Convection Diffusion Reaction Equations Research Articles

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.

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.

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

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

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.

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.

Stable numerical algorithm with localized radial basis function for solution of fractional convection–diffusion–reaction equation

Designing an efficient high-order numerical discretization is one of the challenges in numerical solution of fractional differential equations. Therefore in this paper to overcome the ill-conditioning that often destroys the convergence rate of global RBF methods, we apply a local meshfree method known as radial basis function-generated finite difference (RBF-FD) method equipped with a greedy algorithm to design stable stencil weights and approximate spatial derivatives for parabolic fractional partial differential equations (FPDEs) of convection–diffusion–reaction type. In RBF-FD method to select stable stencils that make weights with desirable properties for differentiation matrix, we apply a greedy algorithm, then convergence and stability of proposed scheme are investigated theoretically and numerically to show efficiency of developed technique. Finally, numerical results are provided by some FPDEs in regular and irregular shaped domains to illustrate the approximation quality and convergence of our approach in comparison with the results presented in literatures.

An H1-Galerkin Space-Time Mixed Finite Element Method for Semilinear Convection–Diffusion–Reaction Equations

In this paper, the semilinear convection–diffusion–reaction equation is split into a lower-order system by introducing the auxiliary variable q=a(x)ux. An H1-Galerkin space-time mixed finite element method for the lower-order system is then constructed. The proposed method applies the finite element method to discretize the time and space directions simultaneously and does not require checking the Ladyzhenskaya–Babusˇka–Brezzi (LBB) compatibility constraints, which differs from the traditional mixed finite element method. The uniqueness of the approximate solutions u and q are proven. The L2(L2) norm optimal order error estimates of the approximate solution u and q are derived by introducing the space-time projection operator. The numerical experiment is presented to verify the theoretical results. Furthermore, by comparing with the classical H1-Galerkin mixed finite element scheme, the proposed scheme can easily improve computational accuracy and time convergence order by changing the basis function.

Convergence analysis of a class of iterative methods for propagation of reaction fronts in porous media

We present an iterative scheme for the numerical analysis of propagating reaction front problems in porous media satisfying an Arrhenius-type law. The governing equations consist of the Darcy equations for the pressure and flow field coupled to two convection–diffusion–reaction equations for the temperature and depth of conversion. Well-posedness, existence and uniqueness of the weak solution are first studied using a fixed-point approach and then, analysis of the proposed iterative scheme is investigated. Numerical results are also presented in order to validate the theoretical estimates and to illustrate the performance of the proposed scheme. The obtained results are in line with our expectations for a good numerical resolution with high accuracy and stability behaviors.

Moving mesh method with variational multiscale finite element method for convection–diffusion–reaction equations

Moving mesh method with variational multiscale finite element method for convection–diffusion–reaction equations

A high resolution Physics-informed neural networks for high-dimensional convection–diffusion–reaction equations

In practical problems, some partial differential equations defined in high-dimensional domains or complex surfaces are difficult to calculate by traditional methods. In this paper, a novel data-driven deep learning algorithm is proposed to solve high-dimensional convection–diffusion–reaction equations. The main idea of the method is to use the neural network which combines the physical characteristics of the equation to get high accuracy numerical solution. The proposed method not only avoids the high cost of mesh generation, but also effectively reduces the numerical oscillation caused by the domination of the convection. In addition, two types of loss functions are designed to force physical properties, such as the positivity or maximum principle of the solution. Various numerical examples are performed to demonstrate the validity and accuracy of the proposed method.

A streamline-derivative-based local projection stabilization virtual element method for nonlinear convection–diffusion–reaction equation

A streamline-derivative-based local projection stabilization virtual element method for nonlinear convection–diffusion–reaction equation

Optimal control problems for the reaction–diffusion–convection equation with variable coefficients

The solvability of optimal control problems is proved on both weak and strong solutions of a boundary value problem for the nonlinear reaction–diffusion–convection equation with variable coefficients. In the second case, the requirements for smoothness of the multiplicative control are reduced. The study of extremal problems is based on the proof of the solvability of the corresponding boundary value problems and on the qualitative analysis of their solutions properties. The large data existence results for weak solutions, the maximum principle as well as the local existence and uniqueness of a strong solution are established. Moreover, an optimal feedback control problem is considered. Using methods of the theory of topological degree for set-valued perturbations (with aspheric image sets) of generalized monotone operators, we obtain sufficient conditions for the solvability of this problem in the class of weak solutions.

VEM discretization allowing small edges for the reaction–convection–diffusion equation: source and spectral problems

In this paper we analyze a lowest order virtual element method for the classic load reaction–convection–diffusion problem and the convection–diffusion spectral problem, where the assumptions on the polygonal meshes allow to consider small edges for the polygons. Under well defined seminorms depending on a suitable stabilization for this geometrical approach, we derive the well posedness of the numerical scheme and error estimates for the load problem, whereas for the spectral problem we derive convergence and error estimates fo the eigenvalues and eigenfunctions. We report numerical tests to asses the performance of the small edges on our numerical method for both problems under consideration.

A numerical investigation of unsteady space–time dependent coefficients anisotropic-diffusion convection reaction equation

Two dimensional transient problems for anisotropic FGMs governed by a diffusion convection reaction (DCR) equation of time–space dependent coefficients are considered. Using a transformation, the space–time dependent coefficients equation is reduced to a time dependent coefficients equation. Then taking a Laplace transform will result in a constant coefficients equation which can be written in forms of boundary-only integral equations and solved using a standard BEM for the solutions. These BEM solutions are then numerically inverse transformed to get the solutions in the time variable. Some problems of trigonometrically, exponentially and quadratically graded materials with compressible or incompressible flow are solved. The numerical solutions obtained are quite accurate.

A numerical solution of fractional reaction–convection–diffusion for modeling PEM fuel cells based on a meshless approach

The purpose of this contribution is to present or implement generalized finite difference method (GFDM) for the first time in order to solve the reaction convection Diffusion equation (FRCDE) model. The FRCDE equations are nonlinear time fractional differential equations that can be used to describe mathematical models in physics and engineering, such as fuel cells and transport in inhomogeneous media. This model is a generalization of the classical RCDE, with time terms considered by Caputo derivative sense for 0<α(x,t)<1. The meshless GFDM, in conjunction with the ϑ-weighted finite difference method, is developed to approximate processes in the spatial direction. GFDM is based on the Taylor series expansion and the moving least squares (MLS) method to evaluate the derivatives of unknown variables. The proposed method, as a truly meshless approach, is very promising in the numerical approximation of engineering problems within convex, non-convex, irregular, and regular domains (complex domains). The reliability and accuracy of the proposed method are shown by considering a variety of computational domains. The sensitivity of the selection of local collocation points and ϑ is also reported.