Error Analysis of Euler Integration Scheme for Differential Equations Using Taylor Series

In this video, we explore the error of the Forward Euler integration scheme, using the Taylor series. We show that the error at each time step scales with dt^3, where dt is the time-step of the integrator. This basic error analysis technique, based on the Taylor series, applies to much more powerful integrators.
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[@hoseinzahedifar1562]

Thank you very much for this great lecture ❤🤍🖤.
Homework:
In time 7:29: That's a great homework exercise for you to do the steps calculating the error order of the Forward Euler method for the Backward Euler method, and convince yourself that the order of the error remains 2.
Hints:
1- you can use Taylor series expansion for f(x(k+1)) over xk (e.g. f(x(k+1)) = f(xk) + f'(xk)dx + ..., where dx = x(k+1)-xk.
2- To calculate dx based on dt, you may use the Taylor expansion of the "true" x(k+1).

[@benceweisz1063]

I think another method is to consider x(t_k + (dt) + (-dt)) = x(t_k+1 - dt) = x(t_k). This will give ensure that the expansion is evaluating the derivatives at t_k+1. Then all you have to do is rearange to get x(t_k+1) on the LHS, after which you essentially get the backward euler step after substituting dx/dt(t_k+1) with f(x_k+1)

[@saras756]

This was precisely the information I was looking for, very-well explained! Thank you! :)

[@Eigensteve]

Thanks for watching!

[@sergeipravosud1848]

Nice video, thanks!

[@user-zq5ne9xb9b]

sometimes I forget to like because I'm in a hurry to watch the next lecture

[@oddRhymes]

Subbed, liked, commented!

[@hoseinzahedifar1562]

In time 4:48: "... this scheme is second order accurate in dt ...":
I agree with you that the order of dt is second for the error term of the Backward Euler method. However, I think the order of the accuracy is 1 (the order of accuracy = the order of error -1). Don't you think so?

[@pranjal86able]

so is he writing on glass with markers? what is the setup here?

[@zildjiandrummer1]

They write everything normally on a clear board with the camera on the other side of the board, then in a video editor they flip the entire frame about the vertical axis so it's not backwards. You're actually seeing the person reversed right now.

[@XenoZeduX]

His colleague Nathan Kutz has a video in his channel explaining the setup and process
I think NancyPi utilizes something similar

[@gemy6188]

thanks, prof for this clarity, I hope you can help me to solve (∂ t T + ⃗u · ∇ T = ∆ T) with Euler forward for R.B. convection, or if you have already done this before can you please redirect me to this video.