Two-dimensional Convection-diffusion Research Articles

An RBF-LOD Method for Solving Stochastic Diffusion Equations

In this study, we introduce an innovative approach to solving stochastic equations in two and three dimensions, leveraging a time-splitting strategy. Our method combines radial basis function (RBF) spatial discretization with the Crank–Nicolson scheme and the local one-dimensional (LOD) method for temporal approximation. To navigate the probabilistic space inherent in these equations, we employ the Monte Carlo method, providing accurate estimates for expectations and variations. We apply our approach to tackle challenging problems, including two-dimensional convection-diffusion and Burgers’ equations, resulting in reduced computational and memory requirements. Through rigorous testing against diverse problem sets, our methodology demonstrates efficiency and reliability, underscoring its potential as a valuable tool in solving complex multidimensional stochastic equations. We have validated the method’s stability and showcased its convergence as the number of collocation points increases. These findings serve as compelling evidence of the suggested method’s convergence properties.

Investigating the chloride ion resistance of cracked concrete through Reverse-seepage and Saturation based Action Anti-corrosion Tech (RS-AAT): Experimental and numerical analysis

This study investigates the chloride ions transport in cracked concrete with the impact of RS-AAT (Reverse-seepage and Saturation based Action Anti-corrosion Tech) in the environment of wet-dry cycling and soaking conditions. With the effect of reverse water pressure, the chloride ions accumulation surrounding the crack on the concrete surface and the chloride ions penetration within the concrete along the cracks are demonstrated by the indoor experiments. A two-dimensional convective diffusion model of chloride ions in cracked concrete is established to simulate the correlation between crack size and the change of reverse water pressure. Four types of cracks with widths of 0 mm, 0.2 mm, 0.4 mm and 1 mm, and a depth of 20 mm were selected. Three types of water pressures with values of 0.2 MPa, 0.3 MPa and 0.4 MPa were used for the technique. The experimental results show that the anti-convection effect induced by the reverse water pressure can inhibit the accumulation of chloride ions around the cracks and effectively limit the intrusion of chloride ions into the concrete protection layer. With the increase of reverse water pressure, the inhibition effect of seepage flow on chloride ions is more obvious. The experimental findings provide valuable references for investigating marine concrete durability and resistance to chloride ion erosion.

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.

Investigation of the reverse-seepage technique to enhance the chloride ion invasion resistance of rectangular section concrete: Experiments and numerical analysis

This paper looks at the application of the reverse-seepage technique in reinforced concrete (RC) structures with rectangular cross-sectional forms under chloride ion attack. Experiments and numerical analyses of the chloride invasion within rectangular section concrete which applied the reverse-seepage technique are performed. A two-dimensional convection–diffusion model of chloride is developed, which can be used to further investigate the effect of the shape of permeable tube arrangement and ratio of water permeable area. The results indicate that the reverse-seepage technique can greatly inhibit the chlorine accumulation effect in the corner of the rectangular cross-section under the effect of wetting–drying cycles. In addition, it is recommended that the water permeable area ratio should be set at about 0.6, and the permeable tubes should