We have the linear multi-step process
$-y_k-y_{k+1}+y_{k+2}+y_{k+3}=4hf(t_k,y_k)$
for the initial value problem $y'(t)=f(t,y(t)), y(0)=0$
Check the consistency of the process and the criterion of Dahlquist of the characteristical polynomial.
Checking the criterion of Dahlquist is easy. The characteristical polynomial of this process is given by:
$\rho(x)=x^3+x^2-x-1=(x-1)(x+1)^2$ with the roots $\lambda_1=1$ and $\lambda_{2,3}=-1$.
So $|\lambda_i|\leq 1$ but we have a double root with $|\lambda_2|=1$ so the criterion of Dahlquist does not apply here.
I struggle with the consistency of this process.
I have to show that for
$\tau_h(t,y(t))=\frac1h\sum_{j=0}^m \alpha_j y(t+jh)-\sum_{j=0}^m \beta_j f(t+jh,y(t+jh))$
it is
$\lim_{h\to\infty} \tau_h =0$
Correct?
Is there an easier way? Thanks in advance.
A linear multi-step method is consistent iff $\rho(0)=0$ and $\rho'(1)-\sigma(1)=0$, in your case $\rho(x)=x^3+x^2-x-1$ and $\sigma(x)=4$ So $\rho(1)=0$ $\rho'(1)-\sigma(1)=3+2-1-4=0$ and and your method IS consistent
PS: Sorry for my previous blunder, I had the exam for this a few week ago, and I don't know how I could mess that up :(