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. Kleefeld, B.; Martín-Vaquero, J.: SERK2v2: A new second-order stabilized explicit Runge-Kutta method for stiff problems (2013)
  2. Nazari, Farshid; Mohammadian, Abdolmajid; Zadra, Ayrton; Charron, Martin: A stable and accurate scheme for nonlinear diffusion equations: application to atmospheric boundary layer (2013)
  3. Cai, Zhenning; Li, Ruo; Wang, Yanli: An efficient NR(xx) method for Boltzmann-BGK equation (2012)
  4. González-Pinto, S.; Pérez-Rodríguez, S.: A variable time-step-size code for advection-diffusion-reaction PDEs (2012)
  5. Chen, Shanqin; Zhang, Yong-Tao: Krylov implicit integration factor methods for spatial discretization on high-dimensional unstructured meshes: application to discontinuous Galerkin methods (2011)
  6. Duarte, Max; Massot, Marc; Descombes, Stéphane; Tenaud, Christian; Dumont, Thierry; Louvet, Violaine; Laurent, Frédérique: New resolution strategy for multi-scale reaction waves using time operator splitting and space adaptive multiresolution: application to human ischemic stroke (2011)
  7. Houim, Ryan W.; Kuo, Kenneth K.: A low-dissipation and time-accurate method for compressible multi-component flow with variable specific heat ratios (2011)
  8. Li, Dongfang; Zhang, Chengjian; Wang, Wansheng; Zhang, Yangjing: Implicit-explicit predictor-corrector schemes for nonlinear parabolic differential equations (2011)
  9. Ray, J.; Armstrong, R.; Safta, C.; Debusschere, B. J.; Allan, B. A.; Najm, H. N.: Computational frameworks for advanced combustion simulations (2011)
  10. Zbinden, Christophe J.: Partitioned Runge-Kutta-Chebyshev methods for diffusion-advection-reaction problems (2011)
  11. Alavani, Chitra; Glowinski, Roland; Gomez, Susana; Ivorra, Benjamin; Joshi, Pallavi; Ramos, Angel Manuel: Modelling and simulation of a polluted water pumping process (2010)
  12. El-Amrani, Mofdi; Seaïd, Mohammed: An essentially non-oscillatory semi-Lagrangian method for tidal flow simulations (2010)
  13. Safta, Cosmin; Ray, Jaideep; Najm, Habib N.: A high-order low-Mach number AMR construction for chemically reacting flows (2010)
  14. Van Veldhuizen, S.; Vuik, C.; Kleijn, C. R.: On numerical issues in time accurate laminar reacting gas flow solvers (2010)
  15. Calhoun, Donna A.; Helzel, Christiane: A finite volume method for solving parabolic equations on logically Cartesian curved surface meshes (2009)
  16. Caliari, Marco; Ostermann, Alexander: Implementation of exponential Rosenbrock-type integrators (2009)
  17. Gonzalez-Pinto, S.; Perez-Rodriguez, S.: A time-adaptive integrator based on Radau methods for advection diffusion reaction PDEs (2009)
  18. Jackiewicz, Zdzisław: General linear methods for ordinary differential equations (2009)
  19. Li, Jing-Rebecca; Calhoun, Donna; Brush, Lucien: Efficient thermal field computation in phase-field models (2009)
  20. Martín-Vaquero, J.; Janssen, B.: Second-order stabilized explicit Runge-Kutta methods for stiff problems (2009)