Any desire to further screw around with PDEs would presumably just be for educational purposes, which is fine.

I think that the owner of this thread is asking for educational advises here. We are trying to do our best to help him.

"Screwing around with PDEs" is an interesting turn of phrase.

Given that the exact solution to the valuation problem is given by equation (2), I think any further efforts at numerical solution falls into the screwing around category. Not that there is anything wrong with that! Some of my favorite memories are from screwing around.

Strictly speaking, the value function for a long position in a forward contract with maturity date T and contractual forward price K is given by [$] V(S_t,t) = e^{-r (T-t)} ( F(S_t,T-t) - K )[$], where F() is given by 2. Inspecting its behavior should be helpful in determining boundary conditions.