ORDER RESULTS FOR KRYLOV-W-METHODS

We consider ROW-methods for stiff initial value problems, where the st age equations are solved by Krylov techniques. By using a certain 'mul tiple 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 d epend on the parameters of the method only, not on the dimensions of t he ODE. Stability restrictions usually require larger dimensions, of c ourse, but this can be done adaptivey. These results justify to adopt the step size control of the underlying ROW-method. The widely used RO W-methods of order 4 are discussed in detail and numerical illustratio ns are given. For the special class of semilinear systems with stiffne ss in a constant linear part we establish the order 2 of B-consistency for these Krylov-W-methods.