Frege’s Numbers
Posted 19th November 2021 by Holger Schmitz
In a previous post, I started talking about natural numbers and how the Peano axioms define the relation between natural numbers. These axioms allow you to work with numbers and are good enough for most everyday uses. From a philosophical point of view, the Peano axioms have one big drawback. They only tell us how […]
Computational Physics: Truncation and Rounding Errors
Posted 15th October 2021 by Holger Schmitz
In a previous post, I talked about accuracy and precision in numerical calculations. Ideally one would like to perform calculations that are perfect in these two aspects. However, this is almost never possible in practical situations. The reduction of accuracy or precision is due to two numerical errors. These errors can be classified into two […]
Fifty Solitaires – A Beginning
Posted 22nd September 2021 by Holger Schmitz
Many years ago, when I was a physics student and I was just getting to know the ins and outs of programming, I made a bet with a friend of mine. At the time my mother was into solitaire card games, the ones with real cards you play on the kitchen table. This was before […]
Multi Slit Interference
Posted 11th September 2021 by Holger Schmitz
This week I have been playing around with my electromagnetic wave simulation code MPulse. The code simulates Maxwell’s equations using an algorithm known as finite-difference time-domain, FDTD for short. Here, I am simulating the wave interference pattern behind the screen with n slits. For $n=1$ there is only one slit and no interference at all. […]
Computational Physics Basics: Accuracy and Precision
Posted 24th August 2021 by Holger Schmitz
Problems in physics almost always require us to solve mathematical equations with real-valued solutions, and more often than not we want to find functional dependencies of some quantity of a real-valued domain. Numerical solutions to these problems will only ever be approximations to the exact solutions. When a numerical outcome of the calculation is obtained […]
Computational Physics Basics: Floating Point Numbers
Posted 25th May 2021 by Holger Schmitz
In a previous contribution, I have shown you that computers are naturally suited to store finite length integer numbers. Most quantities in physics, on the other hand, are real numbers. Computers can store real numbers only with finite precision. Like storing integers, each representation of a real number is stored in a finite number of […]
What are Numbers? Or, Learning to Count!
Posted 17th April 2021 by Holger Schmitz
We all use numbers every day and some of us feel more comfortable dealing with them than others. But have you ever asked yourself what numbers really are? For example, what is the number 4? Of course, you can describe the symbol “4”. But that is not really the number, is it? You can use […]
Spherical Blast Wave Simulation
Posted 2nd December 2020 by Holger Schmitz
Here is another animation that I made using the Vellamo fluid code. It shows two very simple simulations of spherical blast waves. The first simulation has been carried out in two dimensions. The second one shows a very similar setup but in three dimensions. You might have seen some youtube videos on blast waves and […]
The Euler Equations: Sod Shock Tube
Posted 30th September 2020 by Holger Schmitz
In the last post, I presented a simple derivation of the Euler fluid equations. These equations describe hydrodynamic flow in the form of three conservation equations. The three partial differential equations express the conservation of mass, the conservation of momentum, and the conservation of energy. These fundamental conservation equations are written in terms of fluxes […]
The Euler Equations
Posted 3rd September 2020 by Holger Schmitz
Leonhard Euler was not only brilliant but also a very productive mathematician. In this post, I want to talk about one of the many things named after the Swiss genius, the Euler equations. These should not be confused with the Euler identity, that famous relation involving $e$, $i$, and $\pi$. No, the Euler equations are […]