Two-dimensional Convection-diffusion Equation Research Articles

OPERATOR-DIFFERENCE APPROXIMATIONS ON NON-STANDARD RECTANGULAR GRIDS

In the approximate solution of boundary value problems for partial differential equations, difference methods are widely used. Grid approximations are most simply constructed when dividing the calculated area into rectangular cells. Usually the grid nodes coincide with the vertices of the cells. In addition to such nodal approximations, grids with nodes in the centers of cells are also used. It is convenient to formulate boundary value problems in terms of invariant operators of vector (tensor) analysis, which are compared with the corresponding grid analogues. The paper builds analogues of gradient and divergence operators on non-standard rectangular grids, the nodes of which consist of both the vertices of the calculated cells and their centers. The proposed approach is illustrated by approximations of the boundary value problem for the stationary two-dimensional convection-diffusion equation. The key features of the construction of approximations for vector problems with orientation to applied problems of solid mechanics are noted.

Radial Basis Function Based Partition of Unity Method for Two-Dimensional Unsteady Convection Diffusion Equations

Radial Basis Function Based Partition of Unity Method for Two-Dimensional Unsteady Convection Diffusion Equations

A novel Cell-Centered nodal integral method for the Convection-Diffusion equation

The nodal integral methods (NIMs) are coarse mesh methods for solving partial differential equations (PDEs) efficiently and accurately. Here, a new variant of the NIM is proposed for the solution of the two-dimensional convection–diffusion equation. The earlier versions of the NIM for fluid flow problems used surface-averaged variables. In contrast, the current scheme is developed in terms of cell-centered values using different treatments of cell interface conditions. The advantage of this scheme is that coupling with other physics is straightforward. Another distinctive feature of the current approach is that the temporal derivative is explicitly addressed, resulting in a system of differential–algebraic equations (DAEs) of index-1. The system of DAEs is then solved using Backward Difference Formula, which provides higher accuracy in time compared to traditional NIM. A few problems with the known analytical solution are solved to demonstrate the efficacy of the current simplified approach.

The Comparison Study of the Hybrid Method for Solving the Unsteady State Two-Dimensional Convection-Diffusion Equations

In this paper, we present a hybrid method combining the reduced differential transform method (RDTM) and a resumption method based on Yang transform and Padé approximant to find analytical solutions for three test problems for the unsteady state two-dimensional convection-diffusion equation. The proposed method significantly improves the approximate solution series and broadens the convergence field of RDTM. The numerical results obtained are compared to RDTM and other results from previous works. The results show that the proposed method is very efficient and has high accuracy. The main advantage of the proposed method is that it is based on a few straightforward steps and does not generate secular terms or depend on a perturbation parameter. We also provided a powerful and attractive mathematical tool for solving linear and nonlinear equations.

Multiple relaxation time lattice Boltzmann schemes for advection-diffusion equations with application to radar image processing

Motivated by marine radar image processing, we investigate the accuracy of multiple relaxation time lattice Boltzmann schemes designed to simulate two-dimensional convection-diffusion equations. The context of application requires to deal with non-constant advection velocity. Using Taylor expansions, instead of the widely used Chapman-Enskog expansions, we show how to control the accuracy of these schemes when deriving equivalent partial differential equations. On the one hand, a third order analysis is conducted on a scheme involving a constant advection velocity and no source term. First, this analysis derives the stability region through the von Neumann analysis. Second, a numerical convergence rate of three is obtained thanks to an appropriate choice of parameters. On the other hand, non-constant advection velocity together with non-zero source term, introduce additional terms at the second order. Regarding the targeted application, these extra terms are shown to be negligible and experiments on real data show that such multiple relaxation time lattice Boltzmann schemes are relevant for marine radar denoising and enhancement.

