Deriving Forward Euler and Backward/Implicit Euler Integration Schemes for Differential Equations

This video introduces and derives the simples numerical integration scheme for ordinary differential equations (ODEs): the Forward Euler and Backward Euler integration schemes. These integrators are based on the simplest forward and backward finite-different derivative approximations for dx/dt. Although these are not the best all-purpose integrators, they provide a great starting point for understanding sources of error and the stability of numerical integration techniques.

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This video was produced at the University of Washington

%%% CHAPTERS %%%
0:00 Deriving Forward Euler Integration
14:24 Deriving Backward Euler Integration
19:03 Euler Integration for Linear Dynamics

[cerbahsamir]

I really want to say that I just arrived from school, super tired and exhausted and questioning why I even started this. Watching the notification sheered me. It reminds me of how mush I improved over the years and the fact that I can watch such content and appreciate the intuition behind it is everything ✨

[sprmndctrl]

Amazing video. In the past few months I've been searching now and then for explanations about backward Euler - how it works and why it's better. Finally I have a slightly better grasp of the equations. Thank you!

[erikgottlieb]

Thankful for: (1) (t) completeness, (2) Explanation/relation back to Taylor Series, (3) Indexing as a bridge towards code/scripting (4) Summary at 22:38 shows real clarity in your process.

[rushabhyeshwante]

This is present in CFD software, Implicit and Explicit time stepping.

[sehatatlier]

look how many times this great video liked so far just 388. if this was a nonsense video it would have been watched millions of times.

[yuanfrank]

I have a silly question. At 18:50, on the right column, if we plug in f(xk+1) of backward Euler to X_{k+1} = X_k + \delta_t f(X_{k+1}), isn't that the same as forward Euler? the notation of f(x_k+1) also is not consistent with definition of f(x_k) on the left column

[MDNQ-udty]

Is there any study on combining forward and backward such as multiplying them to get x_(k+1) = sqrt((I + dtA)/(I - dtA)) x_k = |I+dtA|/(I - dt^2*A^2) x_k and iterating: x_(k+1) = x_k + dt f(x _k + dt f(x_k)) etc

[MLDawn]

Is this idea of finding trajectory is at all relevant to path integrals?

[mehdykhayamy]

Most of the practical systems have control input u. How to solve the implicit Backward Euler when x dot = Ax + Bu?

[Zahid-]

Hi, i loves your lectures, but I am curious to know how you make these videos.. mirror?

[AgnaktoreX]

excellent but maybe do a simple example to each

[diffgeo]

How is the writing on the board? Is he behind glass? Can't because then he would have to write everything backwards.

[hoseinzahedifar]

Thank you very much... ❤🖤🤍.
you talked about the analysis of error for linear dynamics equations... but how about nonlinear dynamics equations? is there any analytic method to describe the error of numerical integration in the case of nonlinear systems?