I am trying to take a closer look at multi-step methods.
A two-step method is given by $$ \alpha_0 y^{(j)} + \alpha_1 y^{(j+1)} + y^{(j+2)} = \tau ( \beta_0 f^{(j)} +\beta_1 f^{(j+1)} + \beta_2 f^{(j+2)} )$$ $ \alpha_2 =1 $
The aim is to determine the parameters $ \alpha_0 , \alpha_1 ,\beta_0 , \beta_1 , \beta_2 $, so that the method approaches the maximal consistency order $p \in \mathbb{N} $.
A linear multistep method has a consistency order of at least $p$ with $$ \sum_{l=0}^{k} \alpha_l =0, \sum_{l=0}^{k} \alpha_l l^q = q \sum_{l=0}^{k}\beta_l l^{q-1} $$.
I am not sure how to determine those parameters. Don't I need to know what $p$ is, to be able to put up an equation system?
Any Help is very appreciated ! :-)
You have 5 parameters to determine, which means you can impose 5 linear conditions to fix these. As you can confirm by solving the system, the first equation and the second for $q=1,2,3,4$ are independent and thus give a unique solution.