Convection Diffusion Reaction Equations Research Articles

Almost second-order uniformly convergent numerical method for singularly perturbed convection–diffusion–reaction equations with delay

This paper deals with the numerical treatment of time-dependent singularly perturbed convection–diffusion–reaction equations with delay. In the considered equations, the highest order derivative term is multiplied by a perturbation parameter, ε taking arbitrary value in the interval . For small ε, the solution of the equations exhibits a boundary layer on the right side of the spatial domain. The considered equations contain a small delay on the convection and reaction terms of the spatial variable. The terms involving the delay are approximated using the Taylor series approximation. The resulting singularly perturbed parabolic convection–diffusion–reaction equations are treated using the Crank Nicolson method in time derivative discretization and the mid-point upwind finite difference method on piecewise uniform Shishkin mesh for the space variable derivative discretization. The stability and the uniform convergence of the scheme are investigated well. The proposed scheme gives almost first-order convergence in the spatial direction. The Richardson extrapolation technique is applied to accelerate the rate of convergence of the scheme in the spatial direction to order almost two. To validate the applicability of the scheme, numerical examples are presented and solved for different values of the perturbation parameter and delay parameter.

Classical solution for a nonlinear hybrid system modeling combustion in a multilayer porous medium

Combustion processes in porous media have been used by the petroleum engineering industry to extract heavy oil from reservoirs. This study focuses on a one-dimensional nonlinear hybrid system consisting of n reaction–diffusion–convection equations coupled with n ordinary differential equations, which models a combustion front moving through a porous medium with n parallel layers. The state variables are the temperature and fuel concentration in each layer. Coupling occurs in both the reaction function and differential operator coefficients. We prove the existence of a classical solution, first locally and then globally over time, to an initial and boundary value problem for the corresponding system. The proof uses a new approach for combustion problems in porous media. The local solution is obtained by defining an operator in a set of Hölder continuous functions and using Schauder’s fixed-point theorem to find a fixed point as the desired solution. Using Zorn’s lemma, we extend the local solution to a global solution, proving that the first-order spatial derivative of the temperature in each layer is a bounded function.

Numerical solution of convection–diffusion–reaction equations by a finite element method with error correlation

This study contemplates the Finite Element Method (FEM), a well-known numerical method, to find numerical approximations of the Convection–Diffusion–Reaction (CDR) equation. We concentrate on analyzing the convergence and stability of the nonlinear parabolic partial equations. The method is generally applied without truncating the nonlinear terms and avoiding restrictive assumptions. Regular and irregular geometrical shapes are the key objective of this research paper. This study also focuses on the accuracy and acceptance of the FEM method by utilizing dissipation error, dispersion error, and total error analysis. The results are portrayed both graphically and in a tabular form, which virtually ensures the method’s validity and the algorithm’s efficiency to sustain the accuracy, simplicity, and applicability for solving nonlinear CDR equations. The proposed technique may also be applied for solving any nonlinear reaction–diffusion equations.

Stochastic Galerkin Reduced Basis Methods for Parametrized Linear Convection–Diffusion–Reaction Equations

We consider the estimation of parameter-dependent statistics of functional outputs of steady-state convection–diffusion–reaction equations with parametrized random and deterministic inputs in the framework of linear elliptic partial differential equations. For a given value of the deterministic parameter, a stochastic Galerkin finite element (SGFE) method can estimate the statistical moments of interest of a linear output at the cost of solving a single, large, block-structured linear system of equations. We propose a stochastic Galerkin reduced basis (SGRB) method as a means to lower the computational burden when statistical outputs are required for a large number of deterministic parameter queries. Our working assumption is that we have access to the computational resources necessary to set up such a reduced-order model for a spatial-stochastic weak formulation of the parameter-dependent model equations. In this scenario, the complexity of evaluating the SGRB model for a new value of the deterministic parameter only depends on the reduced dimension. To derive an SGRB model, we project the spatial-stochastic weak solution of a parameter-dependent SGFE model onto a reduced basis generated by a proper orthogonal decomposition (POD) of snapshots of SGFE solutions at representative values of the parameter. We propose residual-corrected estimates of the parameter-dependent expectation and variance of linear functional outputs and provide respective computable error bounds. We test the SGRB method numerically for a convection–diffusion–reaction problem, choosing the convective velocity as a deterministic parameter and the parametrized reactivity or diffusivity field as a random input. Compared to a standard reduced basis model embedded in a Monte Carlo sampling procedure, the SGRB model requires a similar number of reduced basis functions to meet a given tolerance requirement. However, only a single run of the SGRB model suffices to estimate a statistical output for a new deterministic parameter value, while the standard reduced basis model must be solved for each Monte Carlo sample.

