Convection Diffusion Reaction Equations Research Articles

Eliminating Gibbs phenomena: A non-linear Petrov–Galerkin method for the convection–diffusion–reaction equation

In this article we consider the numerical approximation of the convection-diffusion-reaction equation. One of the main challenges of designing a numerical method for this problem is that boundary layers occurring in the convection-dominated case can lead to non-physical oscillations in the numerical approximation, often referred to as Gibbs phenomena. The idea of this article is to consider the approximation problem as a residual minimization in dual norms in Lq-type Sobolev spaces, with 1<q<∞. We then apply a non-standard, non-linear Petrov–Galerkin discretization, that is applicable to reflexive Banach spaces such that the space itself and its dual are strictly convex. Similar to discontinuous Petrov–Galerkin methods, this method is based on minimizing the residual in a dual norm. Replacing the intractable dual norm by a suitable discrete dual norm gives rise to a non-linear inexact mixed method. This generalizes the Petrov–Galerkin framework developed in the context of discontinuous Petrov–Galerkin methods to more general Banach spaces. For the convection–diffusion–reaction equation, this yields a generalization of a similar approach from the L2-setting to the Lq-setting. A key advantage of considering a more general Banach space setting is that, in certain cases, the oscillations in the numerical approximation vanish as q tends to 1, as we will demonstrate using a few simple numerical examples.

Complete analytic solutions for convection-diffusion-reaction-source equations without using an inverse Laplace transform

Transient mass-transfer phenomena occurring in natural and engineered systems consist of convection, diffusion, and reaction processes. The coupled phenomena can be described by using the unsteady convection-diffusion-reaction (CDR) equation, which is classified in mathematics as a linear, parabolic partial-differential equation. The availability of analytic solutions is limited to simple cases, e.g., unsteady diffusion and steady convective diffusion. The CDR equation has been considered analytically intractable, depending on the initial and boundary conditions. If spatial adsorption and desorption of matter are super-positioned in the CDR equation as sink and source functions, respectively, then the governing equation becomes an unsteady convection-diffusion-reaction-source (CDRS) equation, of which general solutions are unknown. In this study, a general 1D analytic solution of the CDRS equation is obtained by using a one-sided Laplace transform, by assuming constant diffusivity, velocity, and reactivity. This paper also provides a general formalism to derive 1D analytic solutions for Dirichlet/Dirichlet and Dirichlet/Neumann boundary conditions. Derivations of the analytic solutions are found to be straightforward if a combination of the source function and the initial concentration provide a non-zero singularity pole of inverse Laplace transform.

Classification of Symmetry Properties of the (1 + 2)-Dimensional Reaction–Convection–Diffusion Equation

We present a classification of the symmetry properties of the (1 + 2)-dimensional reaction–convection– diffusion equation depending on the values of nonlinearities in this equation. The Lie symmetry is used for the construction of invariant ansatzes, reduction, and finding the exact solutions of the analyzed equation.

Numerical simulation and applications of the convection–diffusion–reaction equation with the radial basis function in a finite-difference mode

This paper develops two local mesh-free methods for designing stencil weights and spatial discretization, respectively, for parabolic partial differential equations (PDEs) of convection–diffusion–reaction type. These are known as the radial-basis-function generated finite-difference method and the Hermite finite-difference method. The convergence and stability of these schemes are investigated numerically using some examples in two and three dimensions with regularly and irregularly shaped domains. Then we consider the numerical pricing of European and American options under the Heston stochastic volatility model. The European option leads to the solution of a two-dimensional parabolic PDE, and the price of the American option is given by a linear complementarity problem with a two-dimensional parabolic PDE of convection–diffusion–reaction type. Then we use the operator splitting method to perform time-stepping after space discretization. The resulting linear systems of equations are well conditioned and sparse, and by numerical experiments we show that our numerical technique is fast and stable with respect to the change in the shape parameter of the radial basis function. Finally, numerical results are provided to illustrate the quality of approximation and to show how well our approach converges with the results presented in the literature.

New singular travelling waves for convection–diffusion–reaction equations

The present paper studies, strictly within the theory of distributions, the propagation of travelling waves with a distributional profile in one-dimensional models ruled by convection–diffusion–reaction equations. By applying a product of distributions (not defined by approximation processes), a rigorous concept of a solution which extends the classical solution concept is defined. As a consequence we establish necessary and sufficient conditions for the propagation of a distributional profile. The use of these conditions enable us to see the appearance of shock waves, delta waves and also delta shock travelling waves. The main ideas and formulas needed for multiplying distributions are included for the reader’s convenience.

Local knot method for 2D and 3D convection–diffusion–reaction equations in arbitrary domains

In this paper, a novel local knot method (LKM) is presented to solve the 2D and 3D convection–diffusion–reaction equations in arbitrary domains. Contrary to the traditional boundary knot method, the proposed scheme requires the nodes not only on the boundary but also inside the domain. For each node, we can find a local subdomain containing a certain number of neighboring nodes. Utilizing the non-singular general solution of differential operator and the known boundary conditions, a sparse linear system is established to approximate the solutions at all nodes over the physical domain. The present LKM is a local meshless method with the merits of being mathematically simple, numerically accurate and easy to large-scale computation. Two numerical examples, involving 2D and 3D complicated domains, are provided to illustrate the effectiveness and accuracy of the new methodology.

Adaptive Iterative Splitting Methods for Convection-Diffusion-Reaction Equations

This article proposes adaptive iterative splitting methods to solve Multiphysics problems, which are related to convection–diffusion–reaction equations. The splitting techniques are based on iterative splitting approaches with adaptive ideas. Based on shifting the time-steps with additional adaptive time-ranges, we could embedded the adaptive techniques into the splitting approach. The numerical analysis of the adapted iterative splitting schemes is considered and we develop the underlying error estimates for the application of the adaptive schemes. The performance of the method with respect to the accuracy and the acceleration is evaluated in different numerical experiments. We test the benefits of the adaptive splitting approach on highly nonlinear Burgers’ and Maxwell–Stefan diffusion equations.

