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 92 articles , 1 standard article )

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  1. Abdulle, Assyr; Almuslimani, Ibrahim; Vilmart, Gilles: Optimal explicit stabilized integrator of weak order 1 for stiff and ergodic stochastic differential equations (2018)
  2. González-Pinto, S.; Hernández-Abreu, D.; Pérez-Rodríguez, S.: W-methods to stabilize standard explicit Runge-Kutta methods in the time integration of advection-diffusion-reaction PDEs (2017)
  3. Guo, Daniel X.: On stability and convergence of semi-Lagrangian methods for the first-order time-dependent nonlinear partial differential equations in 1D (2017)
  4. Lu, Dong; Zhang, Yong-Tao: Computational complexity study on Krylov integration factor WENO method for high spatial dimension convection-diffusion problems (2017)
  5. Machen, Michael; Zhang, Yong-Tao: Krylov implicit integration factor methods for semilinear fourth-order equations (2017)
  6. Carpio, Jaime; Prieto, Juan Luis; Vera, Marcos: A local anisotropic adaptive algorithm for the solution of low-Mach transient combustion problems (2016)
  7. 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)
  8. Hansen, M. A.; Sutherland, J. C.: Pseudotransient continuation for combustion simulation with detailed reaction mechanisms (2016)
  9. 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)
  10. Jiang, Tian; Zhang, Yong-Tao: Krylov single-step implicit integration factor WENO methods for advection-diffusion-reaction equations (2016)
  11. Kleefeld, B.; Martín-Vaquero, J.: SERK2v3: Solving mildly stiff nonlinear partial differential equations (2016)
  12. Lopez, Luciano; Vacca, Giuseppe: Spectral properties and conservation laws in mimetic finite difference methods for PDEs (2016)
  13. Martín-Vaquero, J.; Kleefeld, B.: Extrapolated stabilized explicit Runge-Kutta methods (2016)
  14. Motheau, E.; Abraham, J.: A high-order numerical algorithm for DNS of low-Mach-number reactive flows with detailed chemistry and quasi-spectral accuracy (2016)
  15. Schneiders, Lennart; Günther, Claudia; Meinke, Matthias; Schröder, Wolfgang: An efficient conservative cut-cell method for rigid bodies interacting with viscous compressible flows (2016)
  16. Weiner, Rüdiger; Bruder, Jürgen: Exponential Krylov peer integrators (2016)
  17. Zhang, Hong; Sandu, Adrian; Blaise, Sébastien: High order implicit-explicit general linear methods with optimized stability regions (2016)
  18. Cheng, Yuanzhen; Kurganov, Alexander; Qu, Zhuolin; Tang, Tao: Fast and stable explicit operator splitting methods for phase-field models (2015)
  19. 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)
  20. 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)

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