Convection Diffusion Reaction Research Articles

Mathematical modelling of a self-oscillating catalytic reaction in a flow reactor

The article is devoted to the analysis of possible spatiotemporal kinetic structures that can arise during catalytic oxidation reactions on metal surfaces at atmospheric pressure. The catalytic oscillatory reaction in a flow reactor is modeled using a 1D system of equations of the reaction–diffusion–convection type. The STM type oscillatory reaction model of catalytic oxidation is used as a kinetic model. The obtained results of mathematical modelling show the decisive influence of an axial mixing in the reactor on the development of spatiotemporal structures. It is also shown that, depending on the ratio of adsorption constants of reacting species, three different isothermal spatiotemporal structures can arise, namely a spatially inhomogeneous stationary state, regular and aperiodic “breathing structures”.

Tensor Network Space-Time Spectral Collocation Method for Time-Dependent Convection-Diffusion-Reaction Equations

Emerging tensor network techniques for solutions of partial differential equations (PDEs), known for their ability to break the curse of dimensionality, deliver new mathematical methods for ultra-fast numerical solutions of high-dimensional problems. Here, we introduce a Tensor Train (TT) Chebyshev spectral collocation method, in both space and time, for the solution of the time-dependent convection-diffusion-reaction (CDR) equation with inhomogeneous boundary conditions, in Cartesian geometry. Previous methods for numerical solution of time-dependent PDEs often used finite difference for time, and a spectral scheme for the spatial dimensions, which led to a slow linear convergence. Spectral collocation space-time methods show exponential convergence; however, for realistic problems they need to solve large four-dimensional systems. We overcome this difficulty by using a TT approach, as its complexity only grows linearly with the number of dimensions. We show that our TT space-time Chebyshev spectral collocation method converges exponentially, when the solution of the CDR is smooth, and demonstrate that it leads to a very high compression of linear operators from terabytes to kilobytes in TT-format, and a speedup of tens of thousands of times when compared to a full-grid space-time spectral method. These advantages allow us to obtain the solutions at much higher resolutions.

Effect of vascular compression and drug injection time on thrombolytic therapy: Numerical simulation and validation

Venous thromboembolism (VTE) has been occurring frequently in human society. There is an urgent need to study the influence of several factors on thrombolytic therapy, such as the effects of vascular pressure levels (VPL) and the drug injection time (DIT). Considering blood as a non-Newtonian fluid, valve as a hyperelastic material, and thrombus as a porous medium, a new numerical simulation model of biofluid mechanics incorporating fluid–solid coupling phenomena and biochemical substance reactions is established based on the N-S equations and the convection–diffusion reaction equations. Then, a unique in vitro experimental platform is established to verify the correctness of the constructed mathematical model. The results showed that vascular compression resulted in significant differences in blood flow status localized within the vessel. Vascular compression causes the blood boosting index to fluctuate and the valve displacement values are 135% and 158% greater than the lower VPL, respectively. At the same time, vascular compression weakened vortex intensity, accelerated material transport and response, and improved the treatment. Compared with low VPL, the therapeutic efficacy increased by 7% and 15%, respectively. In addition, when the dose of the drug is high, different injection times can increase the therapeutic effect to different degrees, with a maximum difference of 12%. Our in vitro experiments are similar to the results obtained by numerical simulation, which can verify the reliability of numerical simulation. The computational model proposed and the experimental platform designed in this study have the potential to assist in clinical medication prediction in different venous thromboembolism patients.

Periodic wave solutions for a generalized reaction–convection–diffusion equation of high-order

In this paper, we investigate the uniqueness of periodic wave solutions for a generalized reaction–convection–diffusion equation with arbitrarily high-order reaction term or convection term. The main technique is to prove the monotonicity of the ratio of two Abelian integrals by a new criterion and Descartes’ rule of signs. A positive answer to a conjecture stated in Wei and Chen (2023) is given and some numeric simulations are carried out to illustrate the obtained theoretical results.

Development of a Reaction–Diffusion–Convection Model and Its Application to NO Oxidation in Reactors

Development of a Reaction–Diffusion–Convection Model and Its Application to NO Oxidation in Reactors

Error Analysis of the Classical Artificial Diffusion Weak Galerkin Finite Element Method for the steady state- convection diffusion-reaction Equation in 2-D

In this study, we modified the error in the weak Galerkin method when solving problems in which diffusion is the dominant convection( h) in two dimensions. This is done by adding the artificial diffusion term (-δΔw, where δ=h-ϵ). The finite element method for discrete functions using a weakly defined gradient operator is presented in this study. The concept of the weak discrete gradient is introduced, which plays an important role when using numerical methods to solve partial differential equations. The goal of this study is to enhance the accuracy and stability of the solutions by studying the ellipticity and stability properties of the method, which works to ensure that the numerical method retains the properties of the original equation while reducing the fluctuations occurring with the weak galerkin finite element method. Specific theories have been used to estimate the error in parameters -norm, and . Practical examples demonstrate how this method improves the handling of partial equations characterized by convection-dominated diffusion, enhancing its potential for advancing numerical simulations in engineering and physics.

