärd, M., Yamaleev, N.: Entropy stable summation-by-parts formulations for compressible computational fluid dynamics. In: Abgrall, R., Shu, C.W. (eds.) Handbook of Numerical Methods for Hyperbolic Problems. Chapter 19, vol. XVII, pp. 495–524. Elsevier, Amsterdam (2016)

    Google Scholar 

  22. Zakerzadeh, H., Fjordholm, U.: High-order accurate, fully discrete entropy stable schemes for scalar conservation laws. IMA J. Numer. Anal. 36(2), 633–654 (2016)

    Article  MathSciNet  Google Scholar 

  23. Tadmor, E.: From semi-discrete to fully discrete: stability of Runge–Kutta schemes by the energy method II. In: Estep D., Tavener S. (eds.) Collected lectures on the preservation of stability under discretization, Lecture notes from Colorado state university conference, Fort Collins, CO, Proceedings in Applied Mathematics 109, SIAM 2002, pp. 25–49 (2001)

  24. Ranocha, H., Glaubitz, J., Öffner, P., Sonar, T.: Time discretisation and L2 stability of polynomial summation-by-parts schemes with Runge–Kutta methods (2016)

  25. Sun, Z., Shu, C.-W.: Stability of the fourth order Runge–Kutta method for time-dependent partial differential equations. Annals of Mathematical Sciences and Applications 2(2), 255–284 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  26. Fjordholm, U., Mishra, S., Tadmor, E.: Energy preserving and energy stable schemes for the shallow water equations. In: Cucker, F., Pinkus, A., Todd, M. (eds.) Proceedings of Foundations of Computational Mathematics, London Mathematical Society, Lecture Note Series 36393-139, Hong Kong (2009)

  27. Gottlieb, S., Ketcheso, D.: Time discretization techniques. In: Abgrall, R., Shu, C.W. (eds.) Handbook of Numerical Methods for Hyperbolic Problems. Chapter 21, vol. XVII, pp. 549–583. Elsevier, Amsterdam (2016)

    Google Scholar 

  28. Tadmor, E.: Entropy functions for symmetric systems of conservation laws. J. Math. Anal. Appl. 122(2), 355–359 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  29. Osher, S., Shu, C.-W.: Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys. 77(2), 439–471 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  30. Tadmor, E., Zhong, W.: Entropy stable approximations of Navier–Stokes equations with no artificial numerical viscosity. J. Hyperbolic Differ. Equ. 3(3), 529–559 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  31. Jameson, A.: Transonic Flow Calculations, MAE Report 1651. Princeton University, Princeton (1983)

    Google Scholar 

  32. Tadmor, E., Zhong, W.: Novel entropy stable schemes for 1D and 2D fluid equations. In: Benzoni-Gavage, S., Serre, D. (eds.) Hyperbolic Problems: Theory, Numerics, Applications, pp. 1111–1119. Springer, New York (2008)

    Chapter  Google Scholar 

  33. Lozano, C.: The Entropy Adjoint for Shocked Flows: Application to the Quasi-One-Dimensional Euler Equations (2014). https://doi.org/10.13140/RG.2.2.27482.54725

  34. Lozano, C., Ponsin, J.: Remarks on the numerical solution of the adjoint quasi-one-dimensional Euler equations. Int. J. Numer. Methods Fluids 69(5), 966–982 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  35. Chandrashekar, P.: Kinetic energy preserving and entropy stable finite volume schemes for compressible euler and Navier–Stokes equations. Commun. Comput. Phys. 14, 1252–1286 (2013)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work has been supported by the Spanish Ministry of Defence/INTA under the research program “Termofluidodinámica” (IGB99001).

Author information

Authors and Affiliations

  1. Fluid Dynamics Group, INTA, Carretera de Ajalvir, km4, Torrejón de Ardoz (Madrid), 28850, Spain

    Carlos Lozano

Authors
  1. Carlos Lozano
    View author publications

    You can also search for this author inPubMed Google Scholar

Corresponding author

Correspondence to Carlos Lozano.

