Numerical algorithms

To analyse partial differential equations on a computer, we must use some way of calculating derivatives numerically and find stable and accurate techniques of modelling the evolution of the equations. Partial differential equations usually involve functions continuous in both spatial and time dimensions, which are impossible to represent on a digital computer, hence we must use some discrete representation of the functions and their derivatives. Such discrete representations are only approximations of the actual evolution of the continuous functions we are interested in, however we can do a pretty good job, and often using a numerical technique is the only way to model some kinds of physical processes. This is why it is important and interesting to study the numerical ways in which we make these approximations and the numerical techniques that can be used to calculate the evolution of differential equations. In this chapter, we shall look at how derivatives are calculated numerically, and several schemes used to numerically solve differential equations.