Skip to main content
SpringerLink
Log in

Adaptive Time Steps for Compressible Flows Based on Dual-Time Stepping and a RK/Implicit Smoother

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

Abstract

We present an adaptive time stepping technique, based on a dual-time stepping scheme together with a RK/implicit smoother for physical time marching. This method is found to be very efficient for transient problem characterized by a large variations of time scales during the physical time and hence a wide range of required physical time steps. We discuss the method and present several examples of compressible flow problems.

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

Access this article

Log in via an institution

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
Fig. 16

Similar content being viewed by others

References

  1. Chiew, J. J., Pulliam, T. H.: Stability analysis of dual-time stepping. In: 46th AIAA Fluid Dynamics Conference (2016)

  2. Darmofal, D., Van Leer, B.: Local preconditioning: manipulating mother nature to fool father time. In: Caughey, D.A., Hafez, M. (eds.) Advances and Prospects in Computational Aerodynamics, Computing the Future II. John Wiley and Sons (1998)

  3. de Brito Alves, L. S.: Dual time stepping with multi-stage schemes in physical time for usteady low Mach number compressible flows. EPTT-VII Escola de Primavera de TransiÃğÃčo e TurbulÃłncia (2010)

  4. Eliasson, P., Weinerfelt, P.: High-order implicit time integration for unsteady turbulent flow simulations. Comput. Fluids 112, 35–49 (2015)

    Article  MathSciNet  Google Scholar 

  5. Gustafsson, K., Lundh, M., Sűderlind, G.: A pi stepsize control for the numerical solution of ordinary differential equations. BIT Numer. Math. 28(2), 270–287 (1988)

    Article  MathSciNet  Google Scholar 

  6. Haas, J.F., Sturtevant, B.: Interaction of weak shock waves with cylindrical and spherical gas inhomogenities. J. Fluid Mech. 181, 41–76 (1987)

    Article  Google Scholar 

  7. Hay, A., Etienne, S., Pelletier, D., Garon, A.: hp-Adaptive time integration based on the BDF for viscous flows. J. Comput. Phys. 291, 151–176 (2015)

    Article  MathSciNet  Google Scholar 

  8. Jameson, A.: Time dependent calculations using multigrid, with applications to unsteady flows past airfoils and wings. AIAA paper, p. 1596 (1991)

  9. Jameson, A.: Evaluation of fully implicit Runge Kutta schemes for unsteady flow calculations. J. Sci. Comput. 73, 1–34 (2017)

    Article  MathSciNet  Google Scholar 

  10. Jameson, A., Schmidt, W., Turkel, E.: Numerical solutions of the Euler equations by finite volume methods using Runge–Kutta time-stepping schemes. AIAA paper, 1259 (1981)

  11. John, V., Rang, J.: Adaptive time step control for the incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 199, 514–524 (2010)

    Article  MathSciNet  Google Scholar 

  12. Kennedy, C.A., Carpenter, M.H.: Additive Runge–Kutta schemes for convection–diffusion–reaction equations. Appl. Numer. Math. 44(1), 139–181 (2003)

    Article  MathSciNet  Google Scholar 

  13. Langer, S.: Accuracy investigations of a compressible second order finite volume code towards the compressible limit. Comput. Fluids 149, 110–137 (2017)

    Article  Google Scholar 

  14. Langer, S.: Agglomeration multigrid methods with implicit Runge–Kutta smoothers applied to aerodynamic simulations on unstructured grids. J. Comput. Phys. 277, 72–100 (2014)

    Article  MathSciNet  Google Scholar 

  15. Peles, O., Yaniv, S., Turkel, E.: Convergence Acceleration of Runge–Kutta Schemes using RK/implicit smoother for Navier–Stokes equations with SST turbulence. In: Proceedings of 52nd Israel Annual Conference on Aerospace Sciences (2012)

  16. Peles, O., Turkel, E., Yaniv, S.: Fast Iterative methods for Navier–Stokes equations with SST turbulence model and chemistry (ICCFD7). In: Seventh International Conference on Computational Fluid Dynamics (2012)

  17. Peles, O., Turkel, E.: Acceleration methods for multi-physics compressible flow. J. Comput. Phys. 358, 201–234 (2018)

    Article  MathSciNet  Google Scholar 

  18. Rossow, C.C.: Efficient computation of compressible and incompressible flows. J. Comput. Phys. 220(2), 879–899 (2007)

    Article  MathSciNet  Google Scholar 

  19. Shi, J., Zhang, Y.T., Shu, C.W.: Resolution of high order WENO schemes for complicated flow structures. J. Comput. Phys. 186), 690–696 (2003)

