Convection Diffusion Reaction Research Articles

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.

Invariant subspace method and exact solutions of the coupled system of time-fractional convection–reaction–diffusion equations

Invariant subspace method and exact solutions of the coupled system of time-fractional convection–reaction–diffusion equations

High-order models for convection–diffusion-reaction transport in multiscale porous media

Developing highly accurate models for predicting the convection–diffusion-reaction (CDR) transport in hierarchical porous media with strong heterogeneities on multiple scales is crucial but not yet available. In this work, an innovative high-order multiscale computational framework is developed to capture the local and global variation characteristics of flow fields and reactant concentration at multiple scales. The homogenized solutions and macro-meso high-order solutions are established by the formal two-scale asymptotic analysis. By directly expanding the mesoscopic cell functions to the microscopic levels, the three-scale high-order models are built by assembling the meso-micro high-order expansions of mesoscopic cell functions and macro-meso low/high-order models. The present approaches follow the reverse thought process of the reiteration homogenization method, and provide a very innovative way to develop highly accurate and efficient solutions for the CDR coupling problems in multiscale porous media. The effectiveness and accuracy of the proposed multiscale models are validated by several representative cases.

An efficient fractional step numerical algorithm for time-delayed singularly perturbed 2D convection-diffusion–reaction problem with two small parameters

An efficient fractional step numerical algorithm for time-delayed singularly perturbed 2D convection-diffusion–reaction problem with two small parameters

Modified Characteristic Finite Element Method with Second-Order Spatial Accuracy for Solving Convection-Dominated Problem on Surfaces.

We present a modified characteristic finite element method that exhibits second-order spatial accuracy for solving convection-reaction-diffusion equations on surfaces. The temporal direction adopted the backward-Euler method, while the spatial direction employed the surface finite element method. In contrast to regular domains, it is observed that the point in the characteristic direction traverses the surface only once within a brief time. Thus, good approximation of the solution in the characteristic direction holds significant importance for the numerical scheme. In this regard, Taylor expansion is employed to reconstruct the solution beyond the surface in the characteristic direction. The stability of our scheme is then proved. A comparison is carried out with an existing characteristic finite element method based on face mesh. Numerical examples are provided to validate the effectiveness of our proposed method.

Adaptive mixed FEM combined with the method of characteristics for stationary convection–diffusion–reaction problems

Adaptive mixed FEM combined with the method of characteristics for stationary convection–diffusion–reaction problems

Very high-order finite difference method on arbitrary geometries with Cartesian grids for non-linear convection diffusion reaction equations

An arbitrary order finite difference method for solving non-linear convection Diffusion Reaction equations in curved boundary domains with Cartesian grid is proposed. Ghost points' values are determined with the Reconstruction Off-Site Data based on a polynomial interpolation using the least square method with constraints to enforce the boundary conditions. We propose a second-, fourth-, and sixth-order schemes for linear non-constant coefficients problem in both the conservative and non-conservative scalar equations. Extensions to non-linear scalar problems and systems are then implemented while preserving the optimal orders. Numerical simulations are carried out to provide evidence about the convergence order and the stability of the method.

Identification of discontinuous parameters in contaminant convection–reaction–diffusion model of recovered fracturing fluid

The primary goal of this paper is to identify the discontinuous parameters in a complicated contaminant convection–reaction–diffusion model of recovered fracturing fluid (RFF model, for short), which is modeled by a nonlinear stationary incompressible Navier–Stokes equation with multivalued and nonsmooth friction boundary condition coupled with a nonlinear convection–reaction–diffusion equation involving mixed Neumann boundary conditions. First, we impose the parameter identification problem which is, exactly, a nonlinear and nonsmooth inverse problem. Then, we show the local boundedness and weakly relative compactness of solution map to RFF model with respect to the discontinuous parameters. Moreover, the generalized continuity of solution map of RFF model is proved. Finally, via employing the theory of nonsmooth analysis and optimization, a sufficiency theorem of existence of solutions to the inverse problem under consideration is established.