Convection Diffusion Reaction Research Articles

A comparison of one-step and two-step W-methods and peer methods with approximate matrix factorization

One- and Two-step W-methods and peer methods are considered combined with an Approximate Matrix Factorization (AMF) for the numerical solution of large systems of stiff Ordinary Differential Equations (ODEs) coming from the spatial discretization of parabolic Partial Differential Equations (PDEs) of convection–diffusion–reaction type. Although one-step AMF W-methods do not require starting values and have larger stability regions, they may suffer from order reduction for such PDE problems when time-dependent Dirichlet boundary conditions are imposed. Here by numerical tests it is illustrated that two-step AMF W-methods and AMF peer methods avoid this order reduction (in stiff sense, i.e. for a fixed spatial grid) due to their high stage order. On the other hand, while AMF W-methods are linearly implicit, AMF peer methods require the solution of nonlinear equations in order to compute an advancing solution. In this case, the predictor considered for the internal stages influences both stability and accuracy. However, with only one Newton iteration, for a given number of stages, the three classes of methods have a similar computational cost.

Diffusion convection-reaction equation in exponentially graded media of incompressible flow: Boundary element method solutions

We setup boundary value problems (BVPs) for exponentially graded media. The BVPs are governed by a diffusion convection-reaction (DCR) equation of spatially varying coefficients (with an anisotropic diffusion coefficient). A boundary element method (BEM) is utilized to find numerical solutions to BVPs. The numerical solutions obtained verify the validity of the analysis used to derive the BEM with accurate and consistent solutions. The computation shows that the BEM procedure elapses very efficient time in producing the solutions. In addition, the results show the effect of anisotropy and inhomogeneity of the media on the solutions. To illustrate the application, an example of a layered material is presented.

On the derivation of a boundary element method for diffusion convection-reaction problems of compressible flow in exponentially inhomogeneous media

Boundary value problems (BVPs) of anisotropic and exponentially-graded media governed by a diffusion convection-reaction (DCR) equation are considered. The governing equation is of spatially varying coefficients (with an anisotropic diffusion coefficient). The variable coefficients equation is firstly transformed into a constant coefficients equation, from which we derive a boundary integral equation. A boundary element method (BEM) is then constructed to be used for finding numerical solutions to the BVPs. For the computation of the solutions a FORTRAN code is developed. Some examples of problems are solved. The numerical solutions obtained verify the validity of the analysis used to construct the boundary element method with accurate and consistent solutions. The results also show that the BEM procedure elapses very efficient time in producing the solutions. In addition, the results indicate the effect of anisotropy of the media on the solutions.

Numerical solution to diffusion convection-reaction equation with trigonometrically variable coefficients of incompressible flow

This paper is concerned with finding numerical solutions to boundary value problems (BVPs) governed by a diffusion convection-reaction (DCR) equation of in-spatial-trigonometrically varying coefficients with an anisotropic diffusion coefficient. The variable coefficients equation is firstly transformed into a constant coefficients equation. A boundary integral equation is the derived from the constant coefficients equation. Consequently, a boundary element method (BEM) is developed and utilized to find numerical solutions to the boundary value problems. For the computation of the solutions for some examples of problems, a FORTRAN script is constructed. The numerical solutions obtained verify the validity of the analysis used to derive the BEM. The results also show that the BEM procedure elapses very efficient time in producing very accurate and consistent solutions. Moreover, the results indicate the effect of anisotropy and inhomogeneity of the media on the solutions.

A numerical study on the effect of the material’s anisotropy in diffusion convection reaction problems

A boundary element method (BEM) is utilized to find numerical solutions to boundary value problems of homogeneous media governed by as anisotropic-diffusion convection-reaction (DCR) equation. Some problems are considered. A FORTRAN script is developed for the computation of the solutions. The numerical solutions verify the validity of the analysis used to derive the boundary element method with accurate and consistent solutions. The computation shows that the BEM procedure elapses very efficient time in producing the solutions. In addition, results obtained for the considered examples show the effect of the anisotropy of the media on the solutions.

