Given the Finite Difference approximation of the Black Scholes equation with time discretization $\Delta t*n, n = 1,2,3,...,N$ and in space $\Delta s*j, j = 1,2,3,...,M$
$$\frac{V^n_j-V^{n-1}_j}{\Delta t} +rS_j\bigg(\frac{V^n_{j+1}-V^{n}_{j-1}}{2\Delta s}\bigg) +\frac{1}{2}\sigma ^2S_j^2\bigg(\frac{V^n_{j+1}-2V^{n}_{j}+V^{n}_{j-1}}{2\Delta s}\bigg) + rV^n_j = 0$$
which has explicit solution backward from final conditions
$$V^{n-1}_j = V^n_j +r\Delta tS_j\bigg(\frac{V^n_{j+1}-V^{n}_{j-1}}{2\Delta s}\bigg) +\frac{1}{2}\sigma ^2\Delta tS_j^2\bigg(\frac{V^n_{j+1}-2V^{n}_{j}+V^{n}_{j-1}}{2\Delta s}\bigg) + \Delta trV^n_j $$
I wish to implement a BFD2 time discretization, (as opposed to the first order approximation in time used above in the first term) i.e
$$\frac{dV}{dt} \approx \frac{3V^{n-2}_j-4V^{n-1}+V^{n}}{2 \Delta t}$$
I do this by replacing the first term in the first equation with the BFD2 approximation and solve for $V^{n-2}_j$ to obtain:
$$V^{n-2}_j = \frac{-2}{3}\bigg(-4V^{n-1}_j + V^n_j +r\Delta tS_j\bigg(\frac{V^n_{j+1}-V^{n}_{j-1}}{2\Delta s}\bigg) +\frac{1}{2}\sigma ^2\Delta tS_j^2\bigg(\frac{V^n_{j+1}-2V^{n}_{j}+V^{n}_{j-1}}{2\Delta s}\bigg) + \Delta trV^n_j\bigg) $$
however this "solution" diverges horribly for all types of discretization. Is my interpretation and implementation of BFD2 correct? I use the final conditions for $V^N_j$ ad then second equation to calculate $V^{N-1}_j$ and then I apply the second order scheme, since it depends on two time steps.
So the reason this did not work was becuse BFD2 is an implicit method in time, and I was trying to implement it as a fully explicit method, i.e explicit in both space and time. In order to utilize BFD2 for the time discretization I thus had to write the whole thing it as an implicit method instead. If anyone has any questions on the details I would be happy to elaborate.