Higher order stable schemes for stochastic convection–reaction–diffusion equations driven by additive Wiener noise

In this paper, we investigate the numerical approximation of stochastic convection–reaction–diffusion equations using two explicit exponential integrators. The stochastic partial differential equation (SPDE) is driven by additive Wiener process. The approximation in space is done via a combination of the standard finite element method and the Galerkin projection method. Using the linear functional of the noise, we construct two accelerated numerical methods, which achieve higher convergence orders. In particular, we achieve convergence rates approximately 1 for trace class noise and for space‐time white noise. These convergence orders are obtained under less regularity assumptions on the nonlinear drift function than those used in the literature for stochastic reaction–diffusion equations. Numerical experiments to illustrate our theoretical results are provided.

Boundary and Extremum Problems for the Nonlinear Reaction–Diffusion–Convection Equation under the Dirichlet Condition

Boundary and Extremum Problems for the Nonlinear Reaction–Diffusion–Convection Equation under the Dirichlet Condition

Mathematical modeling of chemotaxis guided amoeboid cell swimming

Cells and microorganisms adopt various strategies to migrate in response to different environmental stimuli. To date, many modeling research has focused on the crawling-based Dictyostelium discoideum (Dd) cells migration induced by chemotaxis, yet recent experimental results reveal that even without adhesion or contact to a substrate, Dd cells can still swim to follow chemoattractant signals. In this paper, we develop a modeling framework to investigate the chemotaxis induced amoeboid cell swimming dynamics. A minimal swimming system consists of one deformable Dd amoeboid cell and a dilute suspension of bacteria, and the bacteria produce chemoattractant signals that attract the Dd cell. We use the mathematical amoeba model to generate Dd cell deformation and solve the resulting low Reynolds number flows, and use a moving mesh based finite volume method to solve the reaction–diffusion–convection equation. Using the computational model, we show that chemotaxis guides a swimming Dd cell to follow and catch bacteria, while on the other hand, bacterial rheotaxis may help the bacteria to escape from the predator Dd cell.

Analysis of Boundary Value and Extremum Problems for a Nonlinear Reaction–Diffusion–Convection Equation

Analysis of Boundary Value and Extremum Problems for a Nonlinear Reaction–Diffusion–Convection Equation

Solitons and Other Solutions for the Nonlinear Convection–Diffusion–Reaction Equation with Power-Law Nonlinearity by the Extended Simplest Equation Method

Solitons and Other Solutions for the Nonlinear Convection–Diffusion–Reaction Equation with Power-Law Nonlinearity by the Extended Simplest Equation Method

Gradient recovery-based adaptive stabilized mixed FEM for the convection–diffusion–reaction equation on surfaces

In this article, we consider a gradient recovery-based adaptive stabilized mixed finite element method (FEM) for solving the convection–diffusion–reaction equation on surfaces. By defining a new total flux on the surface, we firstly transform the original equation into an equivalent mixed first-order formulation. Then, by using both the convection–convection term and the div–div term to stabilize the weak formulation, we prove the discrete inf–sup condition and the error estimate for the mixed FEM. In this new method, the equal order Lagrangian elements are allowed for the primal variable and the total flux, and a high accurate numerical solution for the flux in the H1 space is obtained. The impact of the curvature on the algorithm is also investigated. Additionally, to overcome the low resolution and the non-physical oscillation in simulating the convection-dominated problem, we study a gradient recovery-based adaptive stabilized mixed FEM. Finally, we show some numerical experiments to verify the theoretical prediction and efficiency of the studied method.

Reaction–diffusion–convection equations in two spatial dimensions; continuous and discrete dynamics

In this paper, we investigate some two-dimensional (with respect to spatial independent variables) reaction–diffusion–convection equations with various nonlinear (reaction) terms. Using Hirota bilinear formalism with a free auxiliary function, we obtain kink solutions and many spatio-temporal discretizations having birational form.

