Convection Diffusion Reaction Equations Research Articles

An assessment of solvers for algebraically stabilized discretizations of convection–diffusion–reaction equations

Abstract We consider flux-corrected finite element discretizations of 3D convection-dominated transport problems and assess the computational efficiency of algorithms based on such approximations. The methods under investigation include flux-corrected transport schemes and monolithic limiters. We discretize in space using a continuous Galerkin method and ℙ1 or ℚ1 finite elements. Time integration is performed using the Crank–Nicolson method or an explicit strong stability preserving Runge–Kutta method. Nonlinear systems are solved using a fixed-point iteration method, which requires solution of large linear systems at each iteration or time step. The great variety of options in the choice of discretization methods and solver components calls for a dedicated comparative study of existing approaches. To perform such a study, we define new 3D test problems for time dependent and stationary convection–diffusion–reaction equations. The results of our numerical experiments illustrate how the limiting technique, time discretization and solver impact on the overall performance.

Diffusion–convection reaction equations with sign-changing diffusivity and bistable reaction term

We consider a reaction–diffusion equation with a convection term in one space variable, where the diffusion changes sign from the positive to the negative and the reaction term is bistable. We study the existence of wavefront solutions, their uniqueness and regularity. The presence of convection reveals several new features of wavefronts: according to the mutual positions of the diffusivity and reaction, profiles can occur either for a single value of the speed or for a bounded interval of such values; uniqueness (up to shifts) is lost; moreover, plateaus of arbitrary length can appear; profiles can be singular where the diffusion vanishes.

A two-level fourth-order approach for time-fractional convection–diffusion–reaction equation with variable coefficients

This paper develops a two-level fourth-order scheme for solving time-fractional convection–diffusion–reaction equation with variable coefficients subjects to suitable initial and boundary conditions. The basis properties of the new approach are investigated and both stability and error estimates of the proposed numerical scheme are deeply analyzed in the L∞(0,T;L2)-norm. The theory indicates that the method is unconditionally stable with convergence of order O(k2−λ2+h4), where k and h are time step and mesh size, respectively, and λ∈(0,1). This result suggests that the two-level fourth-order procedure is more efficient than a large class of numerical techniques widely studied in the literature for the considered problem. Some numerical evidences are provided to verify the unconditional stability and convergence rate of the proposed algorithm.

High-Order Compact Difference Method for Solving Two- and Three-Dimensional Unsteady Convection Diffusion Reaction Equations

In this paper, a type of high-order compact (HOC) finite difference method is developed for solving two- and three-dimensional unsteady convection diffusion reaction (CDR) equations with variable coefficients. Firstly, an HOC difference scheme is derived to solve the two-dimensional (2D) unsteady CDR equation. Discretization in time is carried out by Taylor series expansion and correction of the truncation error remainder, while discretization in space is based on the fourth-order compact difference formulas. The scheme is second-order accuracy in time and fourth-order accuracy in space. The unconditional stability is obtained by the von Neumann analysis method. Then, this scheme is extended to solve the three-dimensional (3D) unsteady CDR equation. It needs only a five-point stencil for 2D problems and a seven-point stencil for 3D problems. Moreover, the present schemes can solve the nonlinear Burgers equation. Finally, numerical experiments are conducted to show the good performances of the new schemes.

The virtual element method for the time fractional convection diffusion reaction equation with non-smooth data

In this paper, we develop an efficient virtual element method to solve the two-dimension time fractional convection diffusion reaction equation involving the Caputo fractional derivative with non-smooth solutions in the time direction. The equation exhibits a weak singularity near the initial time. We first use the exponential transformation to eliminate the convection terms, and obtain the time fractional diffusion reaction equation. By utilizing properties of the energy projection operator and continuous Grönwall inequality, we derive estimates of the approximation error in the L2-norm and H1-norm under semi-discrete on polygonal meshes. Based on the L1 scheme on graded mesh in time, the fully-discrete scheme is proved to be unconditionally stable and the optimal convergence results are derived with regards to the L2-norm and H1-norm. Finally, some numerical experiments are implemented to verify the theoretical results.

