A big AHA moment hit me when you derived continuous time and discrete time system stability! I never had been taught the unit circle intuition. Thanks Dr. Brunton!
Before watching this video, I hadn't quite understood the connection between the stability of discrete systems and stability of numerical integrators, but now it got much more clear. Thanks for the lecture, professor!
This is my first time actually knowing why backward euler is mostly stable!!! In the past all I was taught was just use forward euler for simplicity and a small enough delta t and hope for the best 😂 So much wasted time tinkering the parameters without thinking…
We want our discrete approximation to be stable if our analytic solution is stable, and we want it to be unstable if our analytic solution is unstable. Forward Euler for large dt tends to be inaccurate by giving unstable approximations to stable systems, backward Euler is itself inaccurate because when dt is large it tends to give stable discrete approximations to unstable system.
MAgical as always! I wonder, what happens if we are dealing ith stochastic differential equations? So, basically adding some random process into the ODE. How should one analyse the stability of the equation then?
Thank you. How about the stability of Tustin Method approximation? and How does the original mapping from s plan to z plan by s=ln(z)/T without using the approximation can be applied and checked for stability ?
The FW Euler Integration scheme has , as prooved here, more tendency to be unstable than the BW Euler. The explanation related to the unit circle ' s radius inside the C body and the relationship to stability is great, but I have a question. Has this geometric explanation something to do with the explicit structure of the FW or on the other side with the implicit structure of the BW? In one case you can propagate forward in time and in the last case backward. It s easier to know what happend in the past when things already happend than to predict things on the future. Am I going mad?