Mathematical modeling of thermal decomposition of resins in the process of reversed gasification of plant biomass

THE PURPOSE. Of this work is to estimate parameters for a sufficiently comlete thermal decomposition of tarry products of the fixed-bed gasification of wood fuel. METHODS. To this end, mathematical models are used in different statements: the decomposition of the tar is considered in the approximations of one- and two-reaction kinetic scheme; to assess the influence of the bed height and temperature, the convection-diffusionreaction equation with a given temperature distribution along the length of the reaction zone is used; the temperature of the gasification process is estimated from experimental data and thermodynamic calculations. Along with the numerical model of the decomposition of the tar, a simplified analytical expression (for large Peclet numbers) is applied, the limits of its applicability are determined. RESULTS. Of calculations show that the efficiency of the airblown gasification of wood is determined by the temperature level of the oxidation stage: in the range of modes in which the optimal values of efficiency are achieved, the conversion of tarry products does not proceed sufficiently completely due to kinetic limitations; an increase in the specific consumption of the oxidizer leads to a decrease in efficiency due to stoichiometric reasons. CONCLUSION. Physicochemical limitations do not allow reaching the limiting values of gasification efficiency, shifting the optimal modes towards increasing the specific air consumption; a decrease in the yield of tar requires, first of all, changes in the thermal modes of gasification (for example, external heating or an increase in the oxygen concentration).

A new model for the emergence of blood capillary networks

We propose a new model for the emergence of blood capillary networks. We assimilate the tissue and extra cellular matrix as a porous medium, using Darcy's law for describing both blood and intersticial fluid flows. Oxygen obeys a convection-diffusionreaction equation describing advection by the blood, diffusion and consumption by the tissue. Discrete agents named capillary elements and modelling groups of endothelial cells are created or deleted according to different rules involving the oxygen concentration gradient, the blood velocity, the sheer stress or the capillary element density. Once created, a capillary element locally enhances the hydraulic conductivity matrix, contributing to a local increase of the blood velocity and oxygen flow. No connectivity between the capillary elements is imposed. The coupling between blood, oxygen flow and capillary elements provides a positive feedback mechanism which triggers the emergence of a network of channels of high hydraulic conductivity which we identify as new blood capillaries. We provide two different, biologically relevant geometrical settings and numerically analyze the influence of each of the capillary creation mechanism in detail. All mechanisms seem to concur towards a harmonious network but the most important ones are those involving oxygen gradient and sheer stress. A detailed discussion of this model with respect to the literature and its potential future developments concludes the paper.

Existence of solutions of a finite element flux-corrected-transport scheme

The existence of a solution is proved for a nonlinear finite element flux-corrected-transport (FEM-FCT) scheme with arbitrary time steps for evolutionary convection–diffusion–reaction equations and transport equations.

Variational Multiscale Finite-Element Methods for a Nonlinear Convection–Diffusion–Reaction Equation

This paper is devoted to developing finite-element methods for solving a two-dimensional boundary value problem for a singularly perturbed time-dependent convection–diffusion–reaction equation. The solution to this problem may vary rapidly in thin layers. As a result, spurious oscillations occur in the solution if the standard Galerkin method is used. In the multiscale finite-element methods, the original problem is split into the grid-scale and subgrid-scale problems, which allows capturing the problem features at a scale smaller than the element mesh size. In this study two methods are considered: the variational multiscale method with algebraic sub-scale approximation (VMM-ASA) and the residual-free bubbles (RFB) method. In the first method the subgrid-scale problem is simulated by the residual of the grid-scale equation and intrinsic time scales. In the second method the subgrid-scale problem is approximated by special functions. The grid-scale and subgrid-scale problems are formulated via the linearization procedure on the subgrid component applied to the original problem. The computer implementation of the methods was carried out using a commercial finite-element package. The efficiency of the developed methods is evaluated by solving a test boundary value problem for the nonlinear equation. Cases with different values of the diffusion coefficient have been analyzed. Based on the numerical investigation, it is shown that the multiscale methods enable improving the stability of the numerical solution and decreasing the quantity and the amplitude of oscillations compared to the standard Galerkin method. In the case of a small diffusion coefficient, the developed methods can yield a satisfactory numerical solution on a sufficiently coarse mesh.

