Skip to main content
SpringerLink
Log in

Runge–Kutta Residual Distribution Schemes

Journal of Scientific Computing volume 62, pages 772–802 (2015)Cite this article

Abstract

We are concerned with the solution of time-dependent non-linear hyperbolic partial differential equations. We investigate the combination of residual distribution methods with a consistent mass matrix (discretisation in space) and a Runge–Kutta-type time-stepping (discretisation in time). The introduced non-linear blending procedure allows us to retain the explicit character of the time-stepping procedure. The resulting methods are second order accurate provided that both spatial and temporal approximations are. The proposed approach results in a global linear system that has to be solved at each time-step. An efficient way of solving this system is also proposed. To test and validate this new framework, we perform extensive numerical experiments on a wide variety of classical problems. An extensive numerical comparison of our approach with other multi-stage residual distribution schemes is also given.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

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

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

References

  1. Abgrall, R., Deconinck, H., Sermeus, K.: Status of multidimensional upwind residual distribution schemes and applications in aeronautics. In: AIAA Paper 2000–2328, Fluids 2000/Denver (2000)

  2. Abgrall, R., Mezine, M.: Residual distribution schemes for steady problems. In: Computational Fluid Dynamics, VKI LS 2003–05. von Karman Institute for Fluid Dynamics (2003)

  3. Abgrall, R., Roe, P.L.: High order fluctuation schemes on triangular meshes. J. Sci. Comput. 19(1–3), 3–36 (2003). Special issue in honor of the sixtieth birthday of Stanley Osher

  4. Abgrall, R.: Toward the ultimate conservative scheme: following the quest. J. Comput. Phys. 167(2), 277–315 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  5. Abgrall, R., Mezine, M.: Construction of second order accurate monotone and stable residual distribution schemes for unsteady flow problems. J. Comput. Phys. 188(1), 16–55 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  6. Abgrall, R.: A residual distribution method using discontinuous elements for the computation of possibly non smooth flows. Adv. Appl. Math. Mech. 2(1), 32–44 (2010)

    MathSciNet  Google Scholar 

  7. Abgrall, Ricchiuto M.: Explicit Runge–Kutta residual distribution schemes for time dependent problems: second order case. J. Comput. Phys. 229(16), 5653–5691 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  8. Balay, S., Brown, J., Buschelman, K., Eijkhout, V., Gropp, W.D., Kaushik, D., Knepley, M.G., McInnes, L.C., Smith, B.F., Zhang, H.: PETSc Users Manual. Tech. Rep. ANL-95/11 - Revision 3.3, Argonne National Laboratory (2012)

  9. Balay, S., Brown, J., Buschelman, K., Gropp, W.D., Kaushik, D., Knepley, M.G., Curfman McInnes, L., Smith, B.F., Zhang, H.: PETSc Web page (2012). http://www.mcs.anl.gov/petsc

  10. Brooks, A.N., Hughes, T.J.R.: Streamline upwind/Petrov-Galerkin formulations for convection dominated flows with particular emphasis on the incompressible Navier–Stokes equations. Comput. Methods Appl. Mech. Eng. 32(1–3), 199–259 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  11. Caraeni, D., Fuchs, L.: Compact third-order multidimensional upwind scheme for Navier–Stokes simulations. Theor. Comput. Fluid Dyn. 15(6), 373–401 (2002)

    Article  Google Scholar 

  12. Carette, J.C., Deconinck, H., Paillère, H., Roe, P.L.: Multidimensional upwinding: its relation to finite elements. Int. J. Numer. Methods Fluids 20(8–9), 935–955 (1995)

    Article  MATH  Google Scholar 

  13. Cockburn, B., Shu, C.W.: The Runge–Kutta discontinuous Galerkin method for conservation laws. V. Multidimensional systems. J. Comput. Phys. 141(2), 199–224 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  14. Csík, Á., Deconinck, H., Ricchiuto, M., Poedts, S.: Space-time residual distribution schemes for hyperbolic conservation laws. In: 15th AIAA Computational Fluid Dynamics Conference, Anaheim, CA, USA (2001)

  15. Csík, Á., Deconinck, H.: Space-time residual distribution schemes for hyperbolic conservation laws on unstructured linear finite elements. Int. J. Numer. Methods Fluids 40(3–4), 573–581 (2002)

    Article  MATH  Google Scholar 

  16. Csík, Á., Ricchiuto, M., Deconinck, H.: A conservative formulation of the multidimensional upwind residual distribution schemes for general nonlinear conservation laws. J. Comput. Phys. 179(1), 286–312 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  17. Csík, Á., Deconinck, H., Ricchiuto, M.: Residual distribution for general time-dependent conservation laws. J. Comput. Phys. 209(1), 249–289 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  18. Deconinck, H., Ferrante, A.: Solution of the Unsteady Euler Equations using Residual Distribution and Flux Corrected Transport. Tech. Rep. 97–08, von Karman Institute for Fluid Dynamics (1997)

  19. Deconinck, H., Ricchiuto, M.: Residual distribution schemes: foundations and analysis. In: Encyclopedia of Computational Mechanics, vol. 3. Wiley, London (2007)