Appendix: Tadmor’s Identities

Appendix: Tadmor’s Identities

The following identities are used to obtain the discrete entropy equation for fully discrete schemes:

$$\begin{aligned}&\hbox {v}_i^T \left( F_{i+\frac{1}{2}} -F_{i-\frac{1}{2}}\right) \nonumber \\&\quad =\underbrace{\left( {\bar{\hbox {v}}}_{i+\frac{1}{2}}^T F_{i+\frac{1}{2}} -\bar{{\Theta }}_{i+\frac{1}{2}}\right) }_{\Phi _{i+\frac{1}{2}} }-\underbrace{\left( {\bar{\hbox {v}}}_{i-\frac{1}{2}}^T F_{i-\frac{1}{2}} -\bar{{\Theta }}_{i-\frac{1}{2}}\right) }_{\Phi _{i-\frac{1}{2}}}\nonumber \\&\qquad -\,{\underbrace{\frac{1}{2}\left[ \underbrace{\left( \Delta _{i+\frac{1}{2}} \hbox {v}^T F_{i+\frac{1}{2}} -\Delta _{i+\frac{1}{2}} \Theta \right) }_{\Pi _{i+\frac{1}{2}}}+\underbrace{\left( \Delta _{i-\frac{1}{2}} \hbox {v}^T F_{i-\frac{1}{2}} -\Delta _{i-\frac{1}{2}} \Theta \right) }_{\Pi _{i-\frac{1}{2}}}\right] }_{\Pi _i}}\nonumber \\&\quad (\hbox {v}_i^n )^{T}\left( U_i^{n+1} -U_i^n\right) =\eta _i^{n+1} -\eta _i^n -\underbrace{\int _{-1/2}^{1/2} {\left( {\xi +\frac{1}{2}} \right) \left( \Delta ^{n+\frac{1}{2}}\hbox {v}_i\right) ^{T}H(\hbox {v}(\xi ))\Delta ^{n+\frac{1}{2}}\hbox {v}_i d\xi } }_{{E}_i^{n+\frac{1}{2}} }\nonumber \\&\quad \left( \hbox {v}_i^{n+1}\right) ^{T}\left( U_i^{n+1} -U_i^n\right) =\eta _i^{n+1} -\eta _i^n +\underbrace{\int _{-1/2}^{1/2} {\left( {\frac{1}{2}-\xi } \right) \left( \Delta ^{n+\frac{1}{2}}\hbox {v}_i \right) ^{T}H(\hbox {v}(\xi ))\Delta ^{n+\frac{1}{2}}\hbox {v}_i d\xi } }_{\hbox {B}_i^{n+\frac{1}{2}} }\nonumber \\ \end{aligned}$$
(72)

The first one is straightforward. To prove the second one use

$$\begin{aligned} H(\hbox {v})= & {} \frac{\partial U}{\partial \hbox {v}}=(\eta _{UU} )^{-1} \nonumber \\ \hbox {v}(\xi )= & {} {\bar{\hbox {v}}}_i^{n+\frac{1}{2}} +\xi \Delta ^{n+\frac{1}{2}}\hbox {v}_i \nonumber \\ {\bar{\hbox {v}}}_i^{n+\frac{1}{2}}= & {} \frac{1}{2}\left( \hbox {v}_i^{n+1} +\hbox {v}_i^n\right) \nonumber \\ \Delta ^{n+\frac{1}{2}}\hbox {v}_i= & {} \hbox {v}_i^{n+1} -\hbox {v}_i^n \end{aligned}$$
(73)

