The Fortran 90 code IRKC is intended for the time integration of systems of partial differential equations (PDEs) of diffusion-reaction type for which the reaction Jacobian has real (negative) eigenvalues. It is based on a family of implicit-explicit Runge-Kutta-Chebyshev methods which are unconditionally stable for reaction terms and which impose a stability constraint associated with the diffusion terms that is quadratic in the number of stages. Special properties of the family make it possible for the code to select at each step the most efficient stable method as well as the most efficient step size. Moreover, they make it possible to apply the methods using just a few vectors of storage. A further step towards minimal storage requirements and optimal efficiency is achieved by exploiting the fact that the implicit terms, originating from the stiff reactions, are not coupled over the spatial grid points. Hence, the systems to be solved have a small dimension (viz., equal to the number of PDEs). These characteristics of the code make it especially attractive for problems in several spatial variables. IRKC is a successor to the RKC code of {\itB. P. Sommeijer, L. F. Shampine}, and {\itJ. G. Verwer}, RKC: an explicit solver for parabolic PDEs, J. Comput. Appl. Math. 88, No. 2, 315--326 (1997; Zbl 0910.65067)] that solves similar problems without stiff reaction terms.

References in zbMATH (referenced in 17 articles , 1 standard article )

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  1. Braś, Michał; Izzo, Giuseppe; Jackiewicz, Zdzisław: Accurate implicit-explicit general linear methods with inherent Runge-Kutta stability (2017)
  2. 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)
  3. Ruprecht, Daniel; Speck, Robert: Spectral deferred corrections with fast-wave slow-wave splitting (2016)
  4. 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)
  5. Kuehn, Christian: Multiple time scale dynamics (2015)
  6. O’Sullivan, Stephen: A class of high-order Runge-Kutta-Chebyshev stability polynomials (2015)
  7. Xiao, Aiguo; Zhang, Gengen; Yi, Xing: Two classes of implicit-explicit multistep methods for nonlinear stiff initial-value problems (2014)
  8. González-Pinto, S.; Pérez-Rodríguez, S.: A variable time-step-size code for advection-diffusion-reaction PDEs (2012)
  9. Sgura, Ivonne; Bozzini, Benedetto; Lacitignola, Deborah: Numerical approximation of Turing patterns in electrodeposition by ADI methods (2012)
  10. Duarte, Max; Massot, Marc; Descombes, Stéphane; Tenaud, Christian; Dumont, Thierry; Louvet, Violaine; Laurent, Frédérique: New resolution strategy for multi-scale reaction waves using time operator splitting and space adaptive multiresolution: application to human ischemic stroke (2011)
  11. Zbinden, Christophe J.: Partitioned Runge-Kutta-Chebyshev methods for diffusion-advection-reaction problems (2011)
  12. Gonzalez-Pinto, S.; Perez-Rodriguez, S.: A time-adaptive integrator based on Radau methods for advection diffusion reaction PDEs (2009)
  13. Martín-Vaquero, J.; Janssen, B.: Second-order stabilized explicit Runge-Kutta methods for stiff problems (2009)
  14. Perez-Rodriguez, S.; Gonzalez-Pinto, S.; Sommeijer, B.P.: An iterated Radau method for time-dependent PDEs (2009)
  15. Seaïd, Mohammed: An Eulerian-Lagrangian method for coupled parabolic-hyperbolic equations (2009)
  16. Sommeijer, B.P.; Verwer, J.G.: On stabilized integration for time-dependent PDEs (2007)
  17. Shampine, L.F.; Sommeijer, B.P.; Verwer, J.G.: IRKC: an IMEX solver for stiff diffusion-reaction PDEs (2006)