  20. Deconinck, H.: Upwind methods and multidimensional splittings for the Euler equations. In: Computational Fluid Dynamics, VKI LS 1991–01. von Karman Institute for Fluid Dynamics (1991)

  21. Deconinck, H., Roe, P.L., Struijs, R.: A multidimensional generalization of Roe’s flux difference splitter for the Euler equations. Comput. Fluids 22(2–3), 215–222 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  22. Dobeš, J., Deconinck, H.: Second order blended multidimensional upwind residual distribution scheme for steady and unsteady computations. J. Comput. Appl. Math. 215(2), 378–389 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  23. Emery, A.F.: An evaluation of several differencing methods for inviscid fluid flow problems. J. Comput. Phys. 2, 306–331 (1968)

    Article  MATH  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  25. Guzik, S.M.J., Groth, C.P.T.: Comparison of solution accuracy of multidimensional residual distribution and Godunov-type finite-volume methods. Int. J. Comput. Fluid Dyn. 22(1–2), 61–83 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  26. Hubbard, M., Ricchiuto, M.: Discontinuous fluctuation distribution: a route to unconditional positivity and high order accuracy. In: ICFD 2010 International Conference on Fluid Dynamics, Reading (UK) (2010)

  27. Hubbard, M.E.: Discontinuous fluctuation distribution. J. Comput. Phys. 227(24), 10125–10147 (2008)

  28. Hubbard, M.E.: A framework for discontinuous fluctuation distribution. Int. J. Numer. Methods Fluids 56(8), 1305–1311 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  29. Hughes, T.J.R., Mallet, M.: A new finite element formulation for CFD III: the generalized streamline operator for multidimensional advective-diffusive systems. Comput. Methods Appl. Mech. Eng. 58, 305–328 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  30. Johnson, C.: Numerical Solution of Partial Differential Equations by the Finite Element Method. Dover Publications Inc., Mineola, NY (2009). Reprint of the 1987 edition

  31. LeVeque, R.J.: Numerical Methods for Conservation Laws. Lectures in Mathematics ETH Zürich, 2nd edn. Birkhäuser Verlag, Basel (1992)

    Book  Google Scholar 

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

    Book  Google Scholar 

  33. März, J., Degrez, G.: Improving time accuracy of residual distribution schemes. Tech. Rep. 96–17, von Karman Institute for Fluid Dynamics (1996)

  34. De Palma, P., Pascazio, G., Rossiello, G., Napolitano, M.: A second-order-accurate monotone implicit fluctuation splitting scheme for unsteady problems. J. Comput. Phys. 208(1), 1–33 (2005)

    Article  MATH  Google Scholar 

  35. Roe, P.L.: Fluctuations and signals—a framework for numerical evolution problems. In: Numerical Methods for Fluid Dynamics, pp. 219–257. Academic Press, London (1982)

  36. Roe, P.L.: Linear advection schemes on triangular meshes. Tech. Rep. Technical Report CoA 8720, Cranfield Institute of Technology (1987)

  37. Roe, P.L.: Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43(2), 357–372 (1981)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  39. Struijs, R., Deconinck, H., Roe, P.L.: Fluctuation splitting schemes for the 2D Euler equations. In: Computational Fluid Dynamics, VKI LS 1991–01. von Karman Institute for Fluid Dynamics (1991)

  40. van der Weide, E., Deconinck, H., Issman, E., Degrez, G.: A parallel, implicit, multi-dimensional upwind, residual distribution method for the Navier-Stokes equations on unstructured grids. Comput. Mech. 23(2), 199–208 (1999)

  41. van der Weide, E., Deconinck, H.: Positive matrix distribution schemes for hyperbolic systems, with application to the Euler equations. In: Desideri, J., Hirsch, C., LeTallec, P., Pandolfi, M., Periaux, J. (eds.) Computational Fluid Dynamics ‘96, pp. 747–753 (1996)

  42. Warzyński, A., Hubbard, M.E., Ricchiuto, M.: Discontinuous residual distribution schemes for time-dependent problems. In: Li J., Yang, H.T., Machorro, E. (ed.) Recent Advances In Scientific Computing and Applications, Contemporary Mathematics, vol. 586, pp. 375–382 (2013)

  43. Warzyński, A.: Runge-kutta residual distribution schemes. Ph.D. thesis, University of Leeds (2013)

  44. Woodward, P., Colella, P.: The numerical simulation of two-dimensional fluid flow with strong shocks. J. Comput. Phys. 54(1), 115–173 (1984)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

The first author would like to acknowledge EPSRC who funded this work under grant number EP/G003645/1.

Author information

Authors and Affiliations

  1. School of Computing, University of Leeds, Leeds, LS2 9JT, UK

    Andrzej Warzyński & Matthew E. Hubbard

  2. INRIA Bordeaux Sud-Ouest, 33405 , Talence Cedex, France

    Mario Ricchiuto

Authors
  1. Andrzej Warzyński
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Matthew E. Hubbard
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mario Ricchiuto
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrzej Warzyński.

Rights and permissions

Reprints and Permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Warzyński, A., Hubbard, M.E. & Ricchiuto, M. Runge–Kutta Residual Distribution Schemes. J Sci Comput 62, 772–802 (2015). https://doi.org/10.1007/s10915-014-9879-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10915-014-9879-0

Keywords

search

Navigation