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IRKC: an IMEX solver for stiff diffusion-reaction PDEs. (English) Zbl 1100.65075
Summary: 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 B. P. Sommeijer, L. F. Shampine, and J. 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.

65M20 Method of lines for initial value and initial-boundary value problems involving PDEs
35K57 Reaction-diffusion equations
65L06 Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
Full Text: DOI
[1] Brown, P.N.; Byrne, G.D.; Hindmarsh, A.C., VODE, a variable coefficient ODE solver, SIAM J. sci. statist. comput., 10, 1038-1051, (1989) · Zbl 0677.65075
[2] Byrne, G.D., Pragmatic experiments with Krylov methods in the stiff ODE setting, (), 323-356 · Zbl 0769.65038
[3] A. Guillou, B. Lago, Domaine de stabilité associé aux formules d’intégration numérique d’équations différentielles, a pas séparés et a pas liés, 1er Congr. Assoc. Fran. calcul AFCAL, Grenoble, September 1960, 1961, pp. 43-56.
[4] W. Hundsdorfer, J.G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations, Springer Series in Computational Mathematics, vol. 33, Springer, Berlin, 2003. · Zbl 1030.65100
[5] Mousseau, V.A.; Knoll, D.A.; Rider, W.J., Physics-based preconditioning and the newton – krylov method for non-equilibrium radiation diffusion, J. comput. phys., 160, 743-765, (2000) · Zbl 0949.65092
[6] Saad, Y.; Schultz, M., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. sci. statist. comput., 7, 856-869, (1986) · Zbl 0599.65018
[7] Shampine, L.F., Numerical solution of ordinary differential equations, (1994), Chapman & Hall New York · Zbl 0826.65082
[8] Shampine, L.F.; Baca, L.S., Error estimators for stiff differential equations, J. comput. appl. math., 11, 197-207, (1984) · Zbl 0556.65065
[9] Sommeijer, B.P.; Shampine, L.F.; Verwer, J.G., RKC: an explicit solver for parabolic pdes, J. comput. appl. math., 88, 315-326, (1997) · Zbl 0910.65067
[10] Verwer, J.G.; Sommeijer, B.P., An implicit – explicit runge – kutta – chebyshev scheme for diffusion – reaction equations, SIAM J. sci. comput., 25, 1824-1835, (2003) · Zbl 1061.65090
[11] Zonneveld, J.A., Automatic numerical integration, (1964), Mathematisch Centrum Amsterdam · Zbl 0139.31901
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