The Navier-Stokes equations describe continuous velocity and pressure fields. A computational fluid model must choose how those fields and their conservation laws are represented on a finite grid or mesh.

Spatial formulations

A finite-difference formulation replaces derivatives by discrete approximations at grid points. A finite-volume formulation balances fluxes across the faces of each control volume, making conservation explicit. Finite-element formulations use basis functions and weighted residual statements.

These approaches discretize the same continuum equations but produce different algebraic systems and numerical properties.

Conservation

For fluid flow, preserving mass and momentum balances is especially important. A locally inaccurate-looking formula can still be valuable if its discrete fluxes conserve the quantities that the continuum equations conserve.

Space and time are separate choices

A transient CFD scheme combines a spatial discretization with a time integrator. Refining the grid and reducing the time step address different approximation errors and need not be changed together.

Fluid discretization is therefore the domain-specific step of translating conservation laws into a finite system; the underlying finite-difference, convergence and time-stepping concepts are general numerical methods.