絶対値の微分方式の精密

微分方式の解答を見積もるために二つの絶対値の方法を比較します。
The way be which error in numerical approximations to the solutions of ordinary differential equations is measured involves convergence. Namely, as n goes to infinity, do measured points yn approach actual f(tn) values in the analytic solution to the equation? The particular differential equations under study are initial value problems:
y'=f(t,y) y(t0) = y0
The questions regarding convergence are:
1. does the estimated solution converge
2. how rapidly does it converge (or how small a step size is needed to obtain a particular closeness). The means of expressing this is to state how error increases as the step size (h) increases. Error can be expressed as either global truncation error or local truncation error. Global truncation error is the max over all n of phi(tn)-yn, where phi(tn) is the actual value of the solution function at tn and yn is the approximate value (discounting any rounding error). Local truncation error assumes that yn is exactly phi(n) and computes the error of moving one step due to approximation.
Two ways of estimating the solution are studied: Euler method and two-step Adams-Bashforth.
Euler estimates are found by:
y(n+1) = yn + h(y'(tn,yn))
Adams-Bashforth are:
y(n+1) = yn + h((3/2)y'(tn,yn)-(1/2)y'(t(n-1),y(n-1)))
フォートランと言う言語でプログラムを作って１０回の出力を比較しました。　Using y'=2x y(to)=1 t0=1 as the problem (solution is y=x^2), outputs for Euler and Adams-Bashforth (2-step) were as follows:
t Euler A-B
1 1 1
2 3 4
3 7 9
4 13 16
5 21 25
The above pattern continues up to t=10.
So, while Euler method has some error for each t value, Adams-Bashforth produces exact results. The goal is to prove that, due to an inherent property in its calculation method, Euler's method is consistently less accurate than Adams Bashforth.
In fact, it can be shown that the Euler method's local truncation error is O(h^2), while the same for the Adams Bashforth method is O(h^3). Thus, as h is decreased to be less than one, the error for Adams Bashforth becomes much smaller than that for Euler. If h=1/10, Adams-Bashforth would have local truncation error of about 1/1000, while Euler would have local truncation error of about 1/100.