#computational mathematics

[missbrunettebarbie]

I am desperate enough to ask this on f-ing tumblr, but nothing else has helped so far:

Does anyone know if the 2 equilibrium points (that are not (0,0,0)) of the Lorenz system are stable or not? And what is their classification (eg saddle point etc)?

I am going insane trying to figure it out but I just can't.

[greenglasstudy]

finally got a monitor so I can code easier! Goodbye squinting 😂 Right now I am working on a code for a robust quadratic solver. I’m also working on using spyder and latex for documentation. Really fun stuff!

[mymathematics]

Book Computational mathematics by ROBERT E WHITE pdf Book Computational mathematics by ROBERT E WHITE pdf Pages 388 By ROBERT E.WHITE This book is not just about math or computing or applications, but about all three-in other words, computational science. Each section of the first six chapters is motivated by a specific application. The author applies...

[andranikfakirian]

Project "CompMath.AstrophysicsODEs": Software Package

Finally, I got my hands on this project again. There is definitely a strong progress in the project at the moment: there is an alpha version of the software package. It is an ODE systems configurator, (m, k) - solver and visualizer combined with user-friendly interface. Unfortunately, currently only a version of solver with a constant timestep has been made, since the theory of Novikov methods is freely available only for such an implementation of the algorithm, as far as I know.

To create a solver with automatic timestep sampling, a couple of different algorithms are usually used. Based on the discrepancy in the solutions of the equations produced by these methods, it is decided wether to change the timestep in order to reduce the error by decreasing it, or, conversely, to lessen the total number of steps by increasing it. Among the methods described in the scientific literature with decent accuracy, there is only one calculated one. Thus, to manufacture more advanced solver, it is necessary to devise another algorithm and pair it with the first one.

And that's where the good news begins: after several hours of cumbersome "semi-automatic" calculations, another theoretical technique with all the necessary quotients has already been computed. Consequently, further implementation of the method with an automatic time step will be only a matter of technique.

In the meantime, you can admire the work of the software package with a "constant solver" on a very trivial stiff problem with an exponential analytic solution.

P.S.: The graphics below show the computed solution to a simple task (u(t)) and its relative error (uEps(t)) based on the deviation from the analytical solution. I ask you to pay attention to the fact that the error is really low (maximum about 0.0000025%) for a rather stiff problem (the value reaches 2.5·10^43 in one second).

[andranikfakirian]

Project "CompMath.AstrophysicsODEs": Early Projects, Part 2

Let's continue the series of Early Projects. The next one was from the sphere of analytical mechanics. The task was to simulate the rotational motion of bodies using quaternions. Quaternions make it possible to describe such motions quite succinctly in formulas. Special attention should be paid to modeling the Dzhanibekov effect, aka Tennis racket theorem.

P.S.: Below is a test of the operation of this simulator with a spin of the simple screw around its axis.

P.S.2: Here is a simulation of the Dzhanibekov effect. As it can be seen when the body have three distinct principal moments of inertia, rotation around one of the principal axes (associated with the second-order inertia moment) unstable. That's why the wing screw turns over.

[andranikfakirian]

Project "CompMath.AstrophysicsODEs": Early Projects, Part 1

While the work is underway, and there is not much to show, it was decided to present examples of past projects in the field of computational physics. They were manufactured in the same way as the planned project: an integration solver on C++ language, a visualizer on Python… It's not appropriate to leave this blog for a long time without any content at all.

The first simple project from the field of applied physics was devoted to modeling a double pendulum using Lagrangians. This is a good example from chaos theory, representing the effect when the slightest changes in the initial conditions lead to a completely different behavior of the system.

P.S.: Below is an animation of two double mathematical pendulums with a bit different initial conditions (the second pendulum has a more elevated second end).

P.S.2: Here is an animation of two double pendulums, but with a much heavier second end. Apparently, this causes a more predictable behavior of the double pendulum, allowing it to be interpreted as a pendulum inside a pendulum.

[andranikfakirian]

Project "CompMath.AstrophysicsODEs": Launch

The second semester project in a row. The "Mentor" program is back, however the project form differs. Since I've been working in numerical modeling for the second year now, I found myself eager to take part in the computational physics project. And education in the speciality "Space Research" has shaped the sphere of interests of the listed project in the computational mathematics field. Which is why I started a project "Applications of Computational Mathematics in Astrophysics" (working title).

Currently, I'll be working on a software implementation of a one-step, limplicit, variable-step method for solving Ordinary Differential Equations (ODEs) of the Rosenbrock type, known as (m, k) - Novikov method. Stiff Cauchy problems cannot be solved correctly by explicit methods without rocketing the number of calculations or inaccuracies, because these methods aren't stable enough. Implicit methods are a solution for the problem of stiff systems due to their much larger area of stability, but they are more complex because of their non-linearity. As a rule, systems of differential equations in astrophysics, plasma physics, and the theory of nucleosynthesis are stiff, therefore this specific software is really important for solving this kind of tasks.

I plan to manufacture this software the way I've practiced before and found convenient, professional and profitable. The solver itself is going to be made using the C++ language. I plan to create the interface and visualizer using Python language. The configurations and calculation results will most likely be transferred between these programs using JSON files.

The code itself, highly likely, won't be published on my github for non-disclosure reasons.

[weaponized-mathematics]

The simulation is coming along decently enough I suppose. I’ll have quite a bit more than my classmates to present on Monday I think. 

Still have to actually calculate the fields instead of this just being a magical simulation where E = 0, and B = 1...