An Example of Von Neumann Stability Analysis (For Linear Partial Differential Equations)

Hello! This is an example of von Neumann Stability Analysis used on the Lax-Friedrichs Method for a linear, hyperbolic partial differential equation. Many thanks to Dr. Gardner for sharing this wonderful technique.
I would like to point out something here for the sake of rigor. For a in the set of real numbers, abs(a)^2 is less than or equal to 1 if and only if abs(a) is less than or equal to 1, this is why we can use |G(k)|^2 to make assumptions about |G(k)|. Additionally, I should have said, "The ideal is closer to 1 but not greater than 1." For linear PDEs, 1 actually gets you the exact solution.
Also, I believe that the stencil was cut off in the video, please see www.researchgate.net/figure/S... if you are curious.

[@estellarodriguez-gz4hc]

Hey Mr. Mercer. I’m wondering why we use |G(k)|^2 at the end?

[@MaxMercerPiano]

Hi Estella, thanks for asking! For a number which has magnitude less than 1, the magnitude of the square is less than or equal to the magnitude of the original number. For instance, 0.5^2 is 0.25. This also applies to negative numbers (which is important because even if the growth factor is negative, it’s still technically stable as long as the growth factor is not less than negative one). So if we can show that the square is less than one in magnitude, the original number must also be less than one in magnitude, and so we meet our stability condition. I hope this helps.

[@estellarodriguez-gz4hc]

@@MaxMercerPiano it does, thanks so much!