[Numerical Modeling 14] Stability condition and higher-order methods for numerical solution of PDEs

So far, we saw how the finite difference method work, and we had a general picture of how to use it for ordinary and partial differential equations. But, things can go out of control easily in practice when you want to use the method. Technically speaking, the numerical solution can become unstable. In this video, we see how to apply certain criteria to the simulations to control stability. We will also have a look at how to use higher-order methods to deal with second derivatives, like in a diffusion equation.

Educational Materials:

Topics covered:
🎯 What do convergence and stability mean in numerical computing?
🎯 The impact of not considering stability conditions in numerical simulations
🎯 The concept of the CFL number and its role in stability
🎯 Discretizing second-order derivatives
🎯 Solving a simple diffusion equation using finite difference

Chapters in this video!
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00:00 - Intro
00:30 - Stability and convergence in action
06:53 - CFL number comes into play
08:50 - Higher-order finite difference
11:32 - Second-order numerical solution of the diffusion equation