W-METHODS WITH AUTOMATIC PARTITIONING BY KRYLOV TECHNIQUES FOR LARGE STIFF SYSTEMS

Solving the stage equations in W-methods approximately by a Krylov pro cess may be interpreted as an automatic partitioning method, where som e of the components are integrated by an implicit scheme whereas other s are treated by an explicit Runge-Kutta method. The authors consider an implementation which uses only one family of Krylov spaces for all stages, introducing an additional error of the size of the discretizat ion error. Two main results for the linear autonomous case show that ( i) the method stays at the asymptotic limit solution under mild restri ctions, which may be enforced in the numerical computation, and (ii) t he dimensions of the Krylov spaces need only be slightly larger than t he number of ''fast'' solution components with nonnegligible contribut ion to the local solution. Numerical examples of dimensions 20 to 302 indicate that the method may perform well even for Krylov spaces with low dimension.