Computational Physics

A7 Differential equations 575

Finite difference methods

The Verlet algorithm is a special example of this class of methods. Finite difference
algorithms exist for various numerical problems, like interpolation, numerical dif-
ferentiationandsolvingordinarydifferentialequations[ 12 ].Here we consider the
solution of a differential equation of the form

x ̇(t)=f[x(t),t], (A.55)

and in passing we touch upon the interpolation method.
For any functionxdepending on the coordinatet, one can build the following
table (the entries in this table will be explained below):

− 3 x− 3

δx− 5 / 2

− 2 x− 2 δ^2 x− 2

δx− 3 / 2 δ^3 x− 3 / 2

− 1 x− 1 δ^2 x− 1 δ^4 x− 1

δx− 1 / 2 δ^3 x− 1 / 2 δ^5 x− 1 / 2

0 x 0 δ^2 x 0 δ^4 x 0 δ^6 x 0

δx 1 / 2 δ^3 x 1 / 2 δ^5 x 1 / 2

1 x 1 δ^2 x 1 δ^4 x 1

δx 3 / 2 δ^3 x 3 / 2

2 x 2 δ^2 x 2

δx 5 / 2

3 x 3

The first column contains equidistant time steps at separationh, and the second
one the valuesxtfor the timestof the first column. We measure the time in units
ofh. The third column contains the differences between two subsequent values of
the second one; therefore they are written just halfway between these two values.
The fourth column contains differences of these differences and so on. Formally,
except for the first column, the entries are defined by

δjxi− 1 / 2 =δj−^1 xi−δj−^1 xi− 1

δjxi=δj−^1 xi+ 1 / 2 −δj−^1 xi− 1 / 2 and

δ^0 xi≡xi.

(A.56)

It is possible to build the entire table from the second column only, or from the
valuesδ^2 jx 0 andδ^2 j+^1 x 1 / 2 (that is, the elements on the middle horizontal line and
the line just below), because the table is highly redundant.