An implicit Euler finite-volume scheme for a parabolic reaction-diffusion system modeling biofilm growth is analyzed and implemented. The system consists of a degenerate-singular diffusion equation for the biomass fraction, which is coupled to a diffusion equation for the nutrient concentration, and it is solved in a bounded domain with Dirichlet boundary conditions. By transforming the biomass fraction to an entropy-type variable, it is shown that the numerical scheme preserves the lower and upper bounds of the biomass fraction. The existence and uniqueness of a discrete solution and the convergence of the scheme are proved. Numerical experiments in one and two space dimensions illustrate, respectively, the rate of convergence in space of our scheme and the temporal evolution of the biomass fraction and the nutrient concentration.
翻译:分析和实施了用于模拟生物胶片生长的抛物体反扩散系统模型的隐含的 Euler 有限体积计划,该系统包括生物量分数的降解-脉冲扩散方程式,结合营养浓度的折射方程式,并在一个结合了Drichlet边界条件的封闭域内解决。通过将生物量分数转换成一个entropy型变量,可以证明数字方案保留了生物量分数的上下界。一个离散解决办法的存在和独特性以及该办法的趋同得到了证明。一个和两个空间层面的数值实验分别说明了我们方案空间的趋同速度以及生物量分数和营养浓度的时间演变情况。