American Swaption Pricing with PDE discretization

So I am still trying to price an american swaption. (MC approach here: American Swaption Pricing with Monte-Carlo method)

I've found in Paul Wilmott, The mathematics of financial derivatives, a PDE for swaption:

$$ \partial_tV + \frac{1}{2}w^2\partial_{rr}V+(u-\lambda w) \partial_rV-rV = 0 $$

with terminal condition:

$$ V(r,T)= max(V_S(r,T)-E,0) $$

where $V_S(r,t)$ is the price of a swap of maturity $T_S$. I think I can use the same formula:

$$ V_t = (\delta \sum_{j=n+1}^{M+1} Z_t^{T_j})[R(t,T_n,T_m) \Phi(d_1) - \hat{R} \Phi(d_2)] $$

Source

1) Is it ok to work with that ? Where is handled the fact that we have accrual payments ? what would be the expression of E ? $V_S(\hat{r},T)$ ? I think in my case that $T=T_S$ is this possible ? this would mean that the last value is 0, no ?

The code seems simple to change between European and american option.

2) The notations are not consistants, what would be the link between r,Z,R ?

Then I think we have to choose a model for r (such that $w², u-\lambda w$ are simple).Vasicek for exemple will give:

$$ w^2 = \alpha_0 $$ $$ u - \lambda w = -\gamma_0 r(t) + \eta_0 $$

3) Is this a good approach ? In term of simulating Z,R after r ? In term of calibrating the model ? (I understand that it depends on what data I will have access to, but for the moment I don't know that, I am not working with swaption for financial market).

I think after the choice of the model it won't be a problem to discretize my PDE.