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. 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)
  2. Schneiders, Lennart; Günther, Claudia; Meinke, Matthias; Schröder, Wolfgang: An efficient conservative cut-cell method for rigid bodies interacting with viscous compressible flows (2016)
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  4. Zhang, Hong; Sandu, Adrian; Blaise, Sébastien: High order implicit-explicit general linear methods with optimized stability regions (2016)
  5. Cheng, Yuanzhen; Kurganov, Alexander; Qu, Zhuolin; Tang, Tao: Fast and stable explicit operator splitting methods for phase-field models (2015)
  6. 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)
  7. 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)
  8. Mex, L.; Cruz-Villar, Carlos A.; Peñuñuri, F.: Closed-form solutions to differential equations via differential evolution (2015)
  9. O’Sullivan, Stephen: A class of high-order Runge-Kutta-Chebyshev stability polynomials (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. Meyer, Chad D.; Balsara, Dinshaw S.; Aslam, Tariq D.: A stabilized Runge-Kutta-Legendre method for explicit super-time-stepping of parabolic and mixed equations (2014)
  14. Michoski, C. E.; Evans, J. A.; Schmitz, P. G.: Discontinuous Galerkin (h p)-adaptive methods for multiscale chemical reactors: quiescent reactors (2014)
  15. Nguyen, Dang Van; Li, Jing-Rebecca; Grebenkov, Denis; Le Bihan, Denis: A finite elements method to solve the Bloch-Torrey equation applied to diffusion magnetic resonance imaging (2014)
  16. Niemeyer, Kyle E.; Sung, Chih-Jen: GPU-based parallel integration of large numbers of independent ODE systems (2014)
  17. Niemeyer, Kyle E.; Sung, Chih-Jen: Accelerating moderately stiff chemical kinetics in reactive-flow simulations using GPUs (2014)
  18. Zhang, Hong; Sandu, Adrian; Blaise, Sebastien: Partitioned and implicit-explicit general linear methods for ordinary differential equations (2014)
  19. 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)
  20. Jiang, Tian; Zhang, Yong-Tao: Krylov implicit integration factor WENO methods for semilinear and fully nonlinear advection-diffusion-reaction equations (2013)