Phillip Otto Runge Puns

I am not familiar with implementing Runge-Kutta methods with code. I was wondering if there are any tutorials out there that go over how to implement Runge-Kutta methods in Julia to solve differential equations? Ideally, the tutorial will go over some examples (that is how I learn best).

The reason I ask is I would like to work on some projects that solve DEs using these methods and I also expect this to be a necessary skill to have for my future research.

I just finished teaching my Numerical Analysis class for the Fall. Grades are in, and so I can sit back for a bit.

I always feel that the class gets a bit rushed at the very end. I spend a lot of time on polynomial approximation when I teach the class, and then I view the rest of the class as applications of polynomial approximations. This includes Newton's method, Integration and Differentiation, and the solution to initial value problems.

I felt that my presentation to my students was a bit slap dash for IVPs. So I wrapped up Runge Kutta methods into this video here, if anyone is curious: https://youtu.be/t48a2M27kjM

If you taught a class this semester, how did it go for you? There was such a shuffle working around COVID and the possibility of shutting down all semester, and I feel it took longer than usual to hit a stride. Did you feel you rushed anything at the end?

If you are a student, how was the flip side of all of this?

In the wikipedia article for collocation methods (https://en.wikipedia.org/wiki/Collocation_method) it states

> All these collocation methods are in fact implicit Runge–Kutta methods. The coefficients ck in the Butcher tableau of a Runge–Kutta method are the collocation points. However, not all implicit Runge–Kutta methods are collocation methods.

What's an example of a Runge-Kutta method that's not a collocation method? Would it be an RK method where all the 'c' values in the Butcher tableau are 0?

I'm asking it out of curiosity. It doesn't need to have any pratical purpose. I know there are 14th order RK methods around, but I'm curious if someone developed a 100th or 1000th order method somewhere

The rotting carcass of leadership at the University has affected every aspect. Those of us who take care of patients at the hospital have been shocked at the deterioration of the work environment and our ability to care for our patients. That was pre-Covid. We always knew that Schlissel must be enabling this nightmare via Marschall Runge, and now that chicken has come home to roost.

Why should you care? If you are a patient of the UM or an employee, you have witnessed the deterioration first hand. This began long before Covid but they have doubled down on it using the pandemic as cover. If you are a resident of Washtenaw County, you should care as well. Fortunately, St. Joes still provides excellent health care, but a healthy UM hospital is good for Ann Arbor and it's surrounds.

The changes in the hospital are bad for patients and have destroyed all the good will in faculty and staff. This is really bad for patients.

Those of you who sign up for leadership opportunities in UMMG (the group in charge clinical care) need to take a long hard look in the mirror. Ask yourself, are you propagating the Schlissel/Runge cancer or are you doing what is right for your patients? Are you going to look back on this time and be proud of what you have done?

The sooner Runge is gone and the upper leadership of UMMG is removed the sooner Michigan can heal and we can get back to doing the right thing.

Look in the Mirror.

Peace.

I am modelling dM/dt=gamma*M x B (the simplified version of the Bloch equation) using Runge-Kutta integration and I am struggling to find initial values for B(t) =Bx, By, Bz and M(t)=Mx,My,Mz.

What I tried so far is defining B = (0, 0, B_0) and M=(0,0,1) as my initial values then B_0=1.5 Bxt=B1*cos(w*t), Byt=B1*sin(w*t) , Bzt=B_0, B1=1e-3/sqrt(2) ,    w=gamma*Bknot but my results do not seem accurate

Assuming my code is right, I need help understanding the physics. Has anyone tried to model this before? If I were to plot Mz vs Mx, and Mz vs My should I be getting the Larmor precession?

I need ideas because every website I find says something different and I am struggling to see how they all come together. Thank you all

https://www.cp24.com/news/york-regional-police-officer-charged-in-theft-investigation-1.5673721

One more cop on the paid vacation list!

Can someone who knows about this take a look at my codes? The values I am getting in the final table seem to be wrong.

Not sure how to upload a picture, but the function code looks as follows:

function Sol=RK4(a,b,alpha,f,N)

h=(b-a)/N;

t=a:h:b;

w=zeros(N+1,1);

w(1)=alpha;

for i=1:N

k1=h*f(t(i),w(i));

k2=h*f(t(i)+h/2,w(i)+k1/2);

k3=h*f(t(i)+h/2,w(i)+k2/2);

k4=h*f(t(i)+h,w(i)+k3);

w(i+1)=w(i)+(k1+2*k2+2*k3+k4)/6;

end

Sol=[t',w];

And the specific problem I am working on is:

a=0;

b=1;

alpha=9/5;

f=@(y,t) (t^7)*y^(8/5)-8*t^(7)*y;

N=10;

Sol=RK4(a,b,alpha,f,N)

Then the output looks like:

0 1.800000000000000

0.100000000000000 0.212985933198134

0.200000000000000 0.212983644715527

0.300000000000000 0.212979886602519

0.400000000000000 0.212974693027118

0.500000000000000 0.212968093926933

0.600000000000000 0.212960116556139

0.700000000000000 0.212950786270819

0.800000000000000 0.212940126995763

0.900000000000000 0.212928161529170

1.000000000000000 0.212914911754464

The jump down to 0.212985933198134 does not seem right here as I believe it should be closer to 1.8. Are there any mistakes I have made?

Thanks to anyone who is able to help.

I see a lot of Predictor correctors applied with adam-bashfort/adam moulton, but I feel RK methods on my PoV appear more suited for prediction correction. Just find one explicit and one implicit RK with the same C points. You can even use embedded methods to get an extra truncation error estimate

Is there any disadvantages I'm not aware about?

I didn't want him to be a flat head

I have to write down a finite-difference fourth-order accurate Runge-Kutta approximation of the Brusselator equations on a uniform grid, tn = nh, with the time step h for n = 0, 1, 2 ...

I want to know what this is asking me to do. I know this is probably a dumb question but I don't understand what operations I need to do?

https://twitter.com/slmandel/status/1482029025904119808?s=20