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The original paper by Crank and Nicolson appeared in 1947 in the Mathematical Proceedings of the Cambridge Philosophical Society. It must be one of its most popular papers, with almost 1500 citations on Google Scholar today! The Crank-Nicolson method is very popular to solve parabolic PDEs, and it has found application even in the financial mathematics world, where it is used for options pricing. The main issues to consider when choosing between an explicit and implicit method are the time-step restrictions on explicit methods due to stability constraints, and the computational effort needed for implicit methods to solve the linear systems at each time step. Often the best choice depends on the characteristics of the solution.
The choice between explicit and implicit methods has historically been a source of much debate in CFD!
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