CFD Lesson 5: Explicit / Implicit schemes
Lesson created by Lorena Barba using
Video from Boston University YouTube Channel
The simplest finite-difference schemes for PDEs in space and time result in an algebraic equation that updates all points in space, based on the information at the previous time step. The update is explicit and only one unknown is present in each difference equation. Schemes that also use information from the current step in the update are called implicit. Let's look at schemes for 1D diffusion.
Additional Resources for you to Explore
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!