On solute dispersion in an oscillatory magneto-hydrodynamics porous medium flow under the effect of heterogeneous and bulk chemical reaction

It is well known to all of us that there is a shortage of pure drinking water across the globe. Different types of pollutants (metallic and nonmetallic) mix with the water, and they cause several diseases such as cholera, typhoid, and various kinds of skin diseases, and even it is found that these kinds of particles may cause skin cancer. In the current study, an analytical solution of a two-dimensional convection–diffusion equation is obtained using Mei's multi-scale homogenization technique to investigate the influences of homogeneous and heterogeneous reactions on dispersion phenomena of the solute in an oscillatory magneto-hydrodynamics porous medium flow. In the appearance of the applied transverse magnetic field and oscillatory pressure gradient, a mathematical model of magneto-hydrodynamics dispersion between two parallel plates is presented. The analytical expressions of Taylor dispersivity, longitudinal mean and real concentration distributions, transverse concentration distribution, and transverse uniformity rate of the concentration are obtained. Also, the effect of various flow parameters such as Péclet number, Hartmann number, Schmidt number, Darcy number, oscillatory Reynolds number, porous parameter, dispersion time, downstream and upstream locations, chemical heterogeneous boundary reaction, and bulk reaction is discussed. How the transport phenomena of the solute display different natures with the various ranges of Darcy and Hartmann numbers with the aid of homogeneous and heterogeneous boundary reactions are highlighted. To show the effect of the absorption parameters on the transport coefficient, the third-order approximation of concentration is performed. It is seen that the dispersion coefficient (DT1) corresponding to the purely time-dependent flow increases with the enhancement of the Darcy number (Da). Moreover, it is found that as the Hartmann number (M) enhances, the total dispersivity (DT) decreases. Also, the transverse concentration distribution becomes flat for larger values of the Hartmann number. It is noticed that when Da≥1, the transverse variation curve turns into a trimodal distribution from a bimodal. This model may be helpful for separating various metallic and nonmetallic particles from the water to reduce the water pollution.

A fourth-order unfitted characteristic finite element method for free-boundary problems

A fourth-order unfitted characteristic finite element method (UCFEM) is proposed to solve free-boundary problems of time-dependent partial differential equations (PDEs). The boundary of the domain is implicitly driven by the solution of the PDE, while the PDE is proposed on the time-varying unknown domain. In this way, they form a coupled nonlinear system. The key ingredient of the UCFEM is to trace the domain explicitly with a fourth-order forward flow map and discretize the PDE in time with a fourth-order backward flow map. The PDE is solved with a fourth-order unfitted finite element method on a fixed mesh. Since the boundary of the domain is expressed explicitly with cubic spline functions, quadrature on cut elements can be done accurately and efficiently even the domain undergoes severe deformations. The UCFEM provides a framework of designing high-order numerical methods for free-boundary problems. We apply the UCFEM to a two-dimensional convection-diffusion equation and Navier-Stokes equations, and obtain the overall fourth order of accuracy. With extensive numerical experiments, we show that the proposed method can achieve the fourth-order convergence even on severely deformed domains.

A Samarskii domain decomposition method for two-dimensional convection–diffusion equations

A Samarskii domain decomposition method for two-dimensional convection–diffusion equations

An analytical approximate method for solving unsteady state two-dimensional convection-diffusion equations

In this paper, an analytic approximate method for solving the unsteady two-dimensional convection-diffusion equations is introduced. Also, the convergence of the approximate methods is analyzed. Three test examples are presented, two have exact and one has not exacted solutions. The results obtained show that these methods are powerful mathematical tools for solving linear and nonlinear partial differential equations, moreover, new analytic Taylor method (NATM), reduced differential transform method (RDTM), and homotopy perturbation method (HPM), are more accurate and have less CPU time than the other methods.