A priori estimates to solutions of the time-fractional convection–diffusion–reaction equation coupled with the Darcy system

We consider the Darcy flow equation coupled with the Caputo time-fractional convection–reaction–diffusion equations. This system describes the concentration distribution of a fluid running through a porous medium. A priori estimates are established for the solution by employing the method of energy inequalities. The existence, uniqueness and regularity properties of a weak solution are studied. Our analysis relies on two novel and different methodologies in combination with two alternative variational formulations.

An integrated fluid\u2013structure interaction and thrombosis model for type B aortic dissection

False lumen thrombosis (FLT) in type B aortic dissection has been associated with the progression of dissection and treatment outcome. Existing computational models mostly assume rigid wall behavior which ignores the effect of flap motion on flow and thrombus formation within the FL. In this study, we have combined a fully coupled fluid–structure interaction (FSI) approach with a shear-driven thrombosis model described by a series of convection–diffusion reaction equations. The integrated FSI-thrombosis model has been applied to an idealized dissection geometry to investigate the interaction between vessel wall motion and growing thrombus. Our simulation results show that wall compliance and flap motion can influence the progression of FLT. The main difference between the rigid and FSI models is the continuous development of vortices near the tears caused by drastic flap motion up to 4.45 mm. Flap-induced high shear stress and shear rates around tears help to transport activated platelets further to the neighboring region, thus speeding up thrombus formation during the accelerated phase in the FSI models. Reducing flap mobility by increasing the Young’s modulus of the flap slows down the thrombus growth. Compared to the rigid model, the predicted thrombus volume is 25% larger using the FSI-thrombosis model with a relatively mobile flap. Furthermore, our FSI-thrombosis model can capture the gradual effect of thrombus growth on the flow field, leading to flow obstruction in the FL, increased blood viscosity and reduced flap motion. This model is a step closer toward simulating realistic thrombus growth in aortic dissection, by taking into account the effect of intimal flap and vessel wall motion.

Numerical modeling two natural languages interaction

In this paper, we propose a mathematical model of the interaction of two languages. In our model, we consider two languages, though it can be generalized to multiple languages, which compete in a heterogeneous environment, consisting of highly varying properties related to the dynamics of the interaction. We use coupled convection–diffusion–reaction equations to describe the processes. Each equation describes the dynamics of one of the languages and contains terms related to the stand-alone dynamics and some coupling terms. The coupling terms represent the interaction between languages. We propose a numerical approach for solving the proposed model equation. In particular, we consider various inhomogeneities associated with cities and countryside, where languages are used differently (e.g., Sakha republic). These dynamics are essential for understanding the evolution of languages (one being dominant) and linguistic ecology that studies languages and their use in real/social life. Because of heterogeneities associated with geography, we use a multiscale approach. The proposed multiscale approach designs special basis functions to represent the small-scale information on larger scales. This way, we can solve the problem on a much coarser grid. Numerical results are presented that describe the dynamics and interaction of two languages. The main novelty of the paper consists of the proposed model and a multiscale algorithm.

Continuum modeling of thrombus formation and growth under different shear rates

Obstruction of blood flow due to thrombosis is a major cause of ischemic stroke, myocardial infarction, and in severe cases, mortality. In particular, in blood wetting medical devices, thrombosis is a common reason for failure. The prediction of thrombosis by understanding signaling pathways using computational models, lead to identify the risk of thrombus formation in blood-contacting devices and design improvements. In this study, a mathematical model of thrombus formation and growth is presented. A biochemical model of platelet activation and aggregation is developed to predict thrombus size and shape at the site of vascular injury. Computational fluid dynamics using the finite volume method is employed to compute the velocity and pressure fields which are influenced by the growing thrombi. The passive transport of platelets, agonists, the platelet activation kinetics, their adhesion to the growing thrombi and embolization of platelets are solved by a fully coupled set of convection–diffusion–reaction equations. The thrombogenic surface representing blood-contacting material or injured blood vessel was incorporated into the model as a surface flux boundary condition to initiate thrombus formation. The blood is considered as a Newtonian fluid, while the thrombus is treated as a porous medium. The results are compared with in vitro experiments of a microfluidic chamber at an initial inlet venous shear rate of 200s−1 using a pressure–inlet boundary condition. The thrombus development due to agonist concentrations and change in the shear rate as well as thromboembolism for this benchmark problem is successfully computed. Furthermore, to extend the current model to a physiologically relevant configuration, thrombus formation in a blood tube is simulated. Two different heterogeneous reaction rates for platelet aggregation are used to simulate thrombus growth under a constant inlet flow rate. The findings show that the thrombus shape can be substantially altered by the hemodynamic conditions, increase in the shear rate and due to the combined effects of shear induced platelet activation and the heterogeneous reaction rates. It is also concluded that the model is able to predict thrombus formation in different physiological and pathological hemodynamics.

