Linearisation has many different meanings.. one is spectral methods, it is also used to linearize the system.
The others are numerical schemes like finite difference methods. Like forward, backward,
forward difference: U = [ U_(t+1) - U_t ] / dt
backward difference: U = [ U_t - U_(t-1) ] / dt
Basically, you try to model the higher order terms, derivatives etc with simple algebraic functions.
When you linearize then you get truncation errors and also an artificial viscosity term arises called CFL thanks to the numerical methods and this term has the effect of edge smoothing and determines the convergence or divergence of the problem based on step time. Because we want to reduce the computational time from orderOf(N^2) to N-log(N).
I need to go through my notices. BTW which text books are you following? and is it graduate or undergrad level course?