Runge-Kutta Integrator Overview: All Purpose Numerical Integration of Differential Equations

In this video, I introduce one of the most powerful families of numerical integrators: the Runge-Kutta schemes. These provide very accurate and efficient "all-purpose" numerical integrators for ordinary differential equations. Specifically, we introduce the 2nd-order and 4th-order accurate RK schemes (called RK2 and RK4) and break these algorithms down into simple and intuitive steps. These algorithms are also explained with pictures.
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This video was produced at the University of Washington
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0:00 Overview
3:15 2nd Order Runge-Kutta Integrator
8:07 Geometric intuition for RK2 Integrator
19:59 4th Order Runge-Kutta Integrator

[@user-ms5te5vd1x]

Ladies and gentlemen, what you see is the future of education. I am very excited to witness this.

[@benjtheo414]

I really appreciate your efforts in making all this knowledge available for free, you are doing humanity a great service.

[@laurentthowai3359]

Merci Mr Brunton, c’est toujours un plaisir de vous écouter ! Pour moi le matin en prenant mon petit déjeuner. Impatient d’écouter la suite.

[@diemaschinedieviereckigeei2941]

What a fantastic presentation. You are a great teacher!

[@murtdoc]

Thank you, prof. Brunton. Your style of explaining, at least for me, it's very effective. Please keep up this great work!

[@angelicatorresgarcia5228]

beautifully explained !! thank you! i passed all my first and second year algebra courses with minimum effort during the pandemic so learning this for my scientific computing course has been hell, but you make it so simple i really get it now !!! you're a great teacher !!

[@tanaykumarkarmakar3447]

Extraordinary lecture. Thank you, Steve.

[@jamesdennis6120]

I wonder if there was a hidden joke in the RK2 explanation, or; if it were a complete random occurrence 🤔. These lectures are amazing. They relieve so much stress caused by the heavy reading of the text material. After watching these the reading becomes way easier.

[@bryan-9742]

wow! i'm excited to see how different this will be for stochastic systems.

[@jaihind6472]

thanks for explaining this in such an enthusiastic way. u saved my semester

[@POPO-kk6nh]

An amazing explanation! You nailed it. Thanks

[@sakethmamidi2753]

amazing way of explaining rk4!!

[@Pedritox0953]

Great explanation!

[@stanrunge]

pretty cool to be a long descendant of the author for this numerical integrator (Carl Runge)

[@Eigensteve]

That's super cool!!

[@hoseinzahedifar1562]

Thank you very much for this great stuff. You are amazing...🌺🌺🌺.

[@KitagumaIgen]

Great lecture series! Hopefully you'll get into symplectic integration-schemes too - for dynamical systems with conservation of some properties like energy and momentum.

[@DB-nl9xw]

didn't understand much about it, but i like the video, thanks for sharing and making it

[@whdaffer1]

Your explanation of why the RK four method works is wonderful. It would be equally wonderful, as an example of the history of mathematics, if you could give some discussion on how the formula was derived.

[@saras756]

So grateful for this lecture. It helped make sense of my notes. Thank you very much! :D

[@Eigensteve]

You're very welcome! Glad it was helpful.

[@aakashmaniar9494]

So I dont think it will be better than rk but I am curious about the performance benefits of using previous states to estimate curvature and higher order properties, keeping the call to dynamics function only once or somewhere in between. IE x(n+1) is not just dependant of x(n) but upto x(n-m). Given a pair of position and velocity(m=1) a cubic polynomial can be fitted and taken a time step over. Is there any good comparison for such a techniques?

[@serdar_a]

Does anyone know where to get a transparent board which is used by Mr Brunton?

[@MrNikelborg]

Shouldn't the f_i be first order derivatives or tangents? They become vectors only after you multiply them with delta_t or not?

[@ThenSaidHeUntoThem]

Thou has done unto me that which is good.

[@GeorgeTsiros]

(consider adjusting the compressor in the audio, i _think_ you may need to decrease the attack (?) time?)

[@fabiofarina9579]

would be interesting to have an overview on how to handle RK4 when f(x,t) is not analytical but is only known as a bunch of discrete values (sampled values or data-driven). Then, how to get \Delta{t}/2 values+

[@dexdrurglum]

In my experience when I have only a dataset of discrete values, I just interpolate to get values that don't already exist in the dataset

[@fabiofarina9579]

@@dexdrurglum yeah, I usally do quadratic or spline interpolation but I'd really enjoy a video from Steve on this topic

[@dexdrurglum]

@@fabiofarina9579 I agree! Steve is the king 👑

[@alexistremblay1076]

How about recurrent neural networks? Not entirely sure how it would handle future time steps.

[@joelneto7360]

very helpfull

[@Eigensteve]

Glad you think so!

[@thomaspavelka7335]

Thanks for this great explanation! Just the bracelet I find distracting

[@eig_himanshu]

Can you also keep a live doubt solving session on Numerical Analysis?

[@hoseinzahedifar1562]

It is a good idea...👍

[@pietheijn-vo1gt]

why do all indians call 'problems' as 'doubts'?

[@fabiobiffcg4980]

So, could I keep taking more halves for RK5, RK6, RK7, RK8 ... "RKn" in order to make it more accurate? (Of course, considering only accuracy since it will be to expensive to compute them, probably)

[@chensong254]

I believe after a certain point, the benefit of using a higher order scheme becomes negligible compared to the floating point round-off error.

[@fabiobiffcg4980]

@@chensong254 yeah, after some thinking, that's what I thought

[@sim1_7]

does the runge-kutta 4 method need a root solver for nonlinear ode? if not, why? And instead why, for example, does the forward and backward euler method need a root solver (example: newton's method) for nonlinear ode?

[@chensong254]

I don't think we need a root solver for RK4 or FE. We are just evaluating the derivative function f at certain points, and derivative f can be either linear or nonlinear. As long as the derivative is represented in an explicit form, we don't need a root solver. For BE though, I think we need a root solver if f is nonlinear.

[@sim1_7]

@@chensong254 yes, i think you are right. My knowledge at this point Is that explicit methods are Linear in the variable a time k+1 so there Is no Need for root solver algorithms. Implicit methods, instead, are nonlinear with respect to the variable at time k. I tried explicit RK-4 for some nonlinear odes and it works pretty well. Forward Euler, instead, works bad for these nonlinear odes. It works good if non-linearities are weak but as soon as they become more complex FE fails

[@user-ce9xq4cl9x]

Amazing！ How can he write in the back of the glass (which is like a mirror effort) but still keep the numbers normal?

[@theintjengineer]

He writes just like you and I usually would. It's the board's mirroring property that reverses it and makes it look like he's writing reverse/backward/whatever.

[@henrydevelopment]

The AI of a thermos flask. When you put hot water in it, it keeps it hot. When you put cold water in it, it keeps it cold. How does it know?

[@pietheijn-vo1gt]

Very simple. Film normally, mirror the video in your recording software...

[@GeorgeTsiros]

f1 and f2 are not _directions_ they are complete values, they do not have length of 1 necessarily?

[@samsara2024]

a real example instead of so many functions, integral, delta etc

[@danitron151]

Are you writing backwards!? A short video showing how you use a glass window to make these videos would be really cool!

[@ac2italy]

interesting that glass board? Is he writing left-to-right?

[@ac2italy]

kzbin.info/www/bejne/m4eygXeHarCMqdE&ab_channel=ScopeTraining

[@pietheijn-vo1gt]

Ofcourse. Just mirror the video in your recording software