By J. Hymers
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4, this optimistic prediction is invalid. What is the fallacy in the random walk argument? Be speci c, citing an equation in the text to support your answer. 6. Prove the lemma alluded to on p. 3. ACCURACY AND CONSISTENCY TREFETHEN 1994 29 of degree p +1. ) . 7. If you have access to a symbolic computation system, carry out the suggestion on p. 26: write a short program which, given the parameters j and j for a linear multistep formula, computes the coe cients C0 C1 :::. 2, and then explore other linear multistep formulas that interest you.
3. 1, CONTINUED. First let us analyze the local accuracy of the Euler formula in a lowbrow fashion that is equivalent to the de nition above: we suppose that v0 ::: vn are exactly equal to u(t0 ) ::: u(tn ), and ask how close vn+1 will then be to u(tn+1 ). 7) gives u(tn+1 ) = u(tn )+ kut (tn )+ k2 utt (tn )+ O(k3 ): 2 Subtraction gives 2 u(tn+1 ) vn+1 = k2 utt (tn )+ O(k3 ) ; as the formal local discretization error of the Euler formula. Now let us restate the argument in terms of the operator .
2. 6. 4. DERIVATION OF LINEAR MULTISTEP FORMULAS TREFETHEN 1994 38 Proof. Let (z) and (z) = zs be the characteristic polynomials corre- sponding to the s-step backwards di erentiation formula. 2, since log z = ; log z 1 , the order of accuracy is p if and only if (z) = ; log z 1 + ((z ; 1)p+1) zs h i = ; (z 1 ; 1) ; 12 (z 1 ; 1)2 + 13 (z 1 ; 1)3 ; + ((z ; 1)p+1) ; ; ; that is, h ; ; (z) = zs (1 ; z 1 )+ 21 (1 ; z 1 )2 + 31 (1 ; z 1 )3 + ; ; ; i + ((z ; 1)p+1): By de nition, (z) is a polynomial of degree at most s with (0) 6= 0 equivalently, it is zs times a polynomial in z 1 of degree exactly s.