In the presentation here about Level set methods
http://www.cs.au.dk/~bang/smokeandwater2006/Lecture9_IntroToWaterAndLS.ppt
the author constructs a linear PDE
$\frac{\partial \phi}{\partial t} + \triangledown \phi \cdot V = 0$
In order to solve this numerically, first the author uses forward Euler time step for discretization. Then, he "discretizes the gradient" and mentions various choices for this, one of which being
$\frac{\phi_i - \phi^n_{i-1}}{\Delta t} = 1$
Other choices he mentions are discretizations equaling 0, or 0.5. I do not understand how these values, 1, 0, 0.5 are chosen.
I know gradient is a vector calculated using partial derivatives, at a specific point x_0, y_0, .. which can also result in a scalar if we want to calculate them for a certain direction, then we take a dot product. It does not make any sense to me discretization of a gradient can automatically be made equal to a specific value.
Any ideas?