IRKC: An IMEX solver for stiff diffusion–reaction PDEs

AbstractThe 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 [B.P. Sommeijer, L.F. Shampine, J.G. Verwer, RKC: an explicit solver for parabolic PDEs, J. Comput. Appl. Math. 88 (1997) 315–326] that solves similar problems without stiff reaction terms