Isolated periodic traveling waves of some reaction convection diffusion equations

AbstractIn this paper, we discuss the existence and uniqueness of isolated periodic traveling wave solutions of a family of nonlinear reaction–convection–diffusion equations using the monotonicity of the ratio of Abelian integrals. A numerical study is also presented.

Numerical solution of Fisher’s equation through the application of Haar wavelet collocation method

This research article presents algorithm that employs the Haar wavelet collocation method (HWCM) to solve the convection diffusion reaction (CDR) model, specifically Fisher’s equation (F.E.). The proposed method is suitable for addressing both linear and nonlinear problems. The Haar wavelet method is a numerical technique that utilizes wavelet basis functions to provide an approximation of the solution for the Fisher’s equations. Additionally, a forward difference scheme is utilized to approximate the time derivative while the highest order space derivative is replaced by Haar function. The performance of the technique is demonstrated through some examples investigated in this study. The maximum absolute errors are computed for varying numbers of collocation points. The HWCM provides superior accuracy, flexibility, convenience, and computational efficiency compared to alternative methods.

A nodally bound-preserving finite element method for reaction–convection–diffusion equations

This paper introduces a novel approach to approximate a broad range of reaction–convection–diffusion equations using conforming finite element methods while providing a discrete solution respecting the physical bounds given by the underlying differential equation. The main result of this work demonstrates that the numerical solution achieves an accuracy of [Formula: see text] in the energy norm, where [Formula: see text] represents the underlying polynomial degree. To validate the approach, a series of numerical experiments had been conducted for various problem instances. Comparisons with the linear continuous interior penalty stabilised method, and the algebraic flux-correction scheme (for the piecewise linear finite element case) have been carried out, where we can observe the favorable performance of the current approach.

Full discretization of the time dependent Navier–Stokes equations with anisotropic slip boundary condition coupled with the convection–diffusion–reaction equation

Full discretization of the time dependent Navier–Stokes equations with anisotropic slip boundary condition coupled with the convection–diffusion–reaction equation

Variational–hemivariational system for contaminant convection–reaction–diffusion model of recovered fracturing fluid

Abstract This work is devoted to study the convection–reaction–diffusion behavior of contaminant in the recovered fracturing fluid which flows in the wellbore from shale gas reservoir. First, we apply various constitutive laws for generalized non-Newtonian fluids, diffusion principles, and friction relations to formulate the recovered fracturing fluid model. The latter is a partial differential system composed of a nonlinear and nonsmooth stationary incompressible Navier-Stokes equation with a multivalued friction boundary condition, and a nonlinear convection–reaction–diffusion equation with mixed Neumann boundary conditions. Then, we provide the weak formulation of the fluid model which is a hemivariational inequality driven by a nonlinear variational equation. We establish existence of solutions to the recovered fracturing fluid model via a surjectivity theorem for multivalued operators combined with an alternative iterative method and elements of nonsmooth analysis.

Analysis of a two phase flow model of biofilm spread

We consider a quasi-stationary problem describing the status of velocities, pressures and chemicals affecting cell behavior within a biofilm. The model couples stationary transport equations and compressible Stokes systems with convection–reaction–diffusion equations. We establish existence, uniqueness and stability of solutions of the different submodels involved and then obtain well posedness results for the quasi-stationary system. Our analysis relies on the construction of weak solutions for the steady transport equations under sign assumptions and the reformulation of the compressible Stokes problem as an elliptic system with enhanced regularity properties on the pressure. We need to consider velocity fields whose divergence and normal boundary components satisfy sign conditions. Applications include the study of cells, biofilms and tissues, where one phase is a liquid solution, whereas the other one is assorted biomass.

