E.1 The Taylor-Galerkin method used for vector-valued variables

The only method which adapts itself easily to the treatment of vector variables is that of the Taylor-Galerkin procedure. Here we can repeat the steps of Sec. 2.7 but now addressed to the vector-valued equation with which we started Chapter 2 [Eq. (2.1)]. Noting that now ? has multiple components, expanding ? by a Taylor series in time we have

where ? is a number such that 0 ? ? ? 1.

From Eq. (2.1),

and differentiating

In the above we can write

where A _i ? ? F _i/ ? ? and if Q = Q( ?, x) and ? Q/ ? ? = S,

We can therefore approximate Eq. (E.1) as

Omitting the second derivatives of G _i and interpolating the n + ? between n and n + 1 values we have

At this stage a standard Galerkin approximation is applied which will result in a discrete, semi-implicit, time-stepping scheme. As the explicit form is of particular interest we shall only give the details of the discretization process for ? = 0. Writing as usual

we have

This can be written in a compact matrix form similar to Eq. (2.107) as

in which, with

we have (on omitting the third derivative terms and the effect of S) matrices of the form of Eq. (2.108), i.e.

With ? =...