A7 Differential equations 569
- Stability. In some methods, errors in the starting values or errors due to the
discrete numerical representation tend to grow during integration, so that the
solutions obtained deviate sometimes strongly from the exact ones. If the errors
tend to grow during integration, the method is calledunstable. It is essentially
the same phenomenon as we encountered in the discussion of recursion in
Appendix A2.
- Implementation. Very complicated algorithms are sometimes less favourable
because of the time needed to implement them correctly. This criterion is of
course irrelevant when using existing routines or programs (e.g. from numerical
libraries).
- Flexibility. In all methods, the coordinates are discretised: some methods
demand a fixed discretisation interval. These are less useful for problems with
solutions having strongly varying behaviour (somewhere very smooth and
elsewhere oscillating rapidly).
- Symmetry. For particular types of differential equations we would like the
numerical method to share symmetry properties of the original equation; an
example is time reversibility which might be present in the equation of motion
of a particle. In Chapter 8, symplectic symmetry properties of Hamiltonian
equations and particular integration schemes are discussed.
There are other criteria, such as analyticity of the functions occurring in the
differential equation, which make some methods more suitable than others.
A7.1 Ordinary differential equations
We now describe a number of numerical algorithms for the solution of this type of
equation. In one dimension, a first order differential equation looks like
x ̇(t)=f[x(t),t]. (A.33)
We call the variablesxandt‘space’ and ‘time’ respectively, although in various
problems they will represent completely different quantities. In the following we
integrate always fromt=0. In practice, one integrates from arbitrary values oft,
but the methods described here are trivially generalised.
By writingy(t)= ̇x(t), a second order equation
̈x(t)=f[x(t),t] (A.34)
can be transformed into two differential equations of the form (A.33):
x ̇(t)=y(t) (A.35a)
y ̇(t)=f[x(t),t] (A.35b)
and the methods that will be discussed are easily generalised to this two-dimensional
case.
