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Let

$x'(t)=f(t,x(t)), t\in(0,T)$ with $x(0)=x_0$

$f$ satifies the Lipschitz-condition $f(t,x)-f(t,y)\le L|x-y|$

$h\in (0,\frac{1}{L})$ is the step size and the approximation $x_k$ for $x(t_k)=hk$ is given by $x_k=x_{k-1}+hf(t_k,x_k)$.

Now I would be very interested how to derive the error

$$|x_k-x(t_k)|\le\frac{1}{1-Lh}\left(|x_{k-1}-x(t_{k-1})|+\frac{h^2}{2} \max_{s\in [0,T]}|x''(s)|\right)$$

I tried to look up it up in some numerical analysis books but it is always different

1 Answers 1