Numerical Experimentation: A Third Way to Study Nature

Abstract

We will outline the history of the numerical approach and trace back the origins of the use of computers to carry out simulations in mathematics and physics. We will then present the techniques used, by taking as example the finite-difference method to solve PDEs and discuss the nature and impact of numerical errors. Finally, we will argue that numerical simulation pertains more to experiment than to theory. Although this may seem a paradox, all exact science is dominated by the idea of approximation (Bertrand Russell) 1 Historical Sketch The numerical approach goes back much further than the appearance of the first computers. In a paper submitted in 1822 [1], Charles Babbage already suggested using numerical machines to calculate astronomical tables. These machines were made up of an 'attic', where data were stored, and of a 'mill', where calculation took place. However, this consisted merely in numerically evaluating some solutions, already known analytically, and not, in fact, per-forming simulations in the modern sense. Numerical simulation is defined as solving the equations that describe the physical laws governing the system studied by using algorithms. One can trace its origin back to the year 1899, with the development of the finite difference method by Sheppard [2]. It was then developed by Richardson [3] who used it, from 1910 onwards to calculate the stress exerted upon a dam. Richardson later had the idea of numerically solving the equations of atmospheric dynamics in order to predict the weather. He designed for this a finite difference scheme, which now bears his name, and applied it to find out the atmospheric situation on May 20th 1910. After six