Enhancing the accuracy and efficiency of two uniformly convergent numerical solvers for singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters

Abstract This study is devoted to designing two hybrid computational algorithms to find approximate solutions for a class of singularly perturbed parabolic convection–diffusion–reaction problems with two small parameters. In our approaches, the time discretization is first performed by the well-known Rothe method and Taylor series procedures, which reduce the underlying model problem into a sequence of boundary value problems (BVPs). Hence, a matrix collocation technique based on novel shifted Delannoy functions (SDFs) is employed to solve each BVP at each time step. We show that our proposed hybrid approximate techniques are uniformly convergent in order O ( Δ τ s + M − 1 2 ) {\mathcal{O}}\left(\Delta {\tau }^{s}+{M}^{-\tfrac{1}{2}}) for s = 1 , 2 s=1,2 , where Δ τ \Delta \tau is the time step and M M is the number of SDFs used in the approximation. Numerical simulations are performed to clarify the good alignment between numerical and theoretical findings. The computational results are more accurate as compared with those of existing numerical values in the literature.

FDM/FEM for nonlinear convection–diffusion–reaction equations with Neumann boundary conditions—Convergence analysis for smooth and nonsmooth solutions

This paper aims to present in a systematic form the stability and convergence analysis of a numerical method defined in nonuniform grids for nonlinear elliptic and parabolic convection–diffusion–reaction equations with Neumann boundary conditions. The method proposed can be seen simultaneously as a finite difference scheme and as a fully discrete piecewise linear finite element method. We establish second convergence order with respect to a discrete H1-norm which shows that the method is simultaneously supraconvergent and superconvergent. Numerical results to illustrate the theoretical results are included.

Simultaneous impacts of H2 and CO2 transport on NO formations of premixed and diffusion opposed flames fueled by biogas-H2 blends

With the numerical simulation, the transport impacts of H2 and CO2 on the NO formations of premixed and diffusion opposed biogas-H2 flames were isolated quantitatively by using the fictious species method. The combined and individual impacts of H2 and CO2 transport on the NO concentrations, radical pools and critical reactions were compared and investigated comprehensively. The results show that the NO concentration of diffusion flame can be increased steadily with either H2 or CO2 transport, while the NO concentration of the premixed flame can be increased and decreased at fuel-lean and fuel-rich conditions respectively owing to the dominant role of H2 transport in the coupling transport influence. The CO2 transport is confirmed to have the effective influence on the NO formations in the fast reaction zones of the premixed flames based on the convection-diffusion-reaction (CDR) budget analysis, but its effects on the NO concentration are weakened effectively owing to that the NO production in the post-flame zone plays a more dominant role in the NO concentration of the premixed flame. Additionally, the H2 and CO2 transport have the opposite impacts on the major NO production/destruction reactions in the premixed flame but they exert negative or positive effects simultaneously in the diffusion flame. In either premixed or diffusion flame, the H2 and CO2 transport can influence the NO production primarily by affecting the NO/NO2 conversion rate, the thermal route (N + OHNO + H) and the NNH intermediate route (HNO + HH2+NO). Particularly, unlike the CO2 transport effects on the thermochemistry identified in our previous studies, the CO2 transport is found to be relatively insignificant to the NO generation of the premixed biogas-H2 flame. Meanwhile, for the NO production in the diffusion biogas-H2 flame, the CO2 transport effects could account for around 25% of total transport effects of H2 and CO2, suggesting that the CO2 transport should be not neglected when investigating the NO production in the diffusion biogas-H2 flame.

Effects of organic carbon and subsurface dams on saltwater intrusion and nitrate pollution in sandy coastal aquifers.

This study explores the impact of a novel approach on the levels of SWI (saltwater intrusion) and NO3- (nitrate) contamination. Some numerical simulations were conducted utilizing a coupled model that incorporates variably saturation and density, as well as convection diffusion reaction within a sandy coastal aquifer. We verified the reliability of the model for SWI based on comparison lab experiments and for chemical reactions based on a comparison of previous in situ observations. Cutoff walls and subsurface dams cannot simultaneously control SWI and reduce NO3- contamination. A novel approach that combines subsurface dams and permeable CH2O (organic carbon) walls (PC-Wall) is proposed. Subsurface dams are utilized to prevent SWI, while PC-Walls are employed to mitigate NO3- pollution. Results demonstrate that the construction of a PC-Wall with a concentration of 1.0mM facilitated a transition from nitrification (Ni)-dominated to denitrification (Dn)-dominated. An increase in CH2O concentration to 1.0mM caused a significant 1942.5 % rise in mDn (the mass of NO3- removed through Dn). Increment of the distance between the PC-Wall and the ocean from 35 to 45m could result in a 103.7 % mDn increase and reduce mN (the compound mass of NO3- remaining in the aquifer) by 11.7 %. The study offers a detailed comprehension of the intricate hydrodynamics of SWI and NO3- pollution. In addition, it provides design guidance for engineering to mitigate contamination by NO3- and controlling SWI, thus fostering the management of groundwater quality.

