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. Marzouk, Youssef M.; Najm, Habib N.: Dimensionality reduction and polynomial chaos acceleration of Bayesian inference in inverse problems (2009)
  2. Seaïd, Mohammed: An Eulerian-Lagrangian method for coupled parabolic-hyperbolic equations (2009)
  3. Wang, P.; Liu, Zhenxin: Stabilized Milstein type methods for stiff stochastic systems (2009)
  4. Zhu, Jianfeng; Zhang, Yong-Tao; Newman, Stuart A.; Alber, Mark: Application of discontinuous Galerkin methods for reaction-diffusion systems in developmental biology (2009)
  5. Abdulle, Assyr; Cirilli, Stephane: S-ROCK: Chebyshev methods for stiff stochastic differential equations (2008)
  6. Calhoun, Donna A.; Helzel, Christiane; LeVeque, Randall J.: Logically rectangular grids and finite volume methods for PDEs in circular and spherical domains (2008)
  7. Hochbruck, Marlis; Niehoff, Jörg: Approximation of matrix operators applied to multiple vectors (2008)
  8. Sommeijer, Ben; Verwer, Jan: A two-step RKC method for time-dependent PDEs (2008)
  9. van Veldhuizen, S.; Vuik, C.; Kleijn, C. R.: Comparison of ODE methods for laminar reacting gas flow simulations (2008)
  10. Zheng, Zheming; Simeon, Bernd; Petzold, Linda: A stabilized explicit Lagrange multiplier based domain decomposition method for parabolic problems (2008)
  11. Abdulle, Assyr; Cirilli, Stéphane: Stabilized methods for stiff stochastic systems (2007)
  12. Bermejo, R.; Carpio, J.: An adaptive finite element semi-Lagrangian implicit-explicit Runge-Kutta-Chebyshev method for convection dominated reaction-diffusion problems (2007)
  13. Ghazaryan, Anna; Sandstede, Björn: Nonlinear convective instability of turing-unstable fronts near onset: a case study (2007)
  14. Lee, J. C.; Najm, H. N.; Lefantzi, S.; Ray, J.; Frenklach, M.; Valorani, M.; Goussis, D. A.: A CSP and tabulation-based adaptive chemistry model (2007)
  15. Lee, Steven L.; Gear, C. William: Second-order accurate projective integrators for multiscale problems (2007)
  16. Marzouk, Youssef M.; Najm, Habib N.; Rahn, Larry A.: Stochastic spectral methods for efficient Bayesian solution of inverse problems (2007)
  17. Sommeijer, B. P.; Verwer, J. G.: On stabilized integration for time-dependent PDEs (2007)
  18. Torrilhon, Manuel; Jeltsch, Rolf: Essentially optimal explicit Runge-Kutta methods with application to hyperbolic-parabolic equations (2007)
  19. Botchev, M. A.; Harutyunyan, D.; van der Vegt, J. J. W.: The Gautschi time stepping scheme for edge finite element discretizations of the Maxwell equations (2006)
  20. Gear, C. W.: Towards explicit methods for differential algebraic equations (2006)