arXiv subjects

Julio Careaga

Publications and source records attributed to Julio Careaga.

14 recordsLinked to original sources

arXiv2026

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.

math.NA↗

arXiv2026

Asymptotic Analysis of Shallow Water Moment Equations

The Shallow Water Moment Equations (SWME) are an extension of the Shallow Water Equations (SWE) for improved modelling of free-surface flows. In contrast to the SWE, the SWME incorporate vertical velocity profile information. The SWME framework approximates vertical velocity profiles using a polynomial expansion with Legendre polynomials and polynomial coefficients, also called moment variables. The SWME have an increased number of variables that must always be incorporated, even when the flow approaches a viscous slip equilibrium state that could be characterised by vanishing moment variables. To reduce the complexity of the SWME in cases proximate to this equilibrium, we conduct an asymptotic analysis of the SWME. This yields the closed form Reduced Shallow Water Moment Equations (RSWME) for deviations from the equilibrium. The RSWME have fewer variables, compared to the SWME. The hyperbolicity of the RSWME is analysed. Numerical tests include a wave with a sharp height gradient, a smoother height gradient and a square root velocity profile. The numerical tests demonstrate that the RSWME reduce computational cost up to 77% compared to the SWME and improves accuracy up to 88% over the SWE.

math.NA↗

arXiv2026

Entropy analysis and entropy stable DG methods for the 1D shallow water moment equations

We demonstrate that the one-dimensional shallow water moment equations satisfy an auxiliary entropy conservation law, where the entropy function corresponds to the total energy. Additionally, we show that the classical Newtonian slip friction and Manning friction terms are entropy dissipative with respect to the developed entropy variables. The results from the continuous entropy analysis are used to construct an entropy stable and well-balanced nodal discontinuous Galerkin spectral element method for the spatial approximation. Key to ensure the entropy stability of the scheme is the derivation of entropy conservative numerical fluxes that satisfy a discrete version of the entropy flux compatibility condition. Finally, numerical examples demonstrate the performance of the scheme and validate the theoretical results.

math.NA↗

arXiv2025

A moment model of shallow granular flows with variable friction laws

In this work, we develop a modelling framework for granular flows based on the shallow water moment equations on inclined planes. Under the assumption of a polynomial expansion of the velocity field, the model extends the classical shallow water equations to vertically variable velocity profiles. The friction effects, which are captured through the strain-rate tensor, are incorporated into the model in two terms, the bulk and bottom friction. We propose a modelling procedure to incorporate general friction laws into our framework and exemplify this combining the Manning, Coulomb, Savage-Hutter, and $\mu(I)$-rheology friction models in our modeling framework. Moreover, we develop a path-conservative finite volume numerical scheme based on the polynomial viscosity matrix method to properly handle the stiffness of the source terms. Numerical simulations are presented for different models of friction, including the case of wet-dry fronts.

math.NA↗

arXiv2025

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

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.

math.NA↗

arXiv2025

Finite element discretization of nonlinear models of ultrasound heating

Heating generated by high-intensity focused ultrasound waves is central to many emerging medical applications, including non-invasive cancer therapy and targeted drug delivery. In this study, we aim to gain a fundamental understanding of numerical simulations in this context by analyzing conforming finite element approximations of the underlying nonlinear models that describe ultrasound-heat interactions. These models are based on a coupling of a nonlinear Westervelt--Kuznetsov acoustic wave equation to the heat equation with a pressure-dependent source term. A particular challenging feature of the system is that the acoustic medium parameters may depend on the temperature. The core of our new arguments in the \emph{a prior} error analysis lies in devising energy estimates for the coupled semi-discrete system that can accommodate the nonlinearities present in the model. To derive them, we exploit the parabolic nature of the system thanks to the strong damping present in the acoustic component. Theoretically obtained optimal convergence rates in the energy norm are confirmed by the numerical experiments. In addition, we conduct a further numerical study of the problem, where we simulate the propagation of acoustic waves in liver tissue for an initially excited profile and under high-frequency sources.

