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.

翻译：我们提出并分析了用于拟线性退化演化方程组（该方程组可建模生物膜生长及多孔介质流动、野火蔓延等过程）的数值格式。方程组中第一个方程为抛物型，包含退化与奇异扩散项；第二个方程要么是均匀抛物型，要么是常微分方程。首先，我们引入一种半隐式时间离散方法，其优势在于能解耦方程组。我们证明了时间离散解的正性、有界性及其向时间连续解的收敛性。接着，我们提出迭代线性化方案来求解所得非线性时间离散问题。在时间步长较弱的假设下，我们证明了该方案的收敛性与空间离散及网格无关。此外，若问题非退化，收敛速度随时间步长减小而加快。最后，采用有限元法进行空间离散，我们研究了该方案的行为，并将其性能与其他常用方案进行比较。这些测试证实了所提方案具有鲁棒性和快速性。