So far in this book the schemes that have been developed to solve partial

differential equations (PDEs) such as the transport, heat or convection equation

have been derived in such a way that there is a single unknown in each equation.

These explicit schemes are easy to code and implement, but there are other ways of

tackling the problem. Firstly, there are other ways of constructing the individual

approximations that lead to different explicit schemes. Secondly, there are schemes

that can be developed where each equation contains more than one unknown. These

schemes, which are called implicit, require a matrix solution for their

implementation. This adds a level of complication for the programmer but there are

significant advantages with this approach.