Finite Difference and Spectral Methods for Ordinary and by Trefethen L.N.

By Trefethen L.N.

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Extra resources for Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations

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7, stability is the condition that all roots of (z) lie in jzj 1, with simple roots only permitted on jzj = 1. Thus the two crucial properties of linear multistep formulas have been reduced completely to algebraic questions concerning a rational function. 9 below, consist of arguments of pure complex analysis of rational functions, having nothing super cially to do with ordinary di erential equations. 3. Stability, consistency, and order of accuracy as algebraic conditions on the rational function (z)= (z).

2) is to ask: if t > 0 is a xed number, and the computation is performed with various step sizes k > 0 in exact arithmetic, will v(t) converge to u(t) as k 0 ? A natural conjecture might be that for any consistent linear multistep formula, the answer must be yes. 3, such a method commits local errors of size O(kp+1 ) with p 1, and there are a total of (k;1 ) time steps. But a simple argument shows that this conjecture is false. Consider a linear multistep formula based purely on extrapolation of previous values vn , such as y !

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.

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