标题： 一种用于慢砂滤生物膜生长的对流-反应-Cahn-Hilliard多相模型的保不变区域格式 显示英文标题

标题： An invariant-region-preserving scheme for a convection-reaction-Cahn-Hilliard multiphase model of biofilm growth in slow sand filters

摘要： 从慢速砂滤器上清液中推导出了一种生物膜生长的多维模型。 该两相模型由固相和液相组成，被表述为一个对流-反应系统，其中包含一个具有退化移动性的 Cahn-Hilliard 型方程，并与混合物速度的 Stokes 流方程耦合。采用上游间断 Galerkin 方法来逼近对流-反应方程，同时为 Stokes 系统提出了一种 primal 形式的$H^1$-兼容方法。 由于反应项的原因，通过分裂过程证明了浓度未知数的一个不变区域原理，即所有相的非负性和固相总浓度的上限。 提出了简化生物膜反应的数值例子以展示模型和数值方案的性能。 显示英文摘要

摘要： A multidimensional model of biofilm growth present in the supernatant water of a Slow Sand Filter is derived. The multiphase model, consisting of solid and liquid phases, is written as a convection-reaction system with a Cahn-Hilliard-type equation with degenerate mobility coupled to a Stokes-flow equation for the mixture velocity. An upwind discontinuous Galerkin approach is used to approximate the convection-reaction equations, whereas an $H^1$-conforming primal formulation is proposed for the Stokes system. By means of a splitting procedure due to the reaction terms, an invariant-region principle is shown for the concentration unknowns, namely non-negativity for all phases and an upper bound for the total concentration of the solid phases. Numerical examples with reduced biofilm reactions are presented to illustrate the performance of the model and numerical scheme.