A kaleidoscopic excursion into numerical calculations of differential equations

Abstract

In this expository paper a sketch is given of some basic problems that arise when differential equations are discretised and calculated numerically. Some analytic techniques are demonstrated for the investigation of these numerical schemes. Numerous pictures illustrate the ideas and the formulae. References to available software packages are given.

For the logistic equation the time asymptotic behaviour depends on the chosen time step, showing period doubling from a correct attraction to the stable equilibrium solution to chaotic behaviour. For the linear harmonic oscillator energy conservation for large times is investigated.

Besides these ordinary differential equations, several aspects of linear and nonlinear wave equations, described by partial differential equations, will also be considered. A finite difference method is demonstrated for a linear wave equation. For the Korteweg-de Vries equation, an equation that combines dispersive and nonlinear effects in wave propagation, Fourier truncations are studied. It is shown that the basic Hamiltonian structure is preserved (implying energy conservation), as well as a continuous translation symmetry (implying conservation of horizontal momentum). As a consequence, travelling waves - i.e. wave profiles that translate with constant speed undeformed in shape-are present in truncations of any order. Numerical evidence, as well as a complete analytical proof, is given.

A preliminary version of a software package WAVEPACK, aimed to familiarize the unexperienced user with many basic concepts from wave theory (dispersion, groupvelocity, mode analysis, etc.) and to perform actual calculations in an easy way, is available upon request.