to rewrite

$$\begin{aligned}&(\hbox {v}_i^n )^{T}\left( U_i^{n+1} -U_i^n\right) =(\hbox {v}_i^n )^{T}\int _{\hbox {v}_i^n }^{\hbox {v}_i^{n+\frac{1}{2}} } {U_\mathrm{v} d\hbox {v}} =(\hbox {v}_i^n )^{T}\int _{\hbox {v}_i^n }^{\hbox {v}_i^{n+\frac{1}{2}} } {H(\hbox {v})d\hbox {v}} \nonumber \\&\quad =(\hbox {v}_i^n )^{T}\int _{-1/2}^{1/2} {H(\hbox {v}(\xi ))\Delta ^{n+\frac{1}{2}}\hbox {v}_i d\xi } \nonumber \\&\quad \hbox {Likewise}, \nonumber \\&\quad \eta _i^{n+1} -\eta _i^n =\int _{\hbox {v}_i^n }^{\hbox {v}_i^{n+\frac{1}{2}} } {(\eta _\mathrm{v} )^{T}d\hbox {v}} =\int _{\hbox {v}_i^n }^{\hbox {v}_i^{n+\frac{1}{2}} } {(\eta _U )^{T}U_\mathrm{v} d\hbox {v}} =\int _{\hbox {v}_i^n }^{\hbox {v}_i^{n+\frac{1}{2}} } {\hbox {v}^{T}H(\hbox {v})d\hbox {v}} \nonumber \\&\quad =\int _{-1/2}^{1/2} {\left( {\bar{\hbox {v}}}_i^{n+\frac{1}{2}} +\xi \Delta ^{n+\frac{1}{2}}\hbox {v}_i\right) ^{T}H(\hbox {v}(\xi ))\Delta ^{n+\frac{1}{2}}\hbox {v}_i d\xi } =(\hbox {v}_i^n )^{T}\int _{-1/2}^{1/2} {H(\hbox {v}(\xi ))\Delta ^{n+\frac{1}{2}}\hbox {v}_i d\xi } \nonumber \\&\qquad +\int _{-1/2}^{1/2} {\left( {\xi +\frac{1}{2}} \right) \left( \Delta ^{n+\frac{1}{2}}\hbox {v}_i\right) ^{T}H(\hbox {v}(\xi ))\Delta ^{n+\frac{1}{2}}\hbox {v}_i d\xi } =(\hbox {v}_i^n )^{T}\left( U_i^{n+1} -U_i^n\right) +{E}_i^{n+\frac{1}{2}}\nonumber \\ \end{aligned}$$
(74)

In a similar fashion,

$$\begin{aligned}&\left( \hbox {v}_i^{n+1}\right) ^{T}\left( U_i^{n+1} -U_i^n\right) =\left( \hbox {v}_i^{n+1}\right) ^{T}\int _{\hbox {v}_i^n }^{\hbox {v}_i^{n+\frac{1}{2}} } {U_\mathrm{v} d\hbox {v}} =(\hbox {v}_i^n )^{T}\int _{\hbox {v}_i^n }^{\hbox {v}_i^{n+\frac{1}{2}} } {H(\hbox {v})d\hbox {v}} \nonumber \\&\quad =\left( \hbox {v}_i^{n+1}\right) ^{T}\int _{-1/2}^{1/2} {H(\hbox {v}(\xi ))\Delta ^{n+\frac{1}{2}}\hbox {v}_i d\xi } \nonumber \\&\quad \hbox {Likewise},\nonumber \\&\quad \eta _i^{n+1} -\eta _i^n =\int _{\hbox {v}_i^n }^{\hbox {v}_i^{n+\frac{1}{2}} } {(\eta _\mathrm{v} )^{T}d\hbox {v}} =\int _{\hbox {v}_i^n }^{\hbox {v}_i^{n+\frac{1}{2}} } {(\eta _U )^{T}U_\mathrm{v} d\hbox {v}} =\int _{\hbox {v}_i^n }^{\hbox {v}_i^{n+\frac{1}{2}} } {\hbox {v}^{T}H(\hbox {v})d\hbox {v}} \nonumber \\&\quad =\int _{-1/2}^{1/2} {\left( {\bar{\hbox {v}}}_i^{n+\frac{1}{2}} +\xi \Delta ^{n+\frac{1}{2}}\hbox {v}_i\right) ^{T}H(\hbox {v}(\xi ))\Delta ^{n+\frac{1}{2}}\hbox {v}_i d\xi } =\left( \hbox {v}_i^{n+1} \right) ^{T}\int _{-1/2}^{1/2} {H(\hbox {v}(\xi ))\Delta ^{n+\frac{1}{2}}\hbox {v}_i d\xi } \nonumber \\&\qquad -\int _{-1/2}^{1/2} {\left( {\frac{1}{2}-\xi } \right) \left( \Delta ^{n+\frac{1}{2}}\hbox {v}_i\right) ^{T}H(\hbox {v}(\xi ))\Delta ^{n+\frac{1}{2}}\hbox {v}_i d\xi } =\left( \hbox {v}_i^{n+1}\right) ^{T}\left( U_i^{n+1} -U_i^n\right) -B_i^{n+\frac{1}{2}}\nonumber \\ \end{aligned}$$
(75)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lozano, C. Entropy Production by Explicit Runge–Kutta Schemes. J Sci Comput 76, 521–564 (2018). https://doi.org/10.1007/s10915-017-0627-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-017-0627-0

Keywords