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When I have found the price of an American option using, say, a finite difference scheme - how do I find the early exercise curve from this solution?

Here is my idea: What I have is the price of the option on a $(t_i, s_j)$-grid. Denote this solution $p_{ij}$. What I can do is for each $i$ to compare the solution to the theoretical payoff $g(s)$. The point where the difference $|g(s_j)- p_{ij}|$ is less than some threshold will trace out the curve.

I guess this will work, but not sure how thorough it is. Are there better methods?

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