Next: 2. Parabolic Equations Up: 1. Introduction Previous: 1.3 Finite differences

## 1.4 Finite differences in polar coordinates

Example 5   Consider in polar coordinates, i.e.,

(See Figure 1.7.)

Then we have

etc.

Example 6   In cylindrical coordinates the PDE can be written

 (1.3)

We can set up a grid system as shown with , and represent derivatives by either central or forward differences. How do we deal with the point ?

Solution
If the solution is independent of then equation (1.3) becomes

 (1.4)

and in Cartesian coordinates it may be written

 (1.5)

We write

and

At a general point

For nonzero values of there are no difficulties using the above formula but how do we deal with , and at ?
e.g.

Therefore at we use the PDE in Cartesian form, i.e. (1.5)

Thus in 2D at we have

 (1.6)

However, the and axes are two perpendicular axes and the generalisation of equation (1.6) gives

where is the mean value of along , i.e., on .
When , equation (1.4) becomes

and the question arises as to how do we deal with the term at ?
The answer is that we use l'Hopital's rule, i.e.

(N.B. at if problem is symmetrical)
Therefore, at

and

Therefore

[You can now do Question 1 of the Examples]

Next: 2. Parabolic Equations Up: 1. Introduction Previous: 1.3 Finite differences
Maintained by: Dr S D Harris
2003-06-16