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AMATH 581 ( c○J. N. Kutz) 17Figure 3: Regions for stable stepping for the forward Euler and backward Eulerschemes.which after N steps leads to( ) N1y N =y 0 . (1.2.26)1 − λ∆tThe round-off error associated with this scheme is given by( ) N1E =e. (1.2.27)1 − λ∆tBy letting z = λ∆t be a complex number, we find the following criteria to yieldunstable behavior based upon (1.2.19) and (1.2.27)forward Euler: |1+z| > 1 (1.2.28a)backward Euler:1∣1 − z ∣ > 1 .(1.2.28b)Figure 3 shows the regions of stable and unstable behavior as afunctionofz.It is observed that the forward Euler scheme has a very small range of stabilitywhereas the backward Euler scheme has a large range of stability. This largestability region is part of what makes implicit methods so attractive. Thusstability regions can be calculated. However, control of the accuracyisalsoessential.1.3 Boundary value problems: the shooting methodTo this point, we have only considered the solutions of differential equations forwhich the initial conditions are known. However, many physical applicationsdo not have specified initial conditions, but rather some given boundary (constraint)conditions. A simple example of such a problem is the second-orderboundary value problemd 2 (ydt 2 = f t, y, dy )(1.3.1)dt