A positivity preserving characteristic finite element method for solving the transport and convection–diffusion–reaction equations on general surfaces

In this paper, a positivity preserving characteristic finite element method is presented to solve the transport and convection–diffusion–reaction equations on general surfaces. The surface finite element method, which solves a variation problem by the linear finite element space on a piecewise triangulated surface, is applied to spatial discretization. For the backtracking in characteristic derivative discretization, unlike the cases on the two-dimensional plane, the foots of approximate characteristics may locate in the outer domain of the surface. To determine the values of solution at the foots of characteristics, a new strategy, which permits larger time steps, is designed instead of the discrete closest point mapping method which has a strict time step restriction. By combining with the mass lumping technique, the proposed numerical scheme is positivity preserving. The proposed method can also be extended to the problems with nonlinear convection terms. Various numerical examples are performed to demonstrate the validity and accuracy of the proposed method.

A Novel Approach for Solving an Inverse Reaction–Diffusion–Convection Problem

In this paper, we consider an inverse reaction–diffusion–convection problem in which one of the boundary conditions is unknown. A sixth-kind Chebyshev collocation method will be proposed to solve numerically this problem and to obtain the unknown boundary function. Since this inverse problem is generally ill-posed, to find an optimal stable solution, we will utilize a regularization method based on the mollification technique with the generalized cross-validation criterion. The error estimate of the numerical solution is investigated. Finally, to authenticate the validity and effectiveness of the proposed algorithm, some numerical test problems are presented.

The weak Galerkin finite element method for the symmetric hyperbolic systems

In this paper, we present and analyze a weak Galerkin finite element (WG) method for solving the symmetric hyperbolic systems. This method is highly flexible by allowing the use of discontinuous finite elements on element and its boundary independently of each other. By introducing special weak derivative, we construct a stable weak Galerkin scheme and derive the optimal L2-error estimate of O(hk+12)-order for the discrete solution when the k-order polynomials are used for k≥0. As application, we discuss this WG method for solving the singularly perturbed convection–diffusion–reaction equation and derive an ε-uniform error estimate of order k+1∕2. Numerical examples are provided to show the effectiveness of the proposed WG method.

Drug delivery mathematical modeling for pressure controlled bioreactor

We analyze a mathematical model for drug delivery monitoring subject to flow-through bioreactor. We use a convection–diffusion–reaction model combined with PID control. Enzyme-containing bioreactor converts a prodrug into an active drug. This approach connects two aspects of drug delivery, mechanical pumps and prodrugs. Drug delivery monitoring is performed by adjusting the prodrug inflow pressure. We model several dynamic treatment regimes. A combined algorithm of treatment regimes can be used for personalized treatment. A control-aided system allows us to reduce the impact of pumping fluctuations, diffusion slowdown, and drug production rate reduction.

Development of the Interfaces for the Nonlinear Reaction-Diffusion equation with Convection

We study the initial development and asymptotics of the interfaces and local solutions near the interfaces for the nonlinear reaction diffusion convection equation with compactly supported initial function. Depending on the relative strength of three competing terms such as diffusion, advection or absorption, the interface may shrink, expand or remain stationary. In this paper we focus only on two cases when the diffusion dominates and the interface expands and the other case when absorption term dominates and the interface shrinks. The significant methods that we used are rescaling and blow-up techniques.

Numerical algorithms for solving the optimal control problem of simple bioreactors

The modified nonlocal feedback controller is used to control the production of drugs in a simple bioreactor. This bioreactor is based on the enzymatic conversion of substrate into the required product. The dynamics of this device is described by a system of two nonstationary nonlinear diffusion–convection–reaction equations. The analysis of the influence of the convection transport is one the aims of this paper. The control loop is defined using the relation, which shows how the amount of the drug produced in the bioreactor and delivered into a human body depends on the substrate concentration specified on the external boundary of the bioreactor. The system of PDEs is solved by using the finite volume and finite difference methods, the control loop parameters are defined from the analysis of stationary linearized equations. The second aim of this paper is to solve the inverse problem and to determine optimal boundary conditions. These results enable us to estimate the potential accuracy of the proposed devices.

