### 3.2 Numerical solution

It is certainly straightforward to numerically solve the two ordinary differential equations of the previous subsection along a characteristic line using say a Runge-Kutta method. You would need to start from some point at which an initial or boundary condition is given.

If you find the solution along each of a densely spaced set of characteristic lines, you have essentially found everywhere.

Of course, if is zero somewhere in the region of interest, it may be a better idea to find and as functions of instead of and as functions of , by taking suitable ratios from (3.4). Or you could just find all three variables as function of the arc length along the characteristic lines, by solving

This allows either or to be zero; it only fails if both are zero at the same point, and that is a true physical problem rather than a mathematical one.