I am trying to determine how to find the accuracy of a LMM. Specifically the BDF2 method, $$y_{n+2}-\frac{4}{3}y_{n+1}+\frac{1}{3}y_n=\frac{2}{3}kf_{n+2}$$ for solving the problem $y'=f(y,x),y(0)=n$ where f is Lipshitz continuous.
I was able to show it was consistent using the characteristic polynomials, but I am not sure how to find the order of accuracy in general for LMM.
Any help on this would be much appreciated. I know how to do this for 1 step methods, but I am struggling on this two step one.
Let $a_i$ be the coefficient of $y_{n+i}$ and $b_i$ be the coefficient of $f_{n+i}$, where $i \in \{0,\dots,k\} = \{0,1,2\}$. Then, define $d_j = \sum_{i=0}^k \frac{i^j}{j!}a_i - \frac{i^{j-1}}{(j-1)!}b_i$, $d_0 = \sum_{i=0}^k a_i$. Then, the least $j$ such that $d_j\neq 0$ is the order of the method.
Edit: To specify, if $k$ is the step size, then the error is $\mathcal{O}(k^j)$ where $j$ is the smallest number such that $d_j\neq 0$, as above. Some consider this to mean $j$ is the order of the method, while others consider $j-1$ to be the order. Consult your text/professor for clarification.