Invariant-region-preserving high-order schemes for a model of reactive sedimentation
Reactive sedimentation of mixtures composed by biological and inert solid matter including liquid substrates is modelled by a system of convection-diffusion-reaction equations. In this process, the small solid particles dispersed in the viscous fluid settle due to gravity while simultaneous chemical reactions take place between solid and liquid components (substrates). Among the applications of reactive sedimentation processes, the principal use is in the simulation and control of secondary settling tanks (SSTs) in water resource recovery facilities (WRRFs). The governing equations include a nonlinear strongly degenerate parabolic equation for the total concentration of solids, and transport equations including reaction terms for the solid components and substrates. To accurately approximate the model equations, a high-order finite volume scheme is developed using polynomial reconstructions. Two reconstruction methods are used, a second-order monotonic upstream-centered scheme (MUSCL), and a third-order central weighted essentially non-oscillatory (CWENO3) method. In addition, for the time discretization, we employ strong stability preserving Runge-Kutta (SSPRK) schemes of second- and third-order. The CWENO3 scheme is complemented with limiters, and in the case of zero diffusion, the designed high-order schemes are shown to preserve an invariant-region property. This property is particularly relevant as it ensures that physically relevant numerical solutions are obtained at each time iteration. In addition, high-order approximations for the diffusion terms are proposed to approximate the complete model equations. Simulations of denitrification in SSTs under a continuously operated regime in WRRFs demonstrate the performance of the model and its discretization. Finally, convergence tests show the order of accuracy of the developed scheme and the invariant-region preservation.