Finite Volume Approximation
- The usual approach is to define control volumes by a suitable grid and assign the computational node to the control volume center
- However, one could as well (for structured grids) define the nodal locations first and construct control volumes around them, so that control volume faces lie midway between nodes
The advantage of the first approach is that the nodal value represents the mean over the control volume volume to higher accuracy (second order) than in the second approach, because the node is located at the centroid of the control volume.
The advantage of the second approach is that central difference scheme approximations of derivatives at control volume faces are more accurate when the face is midway between two nodes.
A typical control volume and the notation used for a Cartesian 2D grid
A typical control volume and the notation used for a Cartesian 3D grid
1Ferziger, Joel & Perić, Milovan & Street, Robert. (2020). Computational Methods for Fluid Dynamics. 10.1007/978-3-319-99693-6.