Fractional step Runge-Kutta methods for time dependent coefficient parabolic problems
For evolution problems in a Hilbert space, with coercive and maximal operator which is a sum of \(n\) simpler, commuting operators, the authors investigate a class of special implicit Runge-Kutta methods and prove consistency, stability and convergence for both the discretization in time only, and for discretization in time and space. In the later case, the approximation of the \(n\) simpler operators is assumed to preserve properties like coerciveness, commutativity and Lipschitz-continuity. To illustrate the theory, the paper presents also numerical results which, however, are disapointing in the sense that they concern the Peaceman-Rachford alternating direction algorithm which, as a numerical method for two-dimensional parabolic equations was fully investigated about thirty years ago (works by D'yakonov, Samarskij, Yanenko, whereas results not assuming commutativity are also known for more than twenty years). Further, the presented numerical results concern also an own, third-order in time method, but for a parabolic equation with first-order derivatives, and here, to compensate for the first-order upwinding, \(h=\text{const}\cdot\tau^3\) is assumed.
scientific article; zbMATH DE number 1915286
scientific article