math.NA↗

arXiv2024

Westervelt-based modeling of ultrasound-enhanced drug delivery

We investigate a nonlinear multiphysics model motivated by ultrasound-enhanced drug delivery. The acoustic pressure field is modeled by Westervelt's quasilinear wave equation to adequately capture the nonlinear effects in ultrasound propagation. The nonlocal attenuation characteristic for soft biological media is modeled by acoustic damping of the time-fractional type. Additionally, acoustic medium parameters are allowed to depend on the temperature of the medium. The wave equation is coupled to the nonlinear Pennes heat equation with a pressure-dependent source to account for ultrasound waves heating up the tissue. Finally, the drug concentration is obtained as the solution to an advection-diffusion equation with a pressure-dependent velocity. Toward gaining a rigorous understanding of this system, we set up a fixed-point argument in the analysis combined with devising energy estimates that can accommodate the time-fractional damping. The energy arguments are, in part, carried out by employing time-weighted test functions to reduce the regularity assumptions on the initial temperature. The analysis reveals that different smoothness of the initial pressure, temperature, and concentration fields is needed as well as smallness of the pressure-temperature data in order to ensure non-degeneracy of the system and establish well-posedness. Our theoretical considerations are complemented by a numerical investigation of the system under more realistic boundary conditions. The numerical experiments, performed in different computational scenarios, underline the importance of considering nonlinear effects when modeling ultrasound-targeted drug delivery.

math.AP↗

arXiv2024

Application of the thin-film equations in modelling of Marangoni flow patterns amongst surfactant source and drain locations

Surfactants that are deposited at aqueous liquid films have the ability to generate surface tension gradients at the air-water interface, and thereby induce Marangoni flow. Combined with the production and depletion of surfactants at different locations of source and drains, out-of-equilibrium surface tension gradients can be sustained, resulting in Marangoni flow patterns that drive e.g., self-organization of amphiphile myelin assemblies. Here, a mathematical model based on the thin-film equations is proposed to simulate these flow patterns. The model equations are based on the surfactant source and drain concentrations, film-height and surfactant bulk concentration. We present a numerical scheme for approximating the model equations and discuss the numerically observed properties of the model.

physics.flu-dyn↗

arXiv2024

A multilayer shallow water model for polydisperse reactive sedimentation

A three-dimensional model of polydisperse reactive sedimentation is developed by means of a multilayer shallow water approach. The model consists of a variety of solid particles of different sizes and densities, and substrates diluted in water, which produce biochemical reactions while the sedimentation process occurs. Based on the Masliyah-Lockett-Bassoon settling velocity, compressibility of the sediment and viscosity of the mixture, the system of governing equations is composed by non-homogeneous transport equations, coupled to a momentum equation describing the mass-average velocity. Besides, the free-surface depicted by the total height of the fluid column is incorporated and fully determined through the multilayer approach. A finite volume numerical scheme on Cartesian grids is proposed to approximate the model equations. Numerical simulations of the denitrification process exemplify the performance of the numerical scheme and model under different scenarios and bottom topographies.

math.NA↗

arXiv2023

Numerical schemes for a moving-boundary convection-diffusion-reaction model of sequencing batch reactors

Sequencing batch reactors (SBRs) are devices widely used in wastewater treatment, chemical engineering, and other areas. They allow for the sedimentation and compression of solid particles of biomass simultaneously with biochemical reactions with nutrients dissolved in the liquid. The kinetics of these reactions may be given by one of the established activated sludge models (ASMx). An SBR is operated in various stages and is equipped with a movable extraction and fill device and a discharge opening. A one-dimensional model of this unit can be formulated as a moving-boundary problem for a degenerating system of convection-diffusion reaction equations whose unknowns are the concentrations of the components forming the solid and liquid phases, respectively. This model is transformed to a fixed computational domain and is discretized by an explicit monotone scheme along with an alternative semi-implicit variant. The semi-implicit variant is based on solving, during each time step, a system of nonlinear equations for the total solids concentration followed by the solution of linear systems for the solid component percentages and liquid component concentrations. It is demonstrated that the semi-implicit scheme is well posed and that both variants produce approximations that satisfy an invariant region principle: solids concentrations are nonnegative and less or equal to a set maximal one, percentages are nonnegative and sum up to one, and substrate concentrations are nonnegative. These properties are achieved undera Courant-Friedrichs-Lewy (CFL) condition that is less restrictive for the semi-implicit than the explicit variant. Numerical examples with realistic parameters illustrate that as a consequence, the semi-implicit variant is more efficient than the explicit one.

math.NA↗

arXiv2022

A model of reactive settling of activated sludge: comparison with experimental data