Virtual element method for nonlinear convection–diffusion–reaction equation on polygonal meshes

ABSTRACT In this paper, we discuss the virtual element formulation for the nonlinear convection–diffusion–reaction equation. We consider the Streamline upwind Petrov–Galerkin stabilization to reduce the non-physical oscillations presented in the solution. The nonlinear term is evaluated after a suitable modification with the help of a polynomial projection operator. We perform the error analysis by showing the existence of the solution and derive the convergence estimates in the energy norm. Numerical experiments are conducted in order to validate the theoretical results.

An efficient characteristic finite difference S-DDM scheme for convection–diffusion equations

In this paper, an efficient splitting domain decomposition method scheme for solving time-dependent convection–diffusion reaction equations is analyzed. A three-step method along each direction is used to solve the solution over each block-divided sub-domain at every time interval. The new solutions are firstly solved by the quadratic interpolation. Then, the intermediate fluxes on the interfaces of sub-domains are computed by local multi-point weighted average from the new solutions. Lastly, the solutions and fluxes in the interiors of sub-domains are computed by the splitting implicit characteristic difference method. By some auxiliary lemmas and the defined intermediate exact solution, the stability and error estimate are given in discrete L2-norm. We further prove that our scheme is of second-order convergence in space and of first-order convergence in time. Numerical experiments are presented to validate theoretical result.

Thermal analysis of loop mediated isothermal DNA amplification (LAMP) based point-of-care diagnostic device

Loop-Mediated Isothermal Amplification reaction or LAMP has been identified as a feasible DNA amplification process. Paper-based LAMP diagnostic devices offer several advantages for point-of-care (POC) applications. LAMP occurs at a constant temperature of higher limits, which remains a major challenge on-site. This calls for an efficient thermal design which is beneficial in maintaining a constant high temperature at POC locations. The present study reports thermal design and energy consumption analysis of a paper-based LAMP POC device for the first time using CFD analysis. The POC design consists of a rectangular porous membrane embedded on a rectangular solid material, heated from the side walls. Isothermal and discrete heating with different horizontal and vertical clearance configurations are used as a design parameter. CFD analysis of different thermal configurations is carried out in COMSOL Multiphysics platform to solve the energy and convection–diffusion-reaction equation in porous media. The thermal analysis demonstrated that two discrete heating configurations with high horizontal clearance are effective in reducing energy consumption. It was inferred that two portable AA batteries are sufficient for the POC device operation. Overall, the current work successfully demonstrates the energy optimization and speed of operation of the POC medical device by detailed thermal analysis.

A Galerkin-characteristic unified finite element method for moving thermal fronts in porous media

We investigate the performance of a unified finite element method for the numerical solution of moving fronts in porous media under non-isothermal flow conditions. The governing equations consist of coupling the Darcy equation for the pressure to two convection–diffusion–reaction equations for the temperature and depth of conversion. The aim is to develop a non-oscillatory unified Galerkin-characteristic method for efficient simulation of moving fronts in porous media. The method is based on combining the modified method of characteristics with a Galerkin finite element discretization of the governing equations. The main feature of the proposed unified finite element method is that the same finite element space is used for all solutions to the problem including the pressure, velocity, temperature and concentration. Analysis of convergence and stability is also presented in this study and error estimates in the L2-norm are established for the numerical solutions. In addition, due to the Lagrangian treatment of convection terms, the standard Courant–Friedrichs–Lewy condition is relaxed and the time truncation errors are reduced in the diffusion–reaction part. We verify the method for the benchmark problem of moving fronts around an array of cylinders. The numerical results obtained demonstrate the ability of the proposed method to capture the main flow features.

Control of the phase formation process in solutions of anionic polyelectrolyte—cationic surfactant complexes in a microfluidic channel

The process of phase formation in solutions of polyacrylic acid and cetyltrimethylammonium bromide was studied in microfluidic channels with a characteristic size of 100–200 µm. The system of convection—diffusion—reaction equations was solved to develop an association model in polyelectrolyte—surfactant solutions in a microchannel with the laminar flow of the reactants. The phase formation conditions in the microchannel volume were considered. The recommendations for control of the phase behavior in the microchannel were proposed, as well as for the design and operation mode of microfluidic devices used for the synthesis of polymer—surfactant complexes and characterization of association in polyelectrolyte—surfactant solutions.