Finite difference methods with discontinuity in the payoff function

I have implemented a finite difference scheme for pricing options using a Black-Scholes-like model. I tested my implementation on a call option, and found that it gave extremely inaccurate results. I investigated intermediate values in my computations, and I suspect that my inaccurate results are caused by the discontinuity in the payoff function. The payoff function: $$\text{Payoff}(S) = \max(S-K, 0) \hspace{.4em} \text{for some strike price } K \text{.} $$ My scheme requires that I calculate $