If the conservation equation is rewritten in a form which resembles the ordinary differential equation, any method of time integration can be used $$ \frac{\partial (\rho \phi)}{\partial t} = - \nabla \cdot (\rho \phi \mathbf{v}) + \nabla \cdot \big( \Gamma \nabla \phi \big) + q_{\phi} = f\big(t, \phi(t)\big) $$ In order to study properties of the explicit Euler and other simple schemes, we consider the 1D version of the conservation equation with constant velocity, constant fluid properties, and no source terms $$ \frac{\partial \phi}{\partial t} = -u \frac{\partial \phi}{\partial x} + \frac{\Gamma}{\rho} \frac{\partial^2 \phi}{\partial x^2} $$ It assumes that the important balance is between advection and streamwise diffusion, a balance that rarely occurs in real flows. For this reason, one must be careful about extending what is learned from this equation to the Navier–Stokes equations

First assume that the spatial derivatives are approximated using central difference scheme and that the grid is uniform in the $x$-direction. In this case the same algebraic equation results from both Finite Difference and Finite Volume discretizations. The new variable value $\phi_i^{n+1}$ is: $$ \phi_i^{n+1} = \phi_i^n + \left[ -u \frac{\phi_{i+1}^n - \phi_{i-1}^n}{2 \Delta x} + \frac{\Gamma}{\rho} \frac{\phi_{i+1}^n + \phi_{i-1}^n - 2 \phi_i^n}{(\Delta x)^2} \right] \Delta t \Rightarrow \phi_i^{n+1} = (1 - 2d)\,\phi_i^n + \left(d - \tfrac{c}{2}\right)\phi_{i+1}^n + \left(d + \tfrac{c}{2}\right)\phi_{i-1}^n $$ where we introduced the dimensionless parameters $$ \overbrace{d \;=\; \underbrace{\frac{\Gamma\,\Delta t}{\rho(\Delta x)^2}}_{\text{time required for a disturbance to diffuse over }\Delta x}}^{\substack{\text{ratio of }\Delta t\text{ to}\\\text{characteristic diffusion time}}} \quad \overbrace{c \;=\; \underbrace{\frac{u\,\Delta t}{\Delta x}}_{\text{time required for a disturbance to convect over }\Delta x}}^{\substack{\text{ratio of }\Delta t\text{ to}\\\text{characteristic convection time}}} $$ The ratio ($c$) is called the Courant number and is one of the key parameters in computational fluid dynamics. This number is often called the CFL number, where CFL stands for the initials of R. Courant, K. Friedrichs and H. Lewy, who first defined it in their paper from 1928.