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 108 articles , 1 standard article )

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  1. Asante-Asamani, E. O.; Kleefeld, A.; Wade, B. A.: A second-order exponential time differencing scheme for non-linear reaction-diffusion systems with dimensional splitting (2020)
  2. Botchev, M. A.; Knizhnerman, L. A.: ART: adaptive residual-time restarting for Krylov subspace matrix exponential evaluations (2020)
  3. Skvortsov, L. M.: Construction and analysis of explicit adaptive one-step methods for solving stiff problems (2020)
  4. Spiteri, Raymond J.; Guo, Wenxian: Efficient partitioned numerical integrators for myocardial cell models (2020)
  5. Zhukov, V. T.; Novikova, N. D.; Feodoritova, O. B.: An approach to time integration of the Navier-Stokes equations (2020)
  6. Ait-Haddou, Rachid: New stability results for explicit Runge-Kutta methods (2019)
  7. Bermejo, R.; del Sastre, P. Galán: An implicit-explicit Runge-Kutta-Chebyshev finite element method for the nonlinear Lithium-ion battery equations (2019)
  8. Carpio, Jaime; Prieto, Juan Luis; Galán del Sastre, Pedro: An anisotropic adaptive, Lagrange-Galerkin numerical method for spray combustion (2019)
  9. Martín-Vaquero, J.; Kleefeld, A.: ESERK5: a fifth-order extrapolated stabilized explicit Runge-Kutta method (2019)
  10. Abdulle, Assyr; Almuslimani, Ibrahim; Vilmart, Gilles: Optimal explicit stabilized integrator of weak order 1 for stiff and ergodic stochastic differential equations (2018)
  11. Bhatt, H. P.; Khaliq, A. Q. M.; Wade, B. A.: Efficient Krylov-based exponential time differencing method in application to 3D advection-diffusion-reaction systems (2018)
  12. Bocher, Philippe; Montijano, Juan I.; Rández, Luis; Van Daele, Marnix: Explicit Runge-Kutta methods for stiff problems with a gap in their eigenvalue spectrum (2018)
  13. Nguyen, Van-Dang; Jansson, Johan; Hoffman, Johan; Li, Jing-Rebecca: A partition of unity finite element method for computational diffusion MRI (2018)
  14. 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)
  15. Guo, Daniel X.: On stability and convergence of semi-Lagrangian methods for the first-order time-dependent nonlinear partial differential equations in 1D (2017)
  16. Hean, Charles R.; Fahrenthold, Eric P.: Discrete Lagrange equations for reacting thermofluid dynamics in arbitrary Lagrangian-Eulerian frames (2017)
  17. Lu, Dong; Zhang, Yong-Tao: Computational complexity study on Krylov integration factor WENO method for high spatial dimension convection-diffusion problems (2017)
  18. Machen, Michael; Zhang, Yong-Tao: Krylov implicit integration factor methods for semilinear fourth-order equations (2017)
  19. Carpio, Jaime; Prieto, Juan Luis; Vera, Marcos: A local anisotropic adaptive algorithm for the solution of low-Mach transient combustion problems (2016)
  20. 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)

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