# Finite volume method

I have question connected with finite volume method. Consider equation

$$\frac{\partial u}{\partial t}=\operatorname{div}A\nabla u +f . \quad x\in \overline{B}_{1} (0)\subset \mathbb{R}^3 -\text{unit ball with its boundary}$$ I solved IBVP for this equation with a help of finite volume method, for grid I took uniform spherical coordinates grid,. (picture lower is related to topic) I have a problem with finding flows for the faces of cell: how to approximate the value $$\oint_{\partial V_{i,j,k}}(A\nabla u, n)dS$$ (where $A$ is a variable matrix)? When $A$ is an identity matrix everything is clear. Could you help me with this issue or to advise literature where this issue is represented?

• Product of A and grad u is a vector quantity - I would try to find value of each component of that vector at cell face. I see that as the main difficulty. The way you will do that will probably be similar to the procedure for discretization of diffusion in general transport eq. I'm not sure you are probably aware of that and look for more specific answer. Feb 4, 2014 at 16:14
• Actually, I thought almost in this way: to find the value of gradient, normal vector and matrix in the center of the cells' face and just write scalar product, and then just add these values for all faces. But also I thought that there are some special methods for doing it for my case.
– cool
Feb 4, 2014 at 17:11
• Or to find the product of the matrix and grad u, and then interpolate. Finding cell face values of any scalar (including individual components of your vector) using least squares gradients at cell centers may be a good idea... Feb 4, 2014 at 17:25
• Isn't the governing equation in Cartesian coordinate system? First you will need to write the equation in spherical coordinate system. Jan 28, 2017 at 9:27