The study of fractional-order convection-reaction-diffusion equation via an Elzake Atangana-Baleanu operator

<abstract><p>The major goal of this research is to use a new integral transform approach to obtain the exact solution to the time-fractional convection-reaction-diffusion equations (CRDEs). The proposed method is a combination of the Elzaki transform and the homotopy perturbation method. He's polynomial is used to tackle the nonlinearity which arise in our considered problems.Three test examples are considered to show the accuracy of the proposed scheme. In order to find satisfactory approximations to the offered problems, this work takes into account a sophisticated methodology and fractional operators in this context. In order to achieve better approximations after a limited number of iterations, we first construct the Elzaki transforms of the Caputo fractional derivative (CFD) and Atangana-Baleanu fractional derivative (ABFD) and implement them for CRDEs. It has been found that the proposed method's solution converges at the desired rate towards the accurate solution. We give some graphical representations of the accurate and analytical results, which are in excellent agreement with one another, to demonstrate the validity of the suggested methodology. For validity of the present technique, the convergence of the fractional solutions towards integer order solution is investigated. The proposed method is found to be very efficient, simple, and suitable to other nonlinear problem raised in science and engineering.</p></abstract>

A High-Order Finite Difference Scheme for 3D Unsteady Convection Diffusion Reaction Equations

A High-Order Finite Difference Scheme for 3D Unsteady Convection Diffusion Reaction Equations

Numerical Simulation of the Photobleaching Process in Laser-Induced Fluorescence Photobleaching Anemometer

At present, a novel flow diagnostic technique for micro/nanofluidics velocity measurement—laser-induced fluorescence photobleaching anemometer (LIFPA)—has been developed and successfully applied in broad areas, e.g., electrokinetic turbulence in micromixers and AC electroosmotic flow. Nevertheless, in previous investigations, to qualitatively reveal the dynamics of the photobleaching process of LIFPA, an approximation of uniform laser distribution was applied. This is different from the actual condition where the laser power density distribution is normally Gaussian. In this investigation, we numerically studied the photobleaching process of fluorescent dye in the laser focus region, according to the convection–diffusion reaction equation. The profiles of effective dye concentration and fluorescence were elucidated. The relationship between the commonly used photobleaching time constant obtained by experiments and the photochemical reaction coefficient is revealed. With the established model, we further discuss the effective spatial resolution of LIFPA and study the influence of the detection region of fluorescence on the performance of the LIFPA system. It is found that at sufficiently high excitation laser power density, LIFPA can even achieve a super-resolution that breaks the limit of optical diffraction. We hope the current investigation can reveal the photobleaching process of fluorescent dye under high laser power density illumination, to enhance our understanding of fluorescent dynamics and photochemistry and develop more powerful photobleaching-related flow diagnostic techniques.

An LT-BEM formulation for problems of anisotropic functionally graded materials governed by transient diffusion–convection–reaction equation

Problems of anisotropic functionally graded media which are governed by the transient diffusion–convection–reaction equation of spatially varying coefficients are discussed in this paper. A mathematical analysis is used to transform the variable coefficient equation into a constant coefficient equation. A boundary-only integral equation is then derived from this constant coefficient equation after being Laplace transformed. Numerical solutions to the problems are sought by using a boundary element method (BEM) which is combined with the Stehfest formula for the numerical Laplace transform inversion. Some problems considered are those of compressible or incompressible flow, and of media which are quadratically, exponentially or trigonometrically graded materials. The results obtained show that the analysis used to transform the variable coefficients equation into the constant coefficients equation is valid, and the mixed LT-BEM is easy to implement and accurate for finding numerical solutions. The numerical solutions of some test problems are justified by showing their accuracy. Some non-test problems of geometrically symmetric systems are also considered to show the effect of the anisotropy and inhomogeneity of the material on the solutions by verifying the symmetry of solutions. In addition, the effect of boundary conditions is also exhibited.

