In this paper, we use three existing schemes namely, Upwind Forward Euler, Non-Standard Finite Difference
(NSFD) and Unconditionally Positive Finite Difference (UPFD) schemes to solve two numerical
experiments described by a linear and a non-linear advection-diffusion-reaction equation with constant
coefficients. These equations model exponential travelling waves and biofilm growth on a medical implant
respectively. We study the exact and numerical dissipative and dispersive properties of the three
schemes for both problems. Moreover, L1 error, dispersion and dissipation errors, at some values of
temporal and spatial step sizes have been computed for the three schemes for both problems.