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.

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  1. González-Pinto, S.; Hernández-Abreu, D.: Splitting-methods based on approximate matrix factorization and Radau-IIA formulas for the time integration of advection diffusion reaction PDEs (2016)
  2. Hansen, M.A.; Sutherland, J.C.: Pseudotransient continuation for combustion simulation with detailed reaction mechanisms (2016)
  3. Huang, Weizhang; Kamenski, Lennard; Lang, Jens: Stability of explicit one-step methods for P1-finite element approximation of linear diffusion equations on anisotropic meshes (2016)
  4. Kleefeld, B.; Martín-Vaquero, J.: SERK2v3: Solving mildly stiff nonlinear partial differential equations (2016)
  5. Lopez, Luciano; Vacca, Giuseppe: Spectral properties and conservation laws in mimetic finite difference methods for PDEs (2016)
  6. Weiner, Rüdiger; Bruder, Jürgen: Exponential Krylov peer integrators (2016)
  7. Zhang, Hong; Sandu, Adrian; Blaise, Sébastien: High order implicit-explicit general linear methods with optimized stability regions (2016)
  8. Gonzalez-Pinto, S.; Hernandez-Abreu, D.; Perez-Rodriguez, S.: AMF-Runge-Kutta formulas and error estimates for the time integration of advection diffusion reaction PDEs (2015)
  9. Kulikov, G.Yu.: Embedded symmetric nested implicit Runge-Kutta methods of Gauss and Lobatto types for solving stiff ordinary differential equations and Hamiltonian systems (2015)
  10. Zhang, Limei; Ma, Fuming: Pouzet-Runge-Kutta-Chebyshev method for Volterra integral equations of the second kind (2015)
  11. Beck, S.; González-Pinto, S.; Pérez-Rodríguez, S.; Weiner, R.: A comparison of AMF- and Krylov-methods in Matlab for large stiff ODE systems (2014)
  12. Martín-Vaquero, J.; Khaliq, A.Q.M.; Kleefeld, B.: Stabilized explicit Runge-Kutta methods for multi-asset American options (2014)
  13. Niemeyer, Kyle E.; Sung, Chih-Jen: GPU-based parallel integration of large numbers of independent ODE systems (2014)
  14. Zhang, Hong; Sandu, Adrian; Blaise, Sebastien: Partitioned and implicit-explicit general linear methods for ordinary differential equations (2014)
  15. Abdulle, Assyr; Vilmart, Gilles: PIROCK: A swiss-knife partitioned implicit-explicit orthogonal Runge-Kutta Chebyshev integrator for stiff diffusion-advection-reaction problems with or without noise (2013)
  16. Kleefeld, B.; Martín-Vaquero, J.: SERK2v2: A new second-order stabilized explicit Runge-Kutta method for stiff problems (2013)
  17. Nazari, Farshid; Mohammadian, Abdolmajid; Zadra, Ayrton; Charron, Martin: A stable and accurate scheme for nonlinear diffusion equations: application to atmospheric boundary layer (2013)
  18. Cai, Zhenning; Li, Ruo; Wang, Yanli: An efficient NR$xx$ Method for Boltzmann-BGK equation (2012)
  19. González-Pinto, S.; Pérez-Rodríguez, S.: A variable time-step-size code for advection-diffusion-reaction PDEs (2012)
  20. Chen, Shanqin; Zhang, Yong-Tao: Krylov implicit integration factor methods for spatial discretization on high dimensional unstructured meshes: Application to discontinuous Galerkin methods (2011)

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