Roughness-induced effects on the convection–diffusion–reaction problem in a thin domain

In this paper, we investigate a convection–diffusion–reaction problem in a thin domain endowed with the Robin-type boundary condition describing the reaction catalyzed by the upper wall. Motivated by the microfluidic applications, we allow the oscillating behavior of the upper boundary and analyze the resonant case where the amplitude and period of the oscillation have the same small order as the domain's thickness. Depending on the magnitude of the reaction mechanism, we rigorously derive three different asymptotic models via the unfolding operator method. In particular, we identify the critical case in which the effects of the domain's geometry and all physically relevant processes become balanced.

Maximum-principle preserving space–time isogeometric analysis

In this work we propose a nonlinear stabilization technique for convection–diffusion–reaction and pure transport problems discretized with space–time isogeometric analysis. The stabilization is based on a graph-theoretic artificial diffusion operator and a novel shock detector for isogeometric analysis. Stabilization in time and space directions are performed similarly, which allow us to use high-order discretizations in time without any CFL-like condition. The method is proven to yield solutions that satisfy the discrete maximum principle (DMP) unconditionally for arbitrary order. In addition, the stabilization is linearity preserving in a space–time sense. Moreover, the scheme is proven to be Lipschitz continuous ensuring that the nonlinear problem is well-posed. Solving large problems using a space–time discretization can become highly costly. Therefore, we also propose a partitioned space–time scheme that allows us to select the length of every time slab, and solve sequentially for every subdomain. As a result, the computational cost is reduced while the stability and convergence properties of the scheme remain unaltered. In addition, we propose a twice differentiable version of the stabilization scheme, which enjoys the same stability properties while the nonlinear convergence is significantly improved. Finally, the proposed schemes are assessed with numerical experiments. In particular, we considered steady and transient pure convection and convection–diffusion problems in one and two dimensions.

Uniformly Convergent Numerical Method for Singularly Perturbed Two Parameter Time Delay Parabolic Problem

This paper discusses the numerical solution of one dimensional parabolic convection–reaction–diffusion time delay problem with two small parameters. For the discretization of the time derivative, we use the implicit Euler scheme on a uniform mesh and for the spatial discretization, we use the upwind difference scheme on the Shishkin type meshes (standard Shishkin mesh, Bakhvalov–Shishkin mesh). We prove that numerically the proposed method is provides a first order convergence, which is optimal for this case. Finally, to support the theoretical results, we present some numerical experiments using the proposed method.

A case study on pricing foreign exchange options using the modified Craig–Sneyd ADI scheme

ABSTRACTOne of the prominent alternating direction implicit (ADI) schemes for numerically pricing financial options, the modified Craig–Sneyd scheme, is put to test for its reliability and efficiency for solving non-trivial problems with empirical market data. The Heston equation for pricing foreign exchange options of European style, a two-dimensional convection–diffusion–reaction equation with a mixed derivative term, is numerically solved for various parameter values observed in the market by employing the said scheme. Numerical stability and convergence issues of this scheme is compared with another popular Hundsdorfer–Verwer ADI scheme. From among a total of 56 options on 8 currency pairs, it is observed that some interesting ones for which the so-called Feller condition is strongly violated, create additional computational challenges. Suggestions on successful implementation of the MCS scheme are made in order to tackle these challenging test cases.

A New Approach for Assessing the LDL Species in Arterial Wall Layers

In the present study, a numerical model for the convective and diffusive transport of LDL in various layers of the arterial wall is developed. The transport in the lumen, endothelium, intima, internal elastic lamina, and media of a straight axisymmetric arterial segment has been considered in the model. The endothelium, intima, internal elastic lamina (iel), and media have been considered as a homogeneous porous media. Navier–stokes equation and Darcy’s equation have been used for modeling the fluid flow in the lumen and the arterial wall, respectively. The mass transport in the lumen has been implemented through the convective–diffusion equation, whereas volume-averaged convective–diffusion–reaction equation has been used for modeling the mass transport through porous layers. The relevant physiological parameters have been taken from various studies. Fluent 12.1 has been used for the solution and analysis of the problem. This study demonstrates that this numerical model can be used efficiently for evaluating the distribution of concentration of LDL in the different layer under the conditions of high LDL concentration in blood (hyperlipidemia) and hypertension. Finally, a new approach for estimating the possible LDL species depositions rate in intima and media layers has been presented. This may demonstrate a possible prediction for extent of progression of the disease with respect to available clinical parameters. The results have shown that rate of accumulation is higher in the intima layer than media layer and pointed out intima layer as a possible prone location of plaque formation. The results also have indicated that both the clinical parameters hypertension and hyperlipidemia are responsible for higher deposition rate of LDL species in intima and media.