Global spectral analysis for convection-diffusion-reaction equation in one and two-dimensions: Effects of numerical anti-diffusion and dispersion

Convection-diffusion-reaction (CDR) equation plays a central role in many disciplines of engineering, science and finances. As a consequence, importance of analysis of numerical methods for the accurate solution of CDR equation has motivated the present research. We have used the global spectral analysis to characterize all the three important physical processes in terms of the non-dimensional numerical parameters, namely, the non-dimensional wavenumber (kh); Courant-Friedrich-Lewy (CFL) number, Nc; the Peclet number (Pe) and the Damkohler number (Da). For the purpose of illustration, we have focused on two space-time discretization schemes known for accuracy and robustness. The basic properties relate to numerical issues arising for the numerical amplification factor, numerical diffusion coefficient, numerical phase speed and numerical group velocity. With the help of model one-dimensional (1D) and two-dimensional (2D) CDR equations, we have reported the numerical property charts for the cases: (i) When all the processes of convection, diffusion and reaction are of same order, with critical numerical behaviour enforcing low values of Da for the 1D CDR equation studied here. (ii) The 2D CDR equation considered is diffusion-reaction dominated, and as a consequence, this enforces Da to be larger. We have thoroughly analyzed these cases to identify the essential roles of anti-diffusion on the critical Nc and Pe values, which in turn decides admissible space and time steps to be used with the discretization schemes. The property charts have been used to calibrate the analysis with two model equations, one of which has an exact solution for a 1D CDR equation, and the second case for the 2D CDR equation has numerical solution available in the literature. These cases help to identify the importance of such analysis in explaining the utility of the choice one can exercise in fixing the numerical parameters. This also identifies and explains some hitherto unknown numerical problems for CDR equation and their alleviation techniques.

A layers capturing type H-adaptive finite element method for convection–diffusion–reaction equations on surfaces

In this work, we study the finite element approximation of surface convection–diffusion–reaction equations which are important fundamental model problems in simulations of complex physical phenomena on moving interfaces, ultra-thin materials and biological films. The optimal error estimates of finite element method are shown. According to the analyzed results, the solutions of convection-dominated diffusion problems have low accuracy in the regions of layers. To overcome such drawback which may lead to non-physical oscillations, the adaptive mesh refinement method is considered. Due to the numerical characteristic of non-physical oscillations and layers, we present an error estimator which specializes in capturing the non-physical oscillations and layers. The mesh refinement marking strategies which take into account the large curvatures of the surfaces are developed. The proposed method can efficiently capture layers, non-physical oscillations and provide high resolution solutions with fewer degree of freedoms. A series of numerical examples are designed to demonstrate the effectiveness of the proposed H-adaptive method and make exploration of the solution behavior of convection-dominated diffusion problems on surfaces.

Full discretization of time dependent convection–diffusion–reaction equation coupled with the Darcy system

In this article, we study the time dependent convection–diffusion–reaction equation coupled with the Darcy equation. We propose and analyze two numerical schemes based on finite element methods for the discretization in space and the implicit Euler method for the discretization in time. An optimal a priori error estimate is then derived for each numerical scheme. Finally, we present some numerical experiments that confirm the theoretical accuracy of the discretization.

A New Solution for Optimal Control of Fractional Convection–Reaction–Diffusion Equation Using Rational Barycentric Interpolation

This paper solves an optimal control problem governed by a fractional convection–reaction–diffusion partial differential equation. Using Lagrangian multipliers, necessary conditions are obtained, and then, Barycentric collocation method are applied for discretizing classical derivatives and Grunwald–Letnikov formula for fractional derivative. Barycentric interpolation is a class of Lagrange polynomial interpolation that is fast and deserves to be known as a method of polynomial interpolation and Grunwald–Letnikov formula is a basic extension of the derivative in fractional calculus. Numerical examples are presented to show the effectiveness of the method.

Optimal Buffer Zone for the Control of Groundwater Pollution from Agricultural Activities

We consider an optimal control model of groundwater pollution due to agricultural activities, the objective of the optimal manager being the optimization of the trade-off between the fertilizer use and the cleaning costs. The size of a buffer zone defined around the water production sites for limiting pollution effects may also be chosen by the manager. The spread of the pollution from crops to captation wells is modeled using a convection–diffusion–reaction equation. The main hydrogeological features of the dynamics are taken into account: the process is convection-dominated, dispersion effects are included, generic nonlinear reaction terms are considered. The existence and the uniqueness of the optimal solution is proven. Using asymptotic analysis, we then rigorously prove that a one-dimensional static model can be substituted to the full dynamic 3D model for the long time study of the optimal solution. We prove that this new optimal control problem is also well-posed. All these theoretical results are used for characterizing the optimal buffer zone.

Local projection stabilization with discontinuous Galerkin method in time applied to convection dominated problems in time-dependent domains

This paper presents the numerical analysis of a stabilized finite element scheme with discontinuous Galerkin (dG) discretization in time for the solution of a transient convection–diffusion–reaction equation in time-dependent domains. In particular, the local projection stabilization and the higher order dG time stepping scheme are used for convection dominated problems. Further, an arbitrary Lagrangian–Eulerian formulation is used to handle the time-dependent domain. The stability and error estimates are given for the proposed numerical scheme. The validation of the proposed local projection stabilization scheme with higher order dG time discretization is demonstrated with appropriate numerical examples.

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.

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.

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.

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.

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.

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.