Two-grid discontinuous Galerkin method for convection–diffusion–reaction equations

In this paper, we study a two-grid method based on discontinuous Galerkin discretization for the convection–diffusion–reaction equation. The two-grid algorithm consists two steps: first solving the original nonsymmetric problem on coarse grid, and then solving the corresponding positive definite diffusion problem on fine grid. Note that the number of degrees of freedom on coarse mesh is less than the ones on fine mesh. Moreover, the bilinear form of positive definite problem on fine mesh only depends on the diffusion coefficient. Therefore, the two-grid algorithm essentially transforms the DG solution of convection–diffusion–reaction equation into the approximation for the DG solution of diffusion equation. The corresponding error estimates of the two-grid solution are also provided. Numerical experiments are performed to verify the theoretical results.

Finite volume element approximation for time dependent convection diffusion reaction equations with memory

Error estimates for element schemes for time-dependent for convection-diffusion-reaction equations with memory are derived and stated. For the spatially discrete scheme, optimal order error estimates in L2 , H1 , and W1,p norms for 2 ≤ p < ∞, are obtained. In this paper, we also study the lumped mass modification. Based on the Crank-Nicolson method, a time discretization scheme is discussed and related error estimates are derived.

Analysis of the SORAS Domain Decomposition Preconditioner for Non-self-adjoint or Indefinite Problems

We analyze the convergence of the one-level overlapping domain decomposition preconditioner SORAS (Symmetrized Optimized Restricted Additive Schwarz) applied to a generic linear system whose matrix is not necessarily symmetric/self-adjoint nor positive definite. By generalizing the theory for the Helmholtz equation developed in Graham et al. (SIAM J Numer Anal 58(5):2515–2543, 2020. https://doi.org/10.1137/19M1272512), we identify a list of assumptions and estimates that are sufficient to obtain an upper bound on the norm of the preconditioned matrix, and a lower bound on the distance of its field of values from the origin. We stress that our theory is general in the sense that it is not specific to one particular boundary value problem. Moreover, it does not rely on a coarse mesh whose elements are sufficiently small. As an illustration of this framework, we prove new estimates for overlapping domain decomposition methods with Robin-type transmission conditions for the heterogeneous reaction–convection–diffusion equation (to prove the stability assumption for this equation we consider the case of a coercive bilinear form, which is non-symmetric, though).

Fast IIF–WENO Method on Non-uniform Meshes for Nonlinear Space-Fractional Convection–Diffusion–Reaction Equations

Fast IIF–WENO Method on Non-uniform Meshes for Nonlinear Space-Fractional Convection–Diffusion–Reaction Equations

Adaptive ADI difference solution of quenching problems based on the 3D convection–reaction–diffusion equation

Abstract The 3D quenching problem reflecting solid-burn scene based on convection–reaction–diffusion equation is creatively concerned in this work. The spatial derivatives of original equation are discretized by Taylor series and the temporal derivatives are approximated by the Crank–Nicolson (CN) method. After the discrete schemes are arranged, an alternating direction implicit (ADI) scheme on adaptive grid is constructed to interpret quenching phenomena of the three-dimension (3D) equation with singularity source. Quenching time, quenching domain, and characteristics relative to temperature as well as variation of temperature over time are achieved via scientific experiment and analysis. Comparing with the 1D or 2D problem, it is harder for the 3D problem to produce quenching phenomena. Regardless of different convection functions, it can form quenching behaviors through experiments when only the elements which include degeneracy parameter, convection parameters, and domain sizes are configured properly. We hope all this can offer references for the 3D engineering problem. At the same time, it will offer support to research the relationship between quenching phenomena and degeneracy parameter, convection parameters, and domain sizes in the future, respectively.

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.