    Article  MathSciNet  Google Scholar 

  20. Shu, C.W., Osher, S.: Efficient implementation of essentially non-oscillatory shock-capturing schemes. J. Comput. Phys. 83(1), 32–78 (1989)

    Article  MathSciNet  Google Scholar 

  21. Sod, G.: A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys. 27(1), 1–31 (1978)

    Article  MathSciNet  Google Scholar 

  22. Swanson, R.C., Rossow, C.C.: An efficient solver for the RANS Equations and a one-equation turbulence model Comput. Fluids 42, 13–25 (2011)

    MathSciNet  MATH  Google Scholar 

  23. Swanson, R.C., Rossow, C.C., Yaniv, S.: Analysis of a RK/implicit smoother for multigrid, computational fluid dynamics. In: Kuzmin, A. (Ed.) The Sixth International Conference on Computational Fluid Dynamics. Springer, St. Petersburg, Russia, July 12–16 (2010)

  24. Swanson, R.C., Turkel, E.: Analysis of a fast iterative method in a dual time algorithm for the Navier–Stokes equations. In: European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 2012), Vienna, Austria, September 10–14, 2012

  25. Swanson, R.C., Turkel, E., Rossow, C.C.: Convergence acceleration of Runge–Kutta schemes for solving the Navier–Stokes equations. J. Comput. Phys. 224(1), 365–388 (2007)

    Article  MathSciNet  Google Scholar 

  26. Turkel, E.: Preconditioned methods for solving the incompressible and low speed compressible equations. J. Comput. Phys. 72(2), 277–298 (1987)

    Article  MathSciNet  Google Scholar 

  27. Turkel, E.: Review of preconditioning methods for fluid dynamics. Appl. Numer. Math. 12, 257–284 (1993)

    Article  MathSciNet  Google Scholar 

  28. Turkel, E., Fiterman, A., Van Leer, B.: Preconditioning and the limit to the incompressible flow equations. In: Caughey, D.A., Hafez, M.M. (eds.) Frontiers of Computational Fluid Dynamics, pp. 215–234. Wiley, New York (1994)

    Google Scholar 

  29. Turkel, E., Radespiel, R., Kroll, N.: Assessment of preconditioning methods for multidimensional aerodynamics. Comput. Fluids 26(2), 613–634 (1997)

    Article  Google Scholar 

  30. Turkel, E., Vatsa, V.N.: Choice of variables and preconditioning for time dependent problems. AIAA paper, 3692 (2003)

  31. Turkel, E., Vatsa, V.N.: Local preconditioners for steady state and dual time stepping. ESAIM: M2AN 39(3), 515–536 (2005)

    Article  Google Scholar 

  32. Van Leer, B., Lee, W., Roe, P.: Characteristic time-stepping or local preconditioning of the Euler equations. In: 10th Computational Fluid Dynamics Conference (1991)

  33. Van Leer, B., Lee, E.T., Roe, P.L., Powell, K.G., Tai, C.H.: Design of optimally smoothing multi-stage schemes for the Euler equations. Commun. Appl. Numer. Methods 8(10), 761–769 (1992)

    Article  Google Scholar 

  34. Wang, B.S., Li, P., Gao, Z., Don, W.S.: An improved fifth order alternative WENO-Z finite difference scheme for hyperbolic conservation laws. J. Comput. Phys. 374, 469–477 (2018)

    Article  MathSciNet  Google Scholar 

  35. Xu, Z., Shu, C.W.: Anti-diffusive flux corrections for high order finite difference WENO schemes. J. Comput. Phys. 205, 458–485 (2005)

    Article  MathSciNet  Google Scholar 

  36. Zhao, G., Sun, M., Xie, S., Wang, H.: Numerical dissipation control in an adaptive WCNS with a new smoothness indicator. Appl. Math. Comput. 330, 239–253 (2018)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. IMI Systems, Ramat Ha-Sharon, Israel

    O. Peles

  2. School of Mathematical Sciences, Tel-Aviv University, Tel Aviv, Israel

    E. Turkel

Authors
  1. O. Peles
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. E. Turkel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to E. Turkel.

Additional information

We dedicate this paper to the memory of Saul (Shalom) Abarbanel who was a mentor to whole generation of students and colleagues.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Peles, O., Turkel, E. Adaptive Time Steps for Compressible Flows Based on Dual-Time Stepping and a RK/Implicit Smoother. J Sci Comput 81, 1409–1428 (2019). https://doi.org/10.1007/s10915-019-01024-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-019-01024-y

Keywords

Mathematics Subject Classification

Search

Navigation