Advection-diffusion Model Research Articles

Stability analysis study for the time-fractional Galilei invariant advection-diffusion model of distributive order using an efficient hybrid approach

Abstract In this manuscript, a new model of the time-fractional Galilei-invariant advection-diffusion model of distributed order is studied. An efficient hybrid numerical approach with high accuracy is used to estimate this equation. The finite difference numerical method is used to approximate the fractional operator in terms of the time variable and to approximate the integral term of distributed order, the Gaussian–Legendre integration is applied. To obtain a fully discrete numerical approach, we used a spectral element numerical approach, in which Legendre polynomials are used as the basis function. For the proposed numerical approach, the error and stability analysis are studied. For the efficiency of the numerical approach, some numerical examples are presented with graphs and tables.

The formation and ventilation of an oxygen minimum zone in a simple model for latitudinally alternating zonal jets

Abstract. An advection–diffusion model coupled to a simple dynamical ocean model is used to illustrate the formation and ventilation of an oxygen minimum zone. The advection–diffusion model carries a passive tracer with a source at the western boundary and a Newtonian damping term to mimic oxygen consumption. The dynamical model is a non-linear one-and-a-half-layer reduced-gravity model. The latter is forced by an annually oscillating mass flux confined to the near-equatorial band that, in turn, leads to the generation of mesoscale eddies and latitudinally alternating zonal jets at higher latitudes. The model uses North Atlantic geometry and develops a tracer minimum zone remarkably similar in location to the observed oxygen minimum zone in the eastern tropical North Atlantic (ETNA). This is despite the absence of wind forcing for a subtropical gyre and the shadow zone predicted by the ventilated thermocline theory. Although the model is forced only at the annual period, the model nevertheless exhibits decadal and multidecadal variability in its spun-up state. The associated trends are comparable to observed trends in oxygen within the ETNA oxygen minimum zone. Notable exceptions are the multi-decadal decrease in oxygen in the lower-oxygen minimum zone and the sharp decrease in oxygen in the upper-oxygen minimum zone between 2006 and 2013.

Analysis of a vector-borne disease model with vector-bias mechanism in advective heterogeneous environment

Abstract This study proposed and analyzed a vector-borne reaction–diffusion–advection model with vector-bias mechanism and heterogeneous parameters in one-dimensional habitat. The basic reproduction number R 0 {{\mathfrak{R}}}_{0} in connection with principal eigenvalue of elliptic eigenvalue problem is characterized as the role of determining the threshold dynamics of the system. The main objective of this study is to investigate the asymptotic profiles and monotonicity of R 0 {{\mathfrak{R}}}_{0} with respect to diffusion rates and advection rates under certain conditions. Through exploring the level set of R 0 {{\mathfrak{R}}}_{0} , we also find that there exists a unique surface separating the dynamics. Our results also reveal that the infected hosts and vectors will aggregate at the downstream end if the ratio of advection rates and diffusion rates is sufficiently large.

Heat and mass transfer visualisation of an exothermic acid-base microfluidic reactor using infrared thermospectroscopy

ABSTRACT Here, a microfluidic reactor that is transparent in the mid-wave infrared spectrum with an effective path length of 38 µm is developed for characterisation via operando Fourier-transform Infrared (FTIR) thermospectroscopy. This imaging technique couples an infrared (IR) camera with an FTIR spectrometer to quantify heat and mass transfer in the exothermic acid-base reaction between hydrochloric acid (HCl) and sodium hydroxide (NaOH). The absorbance fields of the reaction are used to simultaneously derive the molar concentration fields of the reactants (HCl and NaOH) and products (sodium chloride [NaCl]) at the microscale through a linear inverse method. The heat generated at the reaction interface is visualised through a thermal field captured from the measured proper emission. Concentration and thermal fields are then compared to an advection-diffusion model for validation. The results proposed in this work present a refined contactless calorimetry process for characterising the heat and mass transport in an acid-base reaction. Further refinement of this work will enable accurate and non-invasive measurements for the enthalpy of reaction in chemical reactions.