Multi-scale analysis for transport of fine settling particles through an ice-covered channel in a laminar flow condition

The current research deals with the dispersion of fine settling particles in a fluid flowing through an ice-covered channel under the laminar flow condition. An analytical solution of the two-dimensional convection–diffusion equation, based on the multi-scale homogenization technique, is obtained. To validate the current study, analytical results for the dispersion coefficient are compared with the available earlier research work. Moreover, the proposed analytical solutions for mean concentration distributions of the tracers are compared with the numerical results obtained from the finite difference technique. From the industrial and environmental points of view, the vertical concentration distribution provides a very significant information. The downstream evolution of the concentration distribution also is shown for typical time periods at different values of the settling velocity. The approach to the vertical uniformity shows that it is too slow a process in comparison to that of longitudinal normality. It was found that settling velocities of particles disturb the vertical uniformity and the centroid of the solute cloud rises due to the increase in settling velocity. Results illustrate that in the downstream direction, the vertical concentration distribution increases near the bed surface and it decreases in the proximity of the ice-covered surface of the channel with the increase of settling velocity, but the mean concentration of the solute increases. The current study may play an important role to understand the mechanism of the sedimentation process in a closed channel system.

Study of the initial boundary value problem for a two-dimensional convection-diffusion equation with a fractional time derivative in the sense of Caputo-Fabrizio

In this paper, we study an initial boundary value problem for a differential equation with a fractional order derivative in time in the Caputo-Fabrizio sense. This equation is of great practical importance in modeling the processes of fluid motion in porous media and anomalous dispersion. The uniqueness and continuous dependence of the solution of the problem on the input data in differential form is proved. A computationally efficient implicit difference scheme with weights isproposed. A priori estimates are obtained for the solution of the problem under the assumption that the solution exists in the class of sufficiently smooth functions. The uniqueness of the solution and the stability of the difference scheme with respect to the initial data and the right-hand side of the equation follows from the obtained estimates. The convergence of the difference problem solution to the differential problem solution with the second order in time and space variablesis proved. The results of computational experiments confirming the reliability of the theoretical analysis are presented. Key words: Fractional differential equation, fractional derivative in the sense of Caputo-Fabrizio, finite difference method, energy inequality method, stability, convergence, a priori estimate.

Determination of the time-dependent convection coefficient in two-dimensional free boundary problems

PurposeThe purpose of the study is to solve numerically the inverse problem of determining the time-dependent convection coefficient and the free boundary, along with the temperature in the two-dimensional convection-diffusion equation with initial and boundary conditions supplemented by non-local integral observations. From the literature, there is already known that this inverse problem has a unique solution. However, the problem is still ill-posed by being unstable to noise in the input data.Design/methodologyFor the numerical discretization, this paper applies the alternating direction explicit finite-difference method along with the Tikhonov regularization to find a stable and accurate numerical solution. The resulting nonlinear minimization problem is solved computationally using the MATLAB routine lsqnonlin. Both exact and numerically simulated noisy input data are inverted.FindingsThe numerical results demonstrate that accurate and stable solutions are obtained.Originality/valueThe inverse problem presented in this paper was already showed to be locally uniquely solvable, but no numerical solution has been realized so far; hence, the main originality of this work is to attempt this task.

Analytical and approximate solution of two-dimensional convection-diffusion problems

In this work, we have used reduced differential transform method (RDTM) to compute an approximate solution of the Two-Dimensional Convection-Diffusion equations (TDCDE). This method provides the solution quickly in the form of a convergent series. Also, by using RDTM the approximate solution of two-dimensional convection-diffusion equation is obtained. Further, we have computed exact solution of non-homogeneous CDE by using the same method. To the best of my knowledge, the research work carried out in the present paper has not been done, and is new. Examples are provided to support our work.

