Preprint 2026-28
Juan Barajas-Calonge, Julio Careaga, Luis M. Villada:
Invariant-region-preserving high-order schemes for a model of reactive sedimentation
Abstract:
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 force 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 spatially 1D governing equations include a nonlinear and non-homogeneous 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 making use of polynomial reconstructions. Two reconstruction methods are used, a second-order monotonic upstream-centered scheme (MUSCL), and a thirdorder 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 spatial discretization 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 experimentally show the theoretical order of accuracy of the developed schemes and the invariant-region preservation with and without diffusion.
