Journal of Scientific Computing

, Volume 63, Issue 3, pp 633–653 | Cite as

Error Analysis of Explicit Partitioned Runge–Kutta Schemes for Conservation Laws

  • Willem Hundsdorfer
  • David I. Ketcheson
  • Igor Savostianov
Article

Abstract

An error analysis is presented for explicit partitioned Runge–Kutta methods and multirate methods applied to conservation laws. The interfaces, across which different methods or time steps are used, lead to order reduction of the schemes. Along with cell-based decompositions, also flux-based decompositions are studied. In the latter case mass conservation is guaranteed, but it will be seen that the accuracy may deteriorate.

Keywords

Multirate methods Partitioned Runge–Kutta methods  Conservation Stability Convergence 

Mathematics Subject Classification

65L06 65M06 65M20 

References

  1. 1.
    Ascher, U.M., Ruuth, S.J., Spiteri, R.J.: Implicit-explicit Runge–Kutta methods for time-dependent partial differential equations. Appl. Numer. Math. 25, 151–167 (1997)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Brenner, P., Crouzeix, M., Thomée, V.: Single step methods for inhomogeneous linear differential equations. RAIRO Anal. Numer. 16, 5–26 (1982)MATHMathSciNetGoogle Scholar
  3. 3.
    Constantinescu, E.M., Sandu, A.: Multirate timestepping methods for hyperbolic conservation laws. J. Sci. Comput. 33, 239–278 (2007)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Günther, M., Kværnø, A., Rentrop, P.: Multirate partitioned Runge–Kutta methods. BIT 41, 504–514 (2001)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Hairer, E., Nørsett, S.P., Wanner, G.: Solving Ordinary Differential Equations I—Nonstiff Problems, Springer Series Comput. Math. vol. 8, 2nd edn. Springer, Berlin (1993)Google Scholar
  6. 6.
    Hundsdorfer, W., Mozartova, A., Savcenco, V.: Monotonicity conditions for multirate and partitioned explicit Runge–Kutta schemes. In: Ansorge R., et al. (eds.) Recent Developments in the Numerics of Nonlinear Hyperbolic Conservation Laws. NNFM, vol. 120, pp. 177–195. Springer, Berlin (2013)Google Scholar
  7. 7.
    Hundsdorfer, W., Ruuth, S.J.: IMEX extensions of linear multistep methods with general monotonicity and boundedness properties. J. Comput. Phys. 225, 2016–2042 (2007)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Hundsdorfer, W., Verwer, J.G.: Numerical Solution of Advection-Diffusion-Reaction Equations. Springer Series Comput. Math., vol. 33. Springer, Berlin (2003)Google Scholar
  9. 9.
    Ketcheson, D.I., Macdonald, C.B., Ruuth, S.J.: Spatially partitioned embedded Runge–Kutta methods. SIAM J. Numer. Anal. 51, 2887–2910 (2013)CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Mathew, T.P., Polyakov, P.L., Russo, G., Wang, J.: Domain decomposition operator splittings for the solution of parabolic equations. SIAM J. Sci. Comput. 19, 912–932 (1998)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Osher, S., Sanders, R.: Numerical approximations to nonlinear conservation laws with locally varying time and space grids. Math. Comput. 41, 321–336 (1983)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Portero, L., Jorge, J.C., Bujanda, B.: Avoiding order reduction of fractional step Runge–Kutta discretizations for linear time dependent oefficient parabolic problems. Appl. Numer. Math. 48, 409–424 (2004)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Portero, L., Bujanda, B., Jorge, J.C.: A combined fractional step domain decomposition method for the numerical integration of parabolic problems. In: Wyrzykowski R., et al. (eds.) Proceedings PPAM2003, LNCS, vol. 3019, pp. 1034–1041. Springer, Berlin (2004)Google Scholar
  14. 14.
    Savcenco, V., Hundsdorfer, W., Verwer, J.G.: A multirate time stepping strategy for stiff ordinary differential equations. BIT 47, 137–155 (2007)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Schlegel, M., Knoth, O., Arnold, M., Wolke, R.: Multirate Runge–Kutta schemes for advection equations. J. Comput. Appl. Math. 226, 345–357 (2009)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Schlegel, M., Knoth, O., Arnold, M., Wolke, R.: Numerical solution of multiscale problems in atmospheric modeling. Appl. Numer. Math. 62, 1531–1543 (2012)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Shu, C.-W.: High order weighted essentially nonoscillatory schemes for convection dominated problems. SIAM Rev. 51, 82–126 (2009)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Tang, H.-Z., Warnecke, G.: High resolution schemes for conservation laws and convection–diffusion equations with varying time and space grids. J. Comput. Math. 24, 121–140 (2006)MATHMathSciNetGoogle Scholar
  19. 19.
    Wensch, J., Knoth, O., Galant, A.: Multirate infinitesimal step methods for atmospheric flow simulation. BIT Numer. Math. 49, 449–473 (2009)CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Willem Hundsdorfer
    • 1
  • David I. Ketcheson
    • 2
  • Igor Savostianov
    • 1
  1. 1.CWIAmsterdamThe Netherlands
  2. 2.Division Mathematical and Computer Sciences and EngineeringKing Abdullah University of Science and Technology (KAUST)ThuwalSaudi Arabia

Personalised recommendations