2.3.3 Discretization in space by finite volumes

Finite volume methods:

• Restriction: integration over control volume
• Reconstruction: reconstruction (by polynomials)
• Derivatives can become finite differences (or can be avoided)

To go from continuous values of $\rho(x)$ to a discrete set of values $\rho_i$ for a grid with $x_i = \Delta x i + x_0$ we integrate over the ``control volume'' associated to each grid point. In one dimension that might be

$$\rho_i = \frac{1}{\Delta x} \int\limits_{x_i-\Delta x/2}^{x_i+\Delta x/2} \rho \left( x \right) dx \enspace .$$ (94)