PDE equation in spherical coordinates

[vibe3]

I am looking for ideas on how to solve this equation:
$$\nabla \cdot \left( \vec{A} + F \hat{b} \right) = 0$$
where $\vec{A}$ and $\hat{b}$ are known vectors of $(r,\theta,\phi)$ and $F$ is the unknown scalar function to be determined. Also, $\nabla \cdot \hat{b} = 0$. So the equation can also be expressed as
$$(\nabla F) \cdot \hat{b} + \nabla \cdot \vec{A} = 0$$

I am trying to solve this numerically, and so I have b and A on a 3D grid. But I would like to avoid using a 3D finite differencing scheme, if there is a way to simplify this to an algebraic equation.

I'm thinking it may be possible to expand these vectors in some sort of basis (like vector spherical harmonics?) along with some radial basis functions and then solve for the coefficients of $F$ in terms of the coefficients of $\vec{A}$ or something like that.

Does anyone have experience with this type of equation and would know an appropriate basis to use?

 

[jvicens]

One approach to solving this equation could be to use a spectral method, where you expand the unknown scalar function F and the known vectors A and b in a basis of orthogonal functions. This can be done using a technique called Galerkin projection, where you project the equation onto each basis function and solve for the coefficients of F.

One possible basis that could be used is spherical harmonics, as you suggested. These are a set of orthogonal functions that are commonly used in problems involving spherical symmetry. Another option could be to use a set of radial basis functions, which are typically used for problems with radial symmetry.

It may also be helpful to consider the boundary conditions for this equation. Depending on the specific problem you are trying to solve, there may be additional constraints on the solution that could inform the choice of basis functions.

In any case, it would be important to carefully choose a basis that is well-suited to the problem at hand and can accurately represent the solution. This may require some trial and error and experimentation with different basis functions. Additionally, you may need to use a combination of different basis functions to accurately capture the behavior of F.

Overall, solving this equation numerically may require some computational effort and experimentation, but with the right approach and choice of basis functions, it is certainly possible to find a solution without using a 3D finite differencing scheme.