Transient temperature prediction model of horizontal wells during drilling shale gas and geothermal energy

A transient temperature prediction model was established to analyze wellbore temperature distribution of horizontal wells during drilling shale gas and geothermal energy based on unsteady two-dimensional convection-diffusion equation and the model was discretized with finite volume methods and solved by owe relaxation iteration method. The validity of the model has been verified by field temperature data. The influence of drilling parameters on temperature distribution of drilling fluid were discussed. Results indicated that the closer to the bottom of the well, the smaller the rate of change of the drilling fluid temperature with depth. The temperature of the drilling fluid at the same well depth increases with the increasing horizontal section length, drilling fluid density, geothermal gradient and vertical depth. The temperature of the upper part of the wellbore increases with the increasing circulation time and drilling displacement, and the lower part of the wellbore decreases with the increasing circulation time and drilling displacement. Drilling fluid temperature inside the drilling string decreases with the increase in viscosity. However, drilling fluid temperature inside the annulus of the upper part of the wellbore increases with the increase of the viscosity while the drilling fluid temperature of the lower part of the wellbore decreases with the increasing viscosity.

矩形域上的二维对流扩散方程边界流量反问题

矩形域上的二维对流扩散方程边界流量反问题

Difference Schemes on Nonuniform Grids for the Two-Dimensional Convection–Diffusion Equation

New second-order accurate monotone difference schemes on nonuniform spatial grids for two-dimensional stationary and nonstationary convection–diffusion equations are proposed. The monotonicity and stability of the solutions of the computational methods with respect to the boundary conditions, the initial condition, and the right-hand side are proved. Two-sided and corresponding a priori estimates are obtained in the grid norm of C. The convergence of the proposed algorithms to the solution of the original differential problem with the second order is proved.

Integrated framework for solving the convection diffusion equation on 2D Quad mesh relying on internal boundaries

The aim of this research is the design and implementation of an integrated framework to solve numerically the two-dimensional convection diffusion equations on non-rectangular grids formed only by quadrilaterals honoring the internal structures of an oil reservoir. The framework is composed of (1) a preprocessor of the internal boundaries of a 2D oil reservoir structure, (2) a 2D structural quadrilateral grid generator, (3) a solver of the 2D convection diffusion equation (CDE) on the grid, and (4) a visualization module to display the reservoir properties, wells and the simulation results on the grid. An example of typical structures corresponding to an areal view of a hydrocarbon reservoir is presented. Different scenarios were considered varying boundary conditions, source term, and diffusion constant fluid velocity. All the results are consistent with the physical interpretation of each configuration.

Delay-dependent stability analysis of convection-diffusion equations with delay

This paper is concerned with the delay-dependent stability analysis of the one-dimensional and two-dimensional convection-diffusion equations with time delay. Some sufficient and necessary conditions for the asymptotical stability of the equations are proposed. These results lay the foundation for the numerical stability analysis of such kinds of equations.

Computational Solutions of Two Dimensional Convection Diffusion Equation Using Crank-Nicolson and Time Efficient ADI

To develop an efficient numerical scheme for two-dimensional convection diffusion equation using Crank-Nicholson and ADI, time-dependent nonlinear system is discussed. These schemes are of second order accurate in apace and time solved at each time level. The procedure was combined with Iterative methods to solve non-linear systems. Efficiency and accuracy are studied in term of L2, L∞ norms confirmed by numerical results by choosing two test examples. Numerical results show that proposed alternating direction implicit scheme was very efficient and reliable for solving two dimensional nonlinear convection diffusion equation. The proposed methods can be implemented for solving non-linear problems arising in engineering and physics.

POD-Galerkin reduced-order modeling with adaptive finite element snapshots

We consider model order reduction by proper orthogonal decomposition (POD) for parametrized partial differential equations, where the underlying snapshots are computed with adaptive finite elements. We address computational and theoretical issues arising from the fact that the snapshots are members of different finite element spaces. We propose a method to create a POD-Galerkin model without interpolating the snapshots onto their common finite element mesh. The error of the reduced-order solution is not necessarily Galerkin orthogonal to the reduced space created from space-adapted snapshot. We analyze how this influences the error assessment for POD-Galerkin models of linear elliptic boundary value problems. As a numerical example we consider a two-dimensional convection–diffusion equation with a parametrized convective direction. To illustrate the applicability of our techniques to non-linear time-dependent problems, we present a test case of a two-dimensional viscous Burgers equation with parametrized initial data.