18 1. Differential Equations and Their Solutions to define values y0,... , yN in Rd, in such a way that when N is large, each yn is close to the corresponding y(tn). The quantity max0≤n≤N y(tn) yn is called the global error of the algorithm on the interval. If the global error converges to zero as N tends to infinity (for every choice of f satisfying some Lipschitz condition, to, yo, and T T∗), then we say that we have a convergent algorithm. Euler’s Method is a convergent algorithm of this sort. One common way to construct the algorithm that produces the values y1,... , yN uses a recursion based on a so-called (one-step) “stepping procedure”. This is a discrete approximate solution opera- tor, Σ(f, yn,tn,h), having as inputs 1) a time-dependent vector field f on Rd, 2) a time tn in R, 3) a value yn in Rd corresponding to the initial time, and 4) a “time-step” h in R and as output a point of Rd that approximates the solution of the initial value problem y = f(t, y), y(ti) = yi at ti + h well when h is small. (More precisely, the so-called “local truncation error”, σ(f, y(tn),tn,tn + h) Σ(f, y(tn),tn,h) , should approach zero at least superlinearly in the time-step h.) Given such a stepping pro- cedure, the approximations yn of the y(tn) are defined recursively by yn+1 = Σ(f, yn,tn,h). Numerical integration methods that use discrete approximations of derivatives defining the vector field f to obtain the operator Σ are referred to as finite difference methods. 1.2.1. Remark. Notice that there will be two sources that con- tribute to the global error, y(tn) yn . First, at each stage of the recursion there will be an additional local truncation error added to what has already accumulated up to that point. Moreover, because the recursion uses yn rather than y(tn), after the first step there will be an additional error that includes accumulated local truncation er- rors, in addition to amplification or attenuation of these errors by the method. (In practice there is a third source of error, namely machine round-off error from using floating-point arithmetic. Since these are
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