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ISBN-13: 9781884550454

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Example text

DERIVATION OF LINEAR MULTISTEP FORMULAS TREFETHEN 1994 37 Proof. 4. 2. Besides Adams formulas, the most important family of linear multistep formulas dates to Curtiss and Hirschfelder in 1952, and is also associated with the name of C. W. Gear. The s-step backwards di erentiation formula is the optimal implicit linear multistep formula with 0 = = s 1 = 0. ) Unlike the Adams formulas, the backwards differentiation formulas allocate the free parameters to the f j g rather than the f j g. These formulas are \maximally implicit" in the sense that the function f enters the calculation only at the level n +1.

3 we mentioned that an s-step linear multistep formula can have order 2s why not simply use that highorder formula and forget special classes like Adams and Nystrom methods? Dahlquist's famous theorem con rms that the answer is an impassable barrier of stability. 9. The order of accuracy p of a stable s-step linear multistep formula satis es 8 > < s +2 if s is even, p > s +1 if s is odd, : s if the formula is explicit. (1:5:11) Proof. 9 have been published, beginning with the original one by Dahlquist in 1956.

21). 7. 9 in the text covered only the case of an explicit linear multistep formula. Show what modi cations are necessary for the implicit case. 6. 6. Convergence and the Dahlquist equivalence theorem Up to this point we have talked about accuracy, consistency, and stability, but we have yet to establish that a linear multistep formula with these admirable properties will actually work. After one more de nition we shall be able to remedy that. To set the stage, let us return to the footnote on p.