Basics







Numerical Methods

Basics

The subject of this lecture on Numerical Methods in Physics is the solution of real physical problems with methods of numerical mathematics. These methods permit to treat problems which are so complex that they can only be solved with the aid of a computer.

For computationally intensive problems in physics, historically Fortran and c were often used. Fortran has the reputation for delivering the fastest execution speeds but with modern compilers the advantage that can be obtained is not very large. For large projects that require many people to collaborate, a structured language like Java is a good choice. Higher level languages like Matlab or Python are often convenient for calculations, plotting, and manipulating data. Since there is no best computer language for all applications, we will provide example code in various computer languages.

One language that is particulary well suited for web-based course notes is JavaScript. JavaScript was designed to be embedded into web pages. For numerical routines, the syntax of JavaScript is very close to c++ or Java. For example, below on the left is a text area that contains some Javascript code that calculates the squares of some integers. To the right of the JavaScript code is an iframe. An iframe is a webpage within a webpage. When the "Execute" button is pressed, the code in the box on the left is loaded into the iframe. Try this now.


<script> for (i=1; i<20; i++) { document.write(i+" squared is "+i*i+"<br \/>"); } </script>

You can modify the code on the left and see how it changes the output.

Comments in JavaScript

Comments are an important part of any program. They are used to describe how the program works. Since JavaScript is embedded in a web page, the comments can be in Hypertext Markup Language (HTML). This means the comment can contain images, mathematical symbols, or videos. Below are some buttons that load html examples into the textbox which are then rendered in the iframe.


Hello world!

<p>Mathjax uses latex expressions to render mathematics in a web page. You need to include the Mathjax header you see in the texbox on the left and then you can put mathematical expressions inline by putting them between $\$$ signs. For instance  $x^2$ and  $\hbar\omega$ are inline expressions.</p>

<p>Centered equations can be included in an equation environment. See the source code on the left to see how the following equations are coded.</p>

\[ \begin{equation}
-\frac{\hbar^2}{2m}\nabla^2\psi +V(x)\psi=E\psi.
\end{equation} \]

\begin{equation}
\vec{F} = q(\vec{E}+\vec{v}\times\vec{B}).
\end{equation}

\[ \begin{equation}
\chi\left(\omega\right)=\frac{v\left(\omega\right)}{F\left(\omega\right)}=\int\limits_{-\infty}^{\infty}g\left(t\right)e^{-i\omega t}dt.
\end{equation} \]

<p>This example uses a Content Delivery Network (CDN). The code that is needed to display equations is loaded from a remote server. The code is in the Mathjax.js file that is retrieved from cdnjs.cloudflare.com.</p>

For more information about html, visit the w3schools HTML tutorial.