Stability properties of the extrapolation method

Abstract

In comparing the stability regions for the GBS-method to regions for normal one- or multi-step methods one has to keep in mind that they refer to the "basic steps" Δt_n (and not to the individual h_u) so that a large number of "function evaluations" enter into one such step (e.g.112 for m=7 with sequence F₁ !). Thus the most significant conclusion from fig. 2 and 3 is that the regions

of absolute stability do not reach very far into the negative half-plane. For an effective treatment of an equation similar to y′ = −y, e.g., one would hope to be able to use steps Δt of order 10; but this should be dangerous according to our results. On the other hand, one is rather surprised that the stability structure is no worse, considering that down at the bottom of the whole method lies the mid-point rule the absolute stability region of which is empty !

In effect, our results show that the GBS-method — as most other standard methods — should certainly not be used with stiff systems of equations but that in all other situations it will not significantly suffer from stability difficulties (and for its other merits be an exceedingly good method). This conclusion is in agreement with a large body of practical experience gained in various places.

The research reported in this paper has been sponsored in part by the United States Government under Contract 61(052)-960.