  We propose and analyse numerical schemes for a system of quasilinear,
degenerate evolution equations modelling biofilm growth as well as other
processes such as flow through porous media and the spreading of wildfires. The
first equation in the system is parabolic and exhibits degenerate and singular
diffusion, while the second is either uniformly parabolic or an ordinary
differential equation. First, we introduce a semi-implicit time discretisation
that has the benefit of decoupling the equations. We prove the positivity,
boundedness, and convergence of the time-discrete solutions to the
time-continuous solution. Then, we introduce an iterative linearisation scheme
to solve the resulting nonlinear time-discrete problems. Under weak assumptions
on the time-step size, we prove that the scheme converges irrespective of the
space discretisation and mesh. Moreover, if the problem is non-degenerate, the
convergence becomes faster as the time-step size decreases. Finally, employing
the finite element method for the spatial discretisation, we study the
behaviour of the scheme, and compare its performance to other commonly used
schemes. These tests confirm that the proposed scheme is robust and fast.
