IMPLEMENTATION OF EXPLICIT FINITE DIFFERENCE METHODS APPLIED TO DIFUSSION AND WAVE PROBLEMS

Finite difference methods are one of the basic tools in the numerical treatment of problems including partial differential equations. The implementation process generates a large amount of numerical values that must be handled properly. The aim of this paper is to show, from a didactic point of view, the usefulness of Excel spreadsheets to organise and manage properly different results, which are obtained during the entire process.

Firstly, the resulting process of applying these methods to a one-dimensional diffusion problem, which relates two temporal levels, is analysed. After that, the process with a one-dimensional wave problem, which in this case relates three temporal levels, is repeated.

There have been considered different mesh sizes for time and space in order to verify the convergence of the methods used. All calculations have been performed using an Excel spreadsheet, allowing to easily illustrate the main problem that may occur when using a finite difference method: instability.

The Excel sheets that have been built this way offer great easiness as well as high accuracy and speed of calculation, and can help the student to better understand the theoretical concept of convergence of a numerical method. In addition, the use of an Excel spreadsheet also allows the student to actively take part in the resolution of the problem, thus making it easier and encourages him to create his own Excel spreadsheets.