Pooley et al., in Convergence Remedies for non-smooth payoffs in option pricing suggest that placing the strike on the grid for a Vanilla option is good enough:
At the same time, Tavella and Randall show in their book that numerically, placing the strike in the middle of two nodes leads to a more accurate result. My own numerical experiments confirm Tavella and Randall suggestion.
In reality, what Pooley et al. really mean, is that quadratic convergence is maintained if the strike is on the grid for vanilla payoffs, contrary to the case of discontinuous payoffs (like digital options) where the convergence decreases to order 1. So it's ok to place the strike on the grid for a vanilla payoff, but it's not optimal, it's still better to place it in the middle of two nodes. Here are absolute errors in a put option price:
on grid, no smoothing 0.04473021824995271
on grid, Simpson smoothing 0.003942854282069419
on grid, projection smoothing 0.044730218065351934
middle, no smoothing 0.004040359609906119
As expected (and mentioned in Pooley et al.), the projection does not do anything here. When the grid size is doubled, the convergence ratio of all methods is the same (order 2), but placing the strike in the middle still increases accuracy significantly.
Here is are the same results, but for a digital put option:
on grid, no smoothing 0.03781319921461046
on grid, Simpson smoothing 8.289052335705427E-4
on grid, projection smoothing 1.9698293587372406E-4
middle, no smoothing 3.5122153011418744E-4
Here only the 3 last methods are of order-2 convergence, and projection is in deed the most accurate method, but placing the strike in the middle is really not that much worse, and much simpler.
A disadvantage of Simpson smoothing (or smoothing by averaging), is that it breaks put-call parity (see the paper "Exact Forward and Put Call Parity with TR-BDF2")
I think the emphasis in their paper on "no smoothing is required" for vanilla payoffs can be misleading. I hope I have clarified it in this post.