# Shooting method and a shock wave (discontinuity at the inner boundary condition)

1. Jul 7, 2008

### Beeza

I've done searching on the topic, and I really don't know where else to turn, so here it goes. I hope somebody can point me in the right direction.

I've been working on using a shooting method to solve the steady-state spherically symmetric fluid equations for an accreting plasma. Basically, it boils down to solving a system of coupled nonlinear ordinary differential equations. The inner boundary conditions are unknown, and I'm using a shooting method to match the outer boundary conditions.

The inner boundary condition for the three unknowns is at the sonic transition radius for the inward flowing gas. At this point, there is division by zero in one of the equations. Let's say the equation is of the form $$\frac{vdv}{dr} = \frac{A}{B}$$. So, at this inner boundary condition B = 0. Now, based on the condition that B = 0, it must be required that A = 0, and from A = 0, we get another of the inner boundary conditions. Now, this may be a stupid question, but is it typically sufficient to use something really basic like L'Hopitals rule to handle the first few iterations through the integration out to a radius of lets say within 0.1 percent of the sonic radius. Or is it required to use some kind of taylor expansion and keep quite a few terms to achieve smooth integration near the inner boundary condition.

I hope the question is clear enough. Thanks in advance for any help.