A numerical method for singularly perturbed convection–diffusion–reaction equations on polygonal meshes

A numerical method for singularly perturbed convection–diffusion–reaction equations on polygonal meshes

Spatiotemporal modeling of radiopharmaceutical transport in solid tumors: Application to 177Lu-PSMA therapy of prostate cancer

Background and objective177Lu-labeled prostate-specific membrane antigen (PSMA) radiopharmaceutical therapy (RPT) represents a pivotal advancement in addressing prostate cancer. However, existing therapies, while promising, remain incompletely understood and optimized. Computational models offer potential insights into RPTs, aiding in clinical drug delivery enhancement. In this study, we investigate the impact of various physiological parameters on the delivery of 177Lu-PSMA-617 RPT using the convection-diffusion-reaction (CDR) model. MethodsOur investigation encompasses tumor geometry and surrounding tissue, characterized by well-defined boundaries and initial conditions. Utilizing the finite element method, we solve governing equations across a range of parameters: dissociation constant KD (1, 0.1, 0.01 [nM]), internalization rate (0.01–0.0001 [min−1]), diverse tumor shapes, and variable necrotic zone sizes. This model can provide an accurate analysis of radiopharmaceutical delivery from the injection site to the tumor cell, including drug transport in the vascular, interstitial, and intracellular spaces, and considering important parameters (e.g., drug extravasation from microvessels or to lymphatic vessels, the extracellular matrix, receptors, and intracellular space). ResultsOur findings reveal significant enhancements in tumor-absorbed doses as KD decreases. This outcome can be attributed to the higher affinity of radiopharmaceuticals for PSMA receptors as KD diminishes, facilitating a more efficient binding and retention of the therapeutic agent within the tumor microenvironment. Additionally, tumor-absorbed doses for KD ∼ 1 [nM] show an upward trend with higher internalization rates. This observation can be rationalized by considering that a greater internalization rate would result in a higher proportion of radiopharmaceuticals being taken up by tumor cells after binding to receptors on the cell surface. Notably, tumor shape and necrotic zone size exhibit limited influence on tumor absorbed dose. ConclusionsThe present study employs the CDR model to explore the role of physiological parameters in shaping 177Lu-PSMA-617 RPT delivery. These findings provide insights for improving prostate cancer therapy by understanding radiopharmaceutical transport dynamics. This computational approach contributes to advancing our understanding of radiopharmaceutical delivery mechanisms and has implications for enhancing treatment efficacy.

A Generalized Hybrid Method for Handling Fractional Caputo Partial Differential Equations via Homotopy Perturbed Analysis

This article describes a novel hybrid technique known as the Sawi transform homotopy perturbation method for solving Caputo fractional partial differential equations. Combining the Sawi transform and the homotopy perturbation method, this innovative technique approximates series solutions for fractional partial differential equations. The Sawi transform is a recently developed integral transform that may successfully manage recurrence relations and integro-differential equations. Using a homotopy parameter, the homotopy perturbation method is a potent semi-analytical tool for constructing approximate solutions to nonlinear problems. The suggested method offers various advantages over existing methods, including high precision, rapid convergence, minimal computing expense, and broad applicability. The new method is used to solve the convection–reaction–diffusion problem using fractional Caputo derivatives.

An enhanced SUPG-stabilized finite element formulation for simulating natural phenomena governed by coupled system of reaction-convection-diffusion equations

Many phenomena arising in nature, science, and industry can be modeled by a coupled system of reaction-convection-diffusion (RCD) equations. Unfortunately, obtaining analytical solutions to RCD systems is typically not possible and, therefore, usually requires the use of numerical methods. On the other hand, since solutions to RCD-type equations can exhibit rapid changes and may have boundary/inner layers, classical computational tools yield approximations polluted with physically meaningless oscillations when convection dominates the transport process. Towards that end, in order to eliminate such numerical instabilities without sacrificing accuracy, this work employs a stabilized finite element formulation, the so-called streamline-upwind/Petrov-Galerkin (SUPG) method. The SUPG-stabilized formulation is then also supplemented with the YZ$\beta$ shock-capturing mechanism to achieve higher-quality approximations around sharp gradients. A comprehensive set of numerical test experiments, including cross-diffusion systems, the Schnakenberg reaction model, and mussel-algae interactions, is considered to reveal the robustness of the proposed formulation, which we call the SUPG-YZ$\beta$ formulation. Comparisons with reported studies reveal that the proposed formulation performs quite well without introducing excessive numerical dissipation.