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 116 articles , 1 standard article )

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  1. Almuslimani, Ibrahim; Vilmart, Gilles: Explicit stabilized integrators for stiff optimal control problems (2021)
  2. Dawes, A. S.: Some properties of the Runge-Kutta-Legendre super-time-stepping explicit methods (2021)
  3. Eftekhari, Armin; Vandereycken, Bart; Vilmart, Gilles; Zygalakis, Konstantinos C.: Explicit stabilised gradient descent for faster strongly convex optimisation (2021)
  4. 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)
  5. Botchev, M. A.; Knizhnerman, L. A.: ART: adaptive residual-time restarting for Krylov subspace matrix exponential evaluations (2020)
  6. Das, Pratik; Udaykumar, H. S.: A sharp-interface method for the simulation of shock-induced vaporization of droplets (2020)
  7. Munafò, Alessandro; Alberti, Andrea; Pantano, Carlos; Freund, Jonathan B.; Panesi, Marco: A computational model for nanosecond pulse laser-plasma interactions (2020)
  8. Skvortsov, L. M.: Construction and analysis of explicit adaptive one-step methods for solving stiff problems (2020)
  9. Spiteri, Raymond J.; Guo, Wenxian: Efficient partitioned numerical integrators for myocardial cell models (2020)
  10. Tang, Xiao; Xiao, Aiguo: Improved Runge-Kutta-Chebyshev methods (2020)
  11. Zhukov, V. T.; Novikova, N. D.; Feodoritova, O. B.: An approach to time integration of the Navier-Stokes equations (2020)
  12. Ait-Haddou, Rachid: New stability results for explicit Runge-Kutta methods (2019)
  13. Bermejo, R.; del Sastre, P. Galán: An implicit-explicit Runge-Kutta-Chebyshev finite element method for the nonlinear Lithium-ion battery equations (2019)
  14. Carpio, Jaime; Prieto, Juan Luis; Galán del Sastre, Pedro: An anisotropic adaptive, Lagrange-Galerkin numerical method for spray combustion (2019)
  15. Martín-Vaquero, J.; Kleefeld, A.: ESERK5: a fifth-order extrapolated stabilized explicit Runge-Kutta method (2019)
  16. O’Sullivan, Stephen: Runge-Kutta-Gegenbauer explicit methods for advection-diffusion problems (2019)
  17. Vermeire, Brian C.: Paired explicit Runge-Kutta schemes for stiff systems of equations (2019)
  18. Abdulle, Assyr; Almuslimani, Ibrahim; Vilmart, Gilles: Optimal explicit stabilized integrator of weak order 1 for stiff and ergodic stochastic differential equations (2018)
  19. 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)
  20. 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)

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