In this chapter, we present two rather different topics, which we have linked together. First we shall study some initial value problems and initial-boundary value problems, where the forward-in-time integration is handled by the “method of lines”. We can treat the spatial derivatives by one of two methods:

1. finite differencing, the standard approach, which is discussed in every textbook, and some aspects of which will be treated in the next chapter,

2. spectral methods which, for smooth solutions, will give near exponential accuracy.

For simplicity, we shall study only scalar partial differential equations in one spatial dimension, but everything can be generalized to systems of equations in two or more dimensions. We look first at Fourier methods capable of handling problems with periodic spatial dependence. Then in Section 8.6 we suggest a promising approach using Chebyshev transforms for spatial dependencies that are more general. Unfortunately, there is no pre-existing Python black box package to implement this, but there is legacy Fortran77 code which we list in Appendix B.

The second main topic of this chapter is to present the numpy f2py tool in Section 8.7, so that we can re-use the legacy code of Appendix B to construct Python functions to implement the ideas of Section 8.6. If you are interested in reusing legacy code, then you should study the ideas presented in Section 8.7, to find out how to do it, even if you have no interest in Chebyshev transforms.

I have used computers as an aid to scientific research for over 40 years. During that time, hardware has become cheap, fast and powerful. However, software relevant to the working scientist has become progressively more complicated. My favourite textbooks on Fortran90 and C++ run to 1200 and 1600 pages respectively. And then we need documentation on mathematics libraries and graphics packages. A newcomer going down this route is going to have to invest significant amounts of time and energy in order to write useful programmes. This has led to the emergence of “scientific packages” such as Matlab or Mathematica which avoid the complications of compiled languages, separate mathematics libraries and graphics packages. I have used them and found them very convenient for executing the tasks envisaged by their developers. However, I also found them very difficult to extend beyond these boundaries, and so I looked for alternative approaches.

Some years ago, a computer science colleague suggested that I should take a look at Python. At that time, it was clear that Python had great potential but a very flaky implementation. It was however free and open-source, and was attracting what has turned out to be a very effective army of developers. More recently, their efforts have coordinated to produce a formidable package consisting of a small core language surrounded by a wealth of add-on libraries or modules. A select group of these can and do replicate the facilities of the conventional scientific packages.

Although Python is a small language it is a very rich one. It is very tempting, when writing a textbook, to spell out all of the ramifications, concept by concept. The obvious example is the introductory tutorial from the originator of Python, Guido van Rossum. This is available in electronic form as the tutorial in your Python documentation or on-line or as hard copy, van Rossum and Drake Jr. (2011). It is relatively terse at 150 printed pages, and does not mention numpy. My favourite textbook, Lutz (2009) runs to over 1200 pages, a marathon learning curve, and only mentions numpy in passing. It is excellent at explaining the features in detail, but is too expansive for a first course in Python. A similar criticism can be applied to two books with a more scientific orientation, Langtangen (2008) and Langtangen (2009), both around 700 pages with a significant overlap between them. I recommend these various books among many others for reference, but not for learning the language.

Very few people would learn a foreign language by first mastering a grammar textbook and then memorizing a dictionary. Most start with a few rudiments of grammar and a tiny vocabulary. Then by practice they gradually extend their range of constructs and working vocabulary. This allows them to comprehend and speak the language very quickly, and it is the approach to learning Python that is being adopted here.

Introduction

The most venerable and perhaps best-known scientific graphics package is Gnuplot, and downloads of this open-source project can be obtained from its website. The official documentation is Gnuplot Community (2012), some 238 pages, and a more descriptive introduction can be found in Janert (2010). Gnuplot is of course independent of Python. However, there is a numpy interface to it, which provides Python-like access to the most commonly used Gnuplot functions. This is available on line. Although most scientific Python implementations install the relevant code as a matter of course, the documentation and example files from this online source are useful. For many applications requiring two-dimensional graphics, the output from Gnuplot is satisfactory, but only at its best is it of publication quality. Here Matlab is, until recently, the market leader in this respect, but Python aims to equal or surpass it in quality and versatility.

The matplotlib project aims to produce Matlab-quality graphics as an add-on to numpy. Almost certainly, this should be part of your installation. It is installed by default in most Python packages designed for scientists. There is extensive “official documentation” (1255 pages) at Matplotlib Community (2013), and a useful alternative description in Tosi (2009). The reader is strongly urged to peruse the Matplotlib Gallery where a large collection of publication quality figures, and the code to generate them, is displayed.