Pricing Put Options with the Implicit Finite-Difference Method
This Demonstration shows the impact of time to expiry, strike price, volatility, risk-free rate, and dividend:
• on the value of the American put (red line) and the European put (black dashed line) against the starting asset price,
• on the location of the early exercise boundary (blue dots) during the life of the American put. The early exercise is not optimal while the asset price moves above this boundary.
The implicit finite-difference method  is applied to solve the Black–Scholes-Merton partial differential equation, using a uniform price and time grid. The implicit finite-difference method requires the iterative solution of linear equations linking consecutive time steps, whereas the explicit finite-difference method provides an explicit formula for determining future states of the option process in terms of the current state.
 M. Brennan and E. Schwartz, "Finite Difference Methods and Jump Processes Arising in the Pricing of Contingent Claims: A Synthesis", The Journal of Financial and Quantitative Analysis, 13(3), 1978 pp. 461–474.