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：我们针对一组描述生物膜生长及其他过程（如多孔介质中的流动和野火蔓延）的拟线性退化演化方程，提出并分析了数值格式。该系统中的第一个方程为抛物型方程，具有退化与奇异扩散特性；第二个方程要么是一致抛物型，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，其优势在于能够解耦方程。我们证明了时间离散解的正性、有界性及其收敛于时间连续解的性质。随后，我们提出一种迭代线性化格式，用于求解由此产生的非线性时间离散问题。在关于时间步长的弱假设条件下，我们证明了该格式的收敛性不受空间离散化和网格的影响。此外，若问题非退化，收敛速度将随时间步长减小而加快。最后，通过采用有限元方法进行空间离散化，我们研究了该格式的行为，并将其性能与其他常用格式进行比较。这些数值测试证实了所提格式的鲁棒性与高效性。