Solution to advection equation with variable coefficients

[lostidentity]

Hi,

I'm trying to find analytical solution to an advection equation written in Spherical coordinates. It's spherically symmetric so I'm only interested in radial variances.

The equation is:
$$\frac{\partial{c}}{\partial{t}} + \frac{1}{r^2}\frac{\partial}{\partial{r}}(r^2uc) = 0$$

I've seen solutions to advection equation with variable coefficients written in non-conservative form using the method of characteristics. I'm wondering if I could use the same method to solve the above equation, which is written in conservative form. Note that both u and c are functions of both r and t.

Thanks.

 

[jvicens]

Hi there,

Thank you for your question. The advection equation you have provided is indeed written in conservative form, and the method of characteristics can still be used to solve it. However, there are a few additional steps that need to be taken in order to solve it in spherical coordinates.

Firstly, you will need to express the equation in terms of the radial variable r, instead of the Cartesian coordinates x, y, and z. This can be done by using the transformation r = \sqrt{x^2 + y^2 + z^2}.

Next, you will need to convert the partial derivatives with respect to x, y, and z into partial derivatives with respect to r. This can be done using the chain rule, as follows:

\frac{\partial}{\partial{x}} = \frac{\partial}{\partial{r}}\frac{\partial{r}}{\partial{x}} = \frac{x}{r}\frac{\partial}{\partial{r}}

Similarly, \frac{\partial}{\partial{y}} = \frac{y}{r}\frac{\partial}{\partial{r}} and \frac{\partial}{\partial{z}} = \frac{z}{r}\frac{\partial}{\partial{r}}.

Substituting these expressions into the advection equation will give you the equation in terms of r.

Once you have the equation in terms of r, you can then use the method of characteristics to solve it. This involves finding the characteristic curves, which are defined as the curves along which the solution remains constant. In your case, the characteristic curves will be circles centered at the origin, since the equation is spherically symmetric.

Finally, you will need to apply the appropriate boundary and initial conditions to obtain the solution.

I hope this helps. Good luck with your research!