RKC

RKC: An explicit solver for parabolic PDEs. An explicit Runge-Kutta-Chebychev algorithm for parabolic partial differential equations is discussed, implemented and tested. This method exploits some remarkable properties of a class of Runge-Kutta formulas of Chebychev type, proposed almost 20 year ago by P. J. van der Houwen and B. P. Sommeijer [Z. Angew. Math. Mech. 60, 479-485 (1980; Zbl 0455.65052)]. An s-stage (s≥2) method is discussed and analytical expressions for its coefficients are derived. An interesting property of this family makes it possible for the algorithm to select at each step the most efficient stable formula and the most efficient time-step. Various computational results and comparisons with other methods are provided.


References in zbMATH (referenced in 120 articles , 1 standard article )

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  1. McInnes, Lois Curfman; Allan, Benjamin A.; Armstrong, Robert; Benson, Steven J.; Bernholdt, David E.; Dahlgren, Tamara L.; Diachin, Lori Freitag; Krishnan, Manojkumar; Kohl, James A.; Larson, J. Walter; Lefantzi, Sophia; Nieplocha, Jarek; Norris, Boyana; Parker, Steven G.; Ray, Jaideep; Zhou, Shujia: Parallel PDE-based simulations using the common component architecture (2006)
  2. Shampine, L. F.; Sommeijer, B. P.; Verwer, J. G.: IRKC: an IMEX solver for stiff diffusion-reaction PDEs (2006)
  3. van Veldhuizen, S.; Vuik, C.; Kleijn, C. R.: Numerical methods for reacting gas flow simulations (2006)
  4. Zheng, Zheming; Petzold, Linda: Runge-Kutta-Chebyshev projection method (2006)
  5. Lefantzi, Sophia; Ray, Jaideep; Kennedy, Christopher A.; Najm, Habib N.: A component-based toolkit for simulating reacting flows with high order spatial discretisations on structured adaptively refined meshes (2005)
  6. Najm, H. N.; Knio, O. M.: Modeling low Mach number reacting flow with detailed chemistry and transport (2005)
  7. Schmitt, Bernhard A.; Weiner, Rüdiger; Podhaisky, Helmut: Multi-implicit peer two-step W-methods for parallel time integration (2005)
  8. Verwer, J. G.; Sommeijer, B. P.: An implicit-explicit Runge--Kutta--Chebyshev scheme for diffusion-reaction equations (2004)
  9. Verwer, J. G.; Sommeijer, B. P.; Hundsdorfer, W.: RKC time-stepping for advection-diffusion-reaction problems (2004)
  10. Bermejo, Rodolfo; El Amrani, Mofdi: A finite element semi-Lagrangian explicit Runge-Kutta-Chebyshev method for convection dominated reaction-diffusion problems (2003)
  11. Abdulle, Assyr: Fourth order Chebyshev methods with recurrence relation (2002)
  12. Seaid, M.: On the quasi-monotone modified method of characteristics for transport-diffusion problems with reactive sources (2002)
  13. Abdulle, Assyr; Medovikov, Alexei A.: Second order Chebyshev methods based on orthogonal polynomials (2001)
  14. Botchev, Mike A.; van der Vorst, Henk A.: A parallel nearly implicit time-stepping scheme (2001)
  15. Hörnell, Karl; Lötstedt, Per: Time step selection for shock problems (2001)
  16. Verwer, J. G.; Sommeijer, B. P.: A numerical study of mixed parabolic-gradient systems (2001)
  17. Richardson, A. D.; Dormand, J. R.; Shariff, M. H. B. M.: Efficient variable stiffness methods for cooling of hot-rolled steel sections. (2000)
  18. Medovikov, Alexei A.: High order explicit methods for parabolic equations (1998)
  19. Sommeijer, B. P.; Shampine, L. F.; Verwer, J. G.: RKC: An explicit solver for parabolic PDEs (1998)
  20. Verwer, J. G.: Explicit Runge-Kutta methods for parabolic partial differential equations (1996)

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