• Mar 30th 2011, 12:07 PM
hazeleyes
Hey Everyone I could really use some help answering this question as I have tried to answer it several times and I can't seem to understand exactly what to do. so if someone could please show me a step by step calculation(Bow)

The initial value problem y'= f(x,y), y(X0)=Y0 is to be solved numerically using the 2-step Adams-Moulton method

Yi+1 - Yi = h/12 [ (5 Fi+1) + (8 Fi - Fi -1)

where Fi= f(x, y) . Find the local truncation error of this method and hence determine its order.

Thanks for your help everyone, much appreciated :-)
• Apr 3rd 2011, 09:25 AM
TheEmptySet
Quote:

Originally Posted by hazeleyes
Hey Everyone I could really use some help answering this question as I have tried to answer it several times and I can't seem to understand exactly what to do. so if someone could please show me a step by step calculation(Bow)

The initial value problem y'= f(x,y), y(X0)=Y0 is to be solved numerically using the 2-step Adams-Moulton method

Yi+1 - Yi = h/12 [ (5 Fi+1) + (8 Fi - Fi -1)

where Fi= f(x, y) . Find the local truncation error of this method and hence determine its order.

Thanks for your help everyone, much appreciated :-)

You need to expand all of these in a taylor series.

If $h$ is your stepsize then $t_{i+1}=t_i+h$

so

$y(t_{i+1})=y(t_i+h)=y(t_i)+hy'(t_i)+\frac{h^2}{2!} y''(t_i)+\frac{h^3}{3!}y'''(t_i)+\mathcal{O}(h^4)$

Now use the definition of the ODE to get

$5f_{i+1}=5y'(t_i+h)=5\left(y'(t_i)+hy''(t_i)+\frac {h^2}{2!}y'''(t_i)+\frac{h^3}{3!}y^{(4)}(t_i)+\mat hcal{O}(h^5) \right)$

$-f_{i-1}=-y'(t_i-h)=-\left(y'(t_i)-hy''(t_i)+\frac{h^2}{2!}y'''(t_i)-\frac{h^3}{3!}y^{(4)}(t_i)+\mathcal{O}(h^5) \right)$

$8f_{i}=8y'(t_i)$

Now plug all of this in and see how well it matches

On the RHS I get

$hy'(t_i)+\frac{h^2}{2!}y''(t_i)+\frac{7h^3}{24}y'' '(t_i)+\mathcal{O}(h^4)$

so they only match up to the 2nd orderterms