Stabilization of coupled convection–diffusion-reaction equations for continuum dislocation transport

The plasticity of crystalline materials can be described at the meso-scale by dislocations transport models, typically formulated in terms of dislocation densities. This leads to sets of coupled nonlinear partial differential equations involving diffusive and convective transport mechanisms. Since exact solutions for these systems are not available, numerical approximations are needed to efficiently solve them. The properties of these systems of equations cause most traditional numerical methods to fail, even for the case of a single equation. For systems of equations the problem is even more challenging due to the lack of fundamental principles guiding numerical discretization strategies. Special strategies must be developed and carefully applied to obtain physically meaningful and numerically stable approximations. The objective of this paper is to construct a coefficient perturbation-based stabilization technique for general systems of equations and to apply it to the modelling of one-dimensional dislocation transport. A detailed numerical study is carried out in order to demonstrate its ability to render well-behaved and physically admissible numerical approximations.

Visible-light photocatalytic degradation of acid violet 7 dye in a continuous annular reactor using ZnO/PPy photocatalyst: Synthesis, characterization, mass transfer effect evaluation and kinetic analysis

In this work, a composite of ZnO and polypyrrole (ZnO/PPy) was synthesized by polymerization method and characterized by X-ray diffraction (XRD), Ultraviolet–Visible with Near-infrared (UV–Vis NIR) spectroscopy and Fourier Transform Infrared (FT-IR) spectroscopy, both by diffuse reflectance mode. The ZnO/PPy composite was tested as photocatalyst for the degradation of acid violet 7 dye under visible light radiation in two systems: a) by using a batch reactor and b) in a continuous annular reactor. The hole formation in the valence band was evidenced using a hole scavenger test for ZnO/PPy composite and also the hydroxyl radicals formation capacity was determined. A three-resistance model of mass transfer (MT) was evaluated, it considers the traditional external mass transfer coefficient (She) to obtain an overall mass transfer coefficient (Shapp), the model is considered a full 3-D convection–diffusion–reaction applied to the photocatalyst of ZnO/PPy composite attached onto the interior wall of a quartz ring (annular space). The MT analysis showed in the continuous reactor that the degradation reaction was kinetically controlled. A Langmuir-Hinshelwood (L-H) model was analyzed according the fitting of the experimental data in the annular continuous photocatalytic reactor. The adsorption parameter obtained with L-H model was validated based on thermodynamic criteria for the changes in the standard enthalpy of adsorption (ΔHadso) and standard total entropy of adsorption (ΔSadso). The analysis of thermodynamic properties revealed that the adsorbed AV7 molecule on ZnO/PPy photocatalyst still presents mobility in order to react to the very short time OH radicals formed on surface.

Standard-BEM solutions to two types of anisotropic-diffusion convection reaction equations with variable coefficients

Two versions of diffusion–convection–reaction equations for anisotropic functionally graded materials (FGMs) are discussed again in this paper to find their numerical solutions by using the boundary element method (BEM). The numerical results obtained show the consistency and accuracy of the BEM solutions. Also, the solutions exhibit the impact of the anisotropy and inhomogeneity (spatial variability) of the media.

A robust numerical scheme for the simulation of nonlinear convection–diffusion–reaction equation

In this article, the polynomial based differential quadrature method is used to develop a numerical scheme for solving convection–diffusion–reaction model with a general reaction term. The numerical scheme is found applicable to a wide set of problems including diffusion, convection–diffusion and diffusion–reaction. The validation of the method is performed through simulations of Newell-Whitehead, Burger-Fisher, and Burger-Huxley equations. The proposed numerical method is found better than many numerical methods presented in the literature. Vector form of numerical scheme is presented to implement algorithm efficiently. The compatibility, error and computational complexity of the numerical scheme are studied in detail.