Skip to main content
SpringerLink
Log in

Rosenbrock strong stability-preserving methods for convection–diffusion–reaction equations

  • Original Paper
  • Area 2
  • Published:
Japan Journal of Industrial and Applied Mathematics Aims and scope Submit manuscript

Abstract

Rosenbrock methods are normally used for solving moderately stiff problems and strong-stability preserving ones are employed a lot for hyperbolic conservation laws. Their combination called additive Rosenbrock-strong stability preserving (Ros-SSP) schemes are first introduced in this paper to deal with convection– diffusion–reaction equations. Accuracy and stability of the Ros-SSP scheme are considered. Numerical results are given to prove advantages of Ros-SSP methods. A practical application of a three-stage, second order Ros-SSP method solving a spatially discretized angiogenesis model in two-dimensional case is provided as well.

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

Similar content being viewed by others

References

  1. Odanaka, S.: Multidimensional discretization of stationary quantum drift-diffusion model for ultrasmall MOSFET structures. IEEE Trans. Comput. Aided Des. 23, 837–842 (2004)

    Article  Google Scholar 

  2. Peraire, J., Persson, P.-O.: High-order discontinuous Galerkin methods for CFD. In: Adaptive High-Order Methods in Computational Fluid Dynamics, pp. 119–152 (Adv. Comput. Fluid Dyn., 2) (2011)

  3. Hildebrand, M., Kuperman, M., Wio, H., Mikhailov, A.S., Ertl, G.: Self-organized chemical nanoscale microreactors. Phys. Rev. Lett. 83, 1475–1478 (1999)

    Article  Google Scholar 

  4. Anderson, A.R.A., Chaplain, M.A.J.: Continuous and discrete mathematical models of tumor-induced angiogenesis. Bull. Math. Biol. 60, 857–900 (1998)

    Article  MATH  Google Scholar 

  5. Hundsdorfer, W., Verwer, J.G.: Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer Series in Computational Mathematics, vol. 33. Springer, Berlin (2003)

  6. Saito, N.: Conservative upwind finite-element method for a simplified Keller–Segel system modelling chemotaixs. IMA J. Numer. Anal. 27, 332–365 (1997)

    Google Scholar 

  7. Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods. Springer Texts in Applied Mathematics, vol. 54. Springer, New York (2008)

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

  9. Hundsdorfer, W., Koren, B., van Loon, M., Verwer, J.G.: A positive finite-difference advection scheme. J. Comput. Phys. 117, 34–46 (1995)

    Google Scholar 

  10. 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)

    Article  MATH  MathSciNet  Google Scholar 

  11. Tyson, R., Stern, L.G., LeVeque, R.J.: Fractional step methods applied to a chemotaxis model. J. Math. Biol. 41, 455–475 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  12. Persson, P.-O., Peraire, J.: Newton-GMRES preconditioning for discontinuous Galerkin discretizations of the Navier–Stokes equations. SIAM J. Sci. Comput. 30, 2709–2733 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. Verwer, J.G., Spee, E.J., Blom, J.G., Hundsdorfer, W.: A second-order Rosenbrock method applied to photochemical dispersion problems. SIAM J. Sci. Comput. 20, 1456–1480 (1999)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  15. Araújo, A.L., Murua, A., Sanz-Serna, J.M.: Symplectic methods based on decompositions. SIAM J. Numer. Anal. 34, 1926–1947 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  16. Hairer, E.,Wanner, G.: Solving Ordinary Differential Equations. Springer Series in Computational Mathematics, vol. 14. Springer, Berlin (1996)

  17. Hairer, E.: Order conditions for numerical methods for partitioned ordinary differential equations. Numer. Math. 36, 431–445 (1980/81)

  18. Tadmor, E.: Approximate solutions of nonlinear conservation laws. In: Advanced Numerical Approximation of Nonlinear Hyperbolic Equations (Cetraro, 1997), pp. 1–149 (Lecture Notes in Mathematics, 1697) (1998)

  19. Gottlieb, S., Shu, C.W., Tadmor, E.: Strong stability-preserving high-order time discretization methods. SIAM Rev. 43, 89–112 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  21. Gottlieb, S., Shu, C.W.: Total variation diminishing Runge–Kutta schemes. Math. Comput. 67, 73–85 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  22. Arnold, D.N., Brezzi, F., Cockburn, B., Marini, L.D.: Unified analysis of discontinuous Galerkin methods for elliptic problems. SIAM J. Numer. Anal. 39, 1749–1779 (2001/02)

  23. Gustafsson, K., Söderlind, G.: Control strategies for the iterative solution of nonlinear equations in ODE solvers. SIAM J. Sci. Comput. 18, 23–40 (1997)

Download references

Acknowledgments

The authors appreciated Professor S. Odanaka of Osaka University for invaluable discussions on constructing the numerical methods. The authors would like to thank the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.

Author information

Authors and Affiliations

  1. Department of Applied Physics, Osaka University, Suita, Osaka, 565-0871, Japan

    Doan Duy Hai & Atsushi Yagi

Authors
  1. Doan Duy Hai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Atsushi Yagi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Doan Duy Hai.

Additional information

This work is supported by Grant-in-Aid for Scientific Research (No. 20340035) of the Japan Society for the Promotion of Science.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hai, D.D., Yagi, A. Rosenbrock strong stability-preserving methods for convection–diffusion–reaction equations. Japan J. Indust. Appl. Math. 31, 401–417 (2014). https://doi.org/10.1007/s13160-014-0143-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13160-014-0143-7

Keywords

Mathematics Subject Classification (2000)

Search

Navigation