We consider ROW-methods for stiff initial value problems, where the stage equations are solved by Krylov techniques. By using a certain 'multiple Arnoldi process' over all stages the order of the Fully-implicit one-step scheme can be preserved with low Krylov dimensions. Explicit estimates for Minimal order preserving dimensions are derived. They depend on the parameters of the method Only, no ton the dimension of the ODE. Stability restrictions usually require larger dimensions, of Course, but this can be done adaptively.
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