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

[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 let's 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.

 

[lofgran]

Thank you for reaching out for help with your research. I can understand the challenges you are facing in solving the steady-state spherically symmetric fluid equations for an accreting plasma.

Firstly, I would like to commend you for using the shooting method to solve the system of coupled nonlinear ordinary differential equations. This is a common and effective approach for solving such problems.

Regarding your question about handling the division by zero at the inner boundary condition, using L'Hopital's rule or a Taylor expansion can be viable options. However, it ultimately depends on the specific problem and the accuracy required for your solution. In some cases, a simple linear interpolation or extrapolation method can also work well. I would suggest trying out different methods and comparing the results to see which one gives the most accurate and stable solution.

Another important aspect to consider is the choice of your integration method. Depending on the nature of your equations, using a higher-order integration method such as Runge-Kutta or Adams-Bashforth can help in achieving smoother and more accurate results near the inner boundary condition.

It is also worth mentioning that solving the equations numerically can be sensitive to initial conditions and small changes in the parameters. Therefore, it is important to carefully choose your initial conditions and ensure that they are consistent with the physical properties of the system.

I hope this helps in pointing you in the right direction. If you have any further questions or need any assistance, please do not hesitate to ask. Good luck with your research.

 

[quicknote]

I can provide some insight into your question about using a shooting method to solve for the steady-state fluid equations. First of all, a shooting method is a numerical technique used to solve boundary value problems, such as the one you described. It involves guessing initial conditions and then iteratively adjusting them until the solution satisfies the boundary conditions.

In your case, the inner boundary condition at the sonic transition radius presents a challenge because of the discontinuity at this point. This is known as a shock wave, and it is a common occurrence in fluid dynamics when there is a sudden change in the properties of the fluid. In this case, the division by zero in the equation is a result of the sudden change in the velocity of the gas.

Now, to address your question about handling this discontinuity, there are a few approaches you can take. Using L'Hopital's rule can be a good starting point, as it allows you to evaluate the limit of a function as it approaches a point where it is undefined. However, this may not be sufficient to achieve a smooth integration near the inner boundary condition.

Another approach you can consider is using a Taylor expansion, as you mentioned. This involves expressing the function as a series of terms and then using a finite number of terms to approximate the function near the point of interest. The advantage of this approach is that you can control the accuracy of the approximation by including more terms in the series.

In summary, both L'Hopital's rule and a Taylor expansion can be useful techniques for handling the discontinuity at the inner boundary condition in your shooting method. However, the best approach will depend on the specific problem you are trying to solve and the level of accuracy you require. I recommend consulting with a fluid dynamics expert or conducting further research to determine the most appropriate method for your specific case. Good luck with your research!