A non-negligible part of the biological reactions in the activated sludge process for treatment of wastewater takes place in secondary settling tanks that follow biological reactors. It is therefore of interest to develop models of so-called reactive settling that describe the spatial variability of reaction rates caused by the variation of local concentration of biomass due to hindered settling and compression. A reactive-settling model described by a system of nonlinear partial differential equations and a numerical scheme are introduced for the simulation of hindered settling of flocculated particles, compression at high concentrations, dispersion of the flocculated particles in the suspension, dispersion of the dissolved substrates in the fluid, and the mixing that occurs near the feed inlet. The model is fitted to experiments from a pilot plant where the sedimentation tank has a varying cross-sectional area. For the reactions, a modified version of the activated sludge model no. 1 (ASM1) is used with standard coefficients. The constitutive functions for hindered settling and compression are adjusted to a series of conventional batch settling experiments after the initial induction period of turbulence and reflocculation has been transformed away. Further (but not substantial) improvements of prediction of experimental steady-state scenarios can be achieved by also fitting additional terms modelling hydrodynamic dispersion.

math.NA↗

arXiv2021

A moving-boundary model of reactive settling in wastewater treatment. Part 1: Governing equations

Reactive settling is the process of sedimentation of small solid particles in a fluid with simultaneous reactions between the components of the solid and liquid phases. This process is important in sequencing batch reactors (SBRs) in wastewater treatment plants. In that application the particles are biomass (bacteria; activated sludge) and the liquid contains substrates (nitrogen, phosphorus) to be removed through reactions with the biomass. The operation of an SBR in cycles of consecutive fill, react, settle, draw, and idle stages is modelled by a system of spatially one-dimensional, nonlinear, strongly degenerate parabolic convection-diffusion-reaction equations. This system is coupled via conditions of mass conservation to transport equations on a half line, whose origin is located at a moving boundary and that model the effluent pipe. An invariant-region-preserving finite difference scheme is used to simulate operating cycles and the denitrification process within an SBR.

math.NA↗

arXiv2021

A moving-boundary model of reactive settling in wastewater treatment. Part 2: Numerical scheme

A numerical scheme is proposed for the simulation of reactive settling in sequencing batch reactors (SBRs) in wastewater treatment plants. Reactive settling is the process of sedimentation of flocculated particles (biomass; activated sludge) consisting of several material components that react with substrates dissolved in the fluid. An SBR is operated in cycles of consecutive fill, react, settle, draw and idle stages, which means that the volume in the tank varies and the surface moves with time. The process is modelled by a system of spatially one-dimensional, nonlinear, strongly degenerate parabolic convection-diffusion-reaction equations. This system is coupled via conditions of mass conservation to transport equations on a half line whose origin is located at a moving boundary and that models the effluent pipe. A finite-difference scheme is proved to satisfy an invariant-region property (in particular, it is positivity preserving) if executed in a simple splitting way. Simulations are presented with a modified variant of the established activated sludge model no.~1 (ASM1).

math.NA↗

arXiv2020

A method-of-lines formulation for a model of reactive settling in tanks with varying cross-sectional area

Reactive settling denotes the process of sedimentation of small solid particles dispersed in a viscous fluid with simultaneous reactions between the components that constitute the solid and liquid phases. This process is of particular importance for the simulation and control of secondary settling tanks (SSTs) in water resource recovery facilities (WRRFs), formerly known as wastewater treatment plants. A spatially one-dimensional model of reactive settling in an SST is formulated by combining a mechanistic model of sedimentation with compression with a model of biokinetic reactions. In addition, the cross-sectional area of the tank is allowed to vary as a function of height. The final model is a system of strongly degenerate parabolic, nonlinear partial differential equations (PDEs)that include discontinuous coefficients to describe the feed, underflow and overflow mechanisms, as well as singular source terms that model the feed mechanism. A finite difference scheme for the final model is derived by first deriving a method-of-lines formulation (discrete in space, continuous in time), and then passing to a fully discrete scheme by a time discretization. The advantage of this formulation is its compatibility with common practice in development of software for WRRFs. The main mathematical result is an invariant-region property, which implies that physically relevant numerical solutions are produced. Simulations of denitrification in SSTs in wastewater treatment illustrate the model and its discretization.

math.NA↗