Coexistence of two strongly competitive species in a reaction–advection–diffusion system

The main focus of this article is to investigate the behavior of two strongly competitive species in a spatially heterogeneous environment using a Lotka–Volterra-type reaction–advection–diffusion model. The model assumes that one species diffuses at a constant rate, while the other species moves toward a more favorable environment through a combination of constant diffusion and directional movement. The study finds that no stable coexistence can be guaranteed when both species disperse randomly. In contrast, stable coexistence between the two species is possible when one of the species exhibits advection–diffusion. The study also reveals the existence of unstable coexistence imposed by bistability in a strongly competitive system, regardless of the diffusion type. The results are obtained by analyzing the stability of semitrivial solutions. The study concludes that the species moving toward a better environment has a competitive advantage, allowing them to survive even when their population density is initially low. Finally, the study identifies the unique globally asymptotically stable coexistence steady states of the system at high advection rates, particularly for relatively moderate interspecific competition parameters in species with directional movement. These findings underscore the crucial role of directed movement and interspecific competition coefficients in shaping the dynamics and coexistence of strongly competing species.

Structured methods for parameter inference and uncertainty quantification for mechanistic models in the life sciences.

Parameter inference and uncertainty quantification are important steps when relating mathematical models to real-world observations and when estimating uncertainty in model predictions. However, methods for doing this can be computationally expensive, particularly when the number of unknown model parameters is large. The aim of this study is to develop and test an efficient profile likelihood-based method, which takes advantage of the structure of the mathematical model being used. We do this by identifying specific parameters that affect model output in a known way, such as a linear scaling. We illustrate the method by applying it to three toy models from different areas of the life sciences: (i) a predator-prey model from ecology; (ii) a compartment-based epidemic model from health sciences; and (iii) an advection-diffusion reaction model describing the transport of dissolved solutes from environmental science. We show that the new method produces results of comparable accuracy to existing profile likelihood methods but with substantially fewer evaluations of the forward model. We conclude that our method could provide a much more efficient approach to parameter inference for models where a structured approach is feasible. Computer code to apply the new method to user-supplied models and data is provided via a publicly accessible repository.

A new class of efficient high order semi-Lagrangian IMEX discontinuous Galerkin methods on staggered unstructured meshes

In this paper we present a new high order semi-implicit discontinuous-Galerkin (DG) scheme on two-dimensional staggered triangular meshes applied to different nonlinear systems of hyperbolic conservation laws such as advection-diffusion models, incompressible Navier-Stokes equations and natural convection problems. While the temperature and pressure field are defined on a triangular main grid, the velocity field is defined on a quadrilateral edge-based staggered mesh. A semi-implicit time discretization is proposed, which separates slow and fast time scales by treating them explicitly and implicitly, respectively. The nonlinear convection terms are evolved explicitly using a semi-Lagrangian approach, whereas we consider an implicit discretization for the diffusion terms and the pressure contribution. High-order of accuracy in time is achieved using a new flexible and general framework of IMplicit-EXplicit (IMEX) Runge-Kutta schemes specifically designed to operate with semi-Lagrangian methods. To improve the efficiency in the computation of the DG divergence operator and the mass matrix, we propose to approximate the numerical solution with a less regular polynomial space on the edge-based mesh, which is defined on two sub-triangles that split the staggered quadrilateral elements. This allows for a fast computation of the DG operators and matrices without any need to store them for each mesh element. Due to the implicit treatment of the fast scale terms, the resulting numerical scheme is unconditionally stable for the considered governing equations because the semi-Lagrangian approach is the only explicit discretization for convection terms that does not need any stability restriction on the maximum admissible time step. Contrarily to a genuinely space-time discontinuous-Galerkin scheme, the IMEX discretization permits to preserve the symmetry and the positive semi-definiteness of the arising linear system for the pressure that can be solved at the aid of an efficient matrix-free implementation of the conjugate gradient method. We present several convergence results, including nonlinear transport and density currents, up to third order of accuracy in both space and time.

A nonlocal reaction–diffusion–advection model with free boundaries

A nonlocal reaction–diffusion–advection model with free boundaries

The convergence of high-accuracy three-point compact computational method for structural derivative anomalous diffusion–advection model

The convergence of high-accuracy three-point compact computational method for structural derivative anomalous diffusion–advection model

Modeling of brain efflux: Constraints of brain surfaces

Fluid efflux from the brain plays an important role in solute waste clearance. Current experimental approaches provide little spatial information, and data collection is limited due to short duration or low frequency of sampling. One approach shows tracer efflux to be independent of molecular size, indicating bulk flow, yet also decelerating like simple membrane diffusion. In an apparent contradiction to this report, other studies point to tracer efflux acceleration. We here develop a one-dimensional advection-diffusion model to gain insight into brain efflux principles. The model is characterized by nine physiological constants and three efflux parameters for which we quantify prior uncertainty. Using Bayes' rule and the two efflux studies, we validate the model and calculate data-informed parameter distributions. The apparent contradictions in the efflux studies are resolved by brain surface boundaries being bottlenecks for efflux. To critically test the model, a custom MRI efflux assay measuring solute dispersion in tissue and release to cerebrospinal fluid was employed. The model passed the test with tissue bulk flow velocities in the range 60 to 190 [Formula: see text]m/h. Dimensional analysis identified three principal determinants of efflux, highlighting brain surfaces as a restricting factor for metabolite solute clearance.

Physicochemical and Heavy Metal Analysis of Industrial Pollution Model Calibration and Validation Using Field Measurements

This study assessed industrial pollution impacts on rivers near Port Harcourt, Nigeria using physicochemical analysis, heavy metal data, and contaminant transport modeling. Key findings show the rivers are severely degraded with extreme levels of heavy metals, low dissolved oxygen, high conductivity, nutrients, and fecal bacteria exceeding water quality guidelines. Uncontrolled effluent discharges from industries are primarily responsible. A first-order advection-diffusion model reliably predicted the rapid initial dilution and slower downstream attenuation of metals like magnesium and cadmium. The model was successfully calibrated and validated using field measurements to determine key transport parameters including dispersion coefficients. The severe contamination indicates current effluent treatment and regulations are inadequate and require urgent strengthening to control sources, expand monitoring, remediate contamination, restore habitats, and protect ecosystem and public health. Sustained engagement of government, industry, communities and researchers is essential to devise integrated solutions that improve water quality. The modeling provides quantitative guidance on pollution impacts and mitigation needs. Further work should refine predictions, establish ecological thresholds, and validate model results with biomonitoring to support evidence-based, adaptive management.

Analysis of a spatio-temporal advection-diffusion model for human behaviors during a catastrophic event

In this work, using the theory of first-order macroscopic crowd models, we introduce a compartmental advection–diffusion model, describing the spatio-temporal dynamics of a population in different human behaviors (alert, panic and control) during a catastrophic event. For this model, we prove the local existence, uniqueness and regularity of a solution, as well as the positivity and [Formula: see text]-boundedness of this solution. Then, in order to study the spatio-temporal propagation of these behavioral reactions within a population during a catastrophic event, we present several numerical simulations for different evacuation scenarios.

A fast difference scheme for the multi-term time fractional advection–diffusion equation with a non-linear source term

In field experiments involving the movement of solutes through heterogeneous porous and fractured media, it has been observed that the contaminant plumes can transition between various diffusive states. This current research is dedicated to the development of a fast difference scheme designed for a multi-term time-fractional order advection–diffusion model that incorporates a nonlinear source term. This model is an instrumental tool in explaining the underlying dynamics of transport. To effectively address the substantial computational and storage challenges inherent in this problem, we present an efficient algorithm specifically designed for the fast computation of the Caputo derivative. This algorithm is based on the summation of exponentials technique. We thoroughly explore the theoretical aspects of our scheme including its unconditional stability and convergence with rigorous proofs. The proposed numerical scheme minimizes the CPU time and storage requirements when compared to the conventional direct difference scheme.

Weakly nonlinear analysis of a two-species non-local advection–diffusion system

Nonlocal interactions are ubiquitous in nature and play a central role in many biological systems. In this paper, we perform a bifurcation analysis of a widely-applicable advection–diffusion model with nonlocal advection terms describing the species movements generated by inter-species interactions. We use linear analysis to assess the stability of the constant steady state, then weakly nonlinear analysis to recover the shape and stability of non-homogeneous solutions. Since the system arises from a conservation law, the resulting amplitude equations consist of a Ginzburg–Landau equation coupled with an equation for the zero mode. In particular, this means that supercritical branches from the Ginzburg–Landau equation need not be stable. Indeed, we find that, depending on the parameters, bifurcations can be subcritical (always unstable), stable supercritical, or unstable supercritical. We show numerically that, when small amplitude patterns are unstable, the system exhibits large amplitude patterns and hysteresis, even in supercritical regimes. Finally, we construct bifurcation diagrams by combining our analysis with a previous study of the minimizers of the associated energy functional. Through this approach we reveal parameter regions in which stable small amplitude patterns coexist with strongly modulated solutions.

Dynamical behavior of solutions of a reaction–diffusion–advection model with a free boundary

Dynamical behavior of solutions of a reaction–diffusion–advection model with a free boundary

A Lifshitz-Slyozov type model for adipocyte size dynamics: limit from Becker-Döring system and numerical simulation.

Biological data show that the size distribution of adipose cells follows a bimodal distribution. In this work, we introduce a Lifshitz-Slyozov type model, based on a transport partial differential equation, for the dynamics of the size distribution of adipose cells. We prove a new convergence result from the related Becker-Döring model, a system composed of several ordinary differential equations, toward mild solutions of the Lifshitz-Slyozov model using distribution tail techniques. Then, this result allows us to propose a new advective-diffusive model, the second-order diffusive Lifshitz-Slyozov model, which is expected to better fit the experimental data. Numerical simulations of the solutions to the diffusive Lifshitz-Slyozov model are performed using a well-balanced scheme and compared to solutions to the transport model. Those simulations show that both bimodal and unimodal profiles can be reached asymptotically depending on several parameters. We put in evidence that the asymptotic profile for the second-order system does not depend on initial conditions, unlike for the transport Lifshitz-Slyozov model.

Equivalent analytical models for assessment of landfill composite liners

An equivalent analytical model based on time moments is proposed to estimate the properties of landfill composite liners. The service performance of landfill liners composed of a geomembrane layer (GMBL), a geosynthetic clay layer (GCL), and an attenuation layer (AL) and a GMBL, compacted clay layer (CCL), and AL were equivalently analyzed. The equivalent analysis and quantitative correlation of contaminants at the bottom of the GMBL/GCL/AL and GMBL/CCL/AL systems were considered according to the relative concentration (CN), instantaneous flux (JI) and accumulative flux (JA). The relative concentration (CN) was found to be the most conservative equivalent index. The thickness equivalent relation of the AL for the different composite liners was consequently obtained using the pure diffusion equivalent model and the advection–diffusion equivalent model. It was found that the pure diffusion thickness equivalent relationship was a simplified linear correlation and independent of time. The quadratic polynomial equivalent equations of AL thickness for different service times were obtained by the group method of data handling. The results showed that GCL-based composite liner had a better environmental protection performance than the CCL-based composite liner. When the thickness of the GMBL/CCL/AL composite liner was fixed, the corresponding equivalent thickness of AL in the GMBL/GCL/AL composite liner decreased with an increase in the leachate head and the wrinkle length for the same value of CN.

Modeling the impact of directional movement on mosquito population suppress

Due to mosquito habitat heterogeneity, its directional movement has been observed. In this paper, we develop a reaction–advection–diffusion model to explore mosquito populations suppression with sterile insect technique in spatially heterogeneous environments. By employing theoretical analysis, we show that the dynamic behavior of mosquitoes is closely related to their directional movement when the sterile mosquito amount is small. Numerical simulations further verify the theoretical analysis.

Comparison of ensemble‐based data assimilation methods for sparse oceanographic data

AbstractProbabilistic forecasts in oceanographic applications, such as drift trajectory forecasts for search‐and‐rescue operations, face challenges due to high‐dimensional complex models and sparse spatial observations. We discuss localisation strategies for assimilating sparse point observations and compare the implicit equal‐weights particle filter and a localised version of the ensemble‐transform Kalman filter. First, we verify these methods thoroughly against the analytic Kalman filter solution for a linear advection diffusion model. We then use a nonlinear simplified ocean model to do state estimation and drift prediction. The methods are rigorously compared using a wide range of metrics and skill scores. Our findings indicate that both methods succeed in approximating the Kalman filter reference for linear models of moderate dimensions, even for small ensemble sizes. However, in high‐dimensional settings with a nonlinear model, we discover that the outcomes are significantly influenced by the dependence of the ensemble Kalman filter on relaxation and the particle filter's sensitivity to the chosen model error covariance structure. Upon proper relaxation and localisation parametrisation, the ensemble Kalman filter version outperforms the particle filter in our experiments.

Structure preservation in high-order hybrid discretisations of potential-driven advection-diffusion: linear and nonlinear approaches

<abstract><p>We are interested in the high-order approximation of anisotropic, potential-driven advection-diffusion models on general polytopal partitions. We study two hybrid schemes, both built upon the Hybrid High-Order technology. The first one hinges on exponential fitting and is linear, whereas the second is nonlinear. The existence of solutions is established for both schemes. Both schemes are also shown to possess a discrete entropy structure, ensuring that the long-time behaviour of discrete solutions mimics the PDE one. For the nonlinear scheme, the positivity of discrete solutions is a built-in feature. On the contrary, we display numerical evidence indicating that the linear scheme violates positivity, whatever the order. Finally, we verify numerically that the nonlinear scheme has optimal order of convergence, expected long-time behaviour, and that raising the polynomial degree results, also in the nonlinear case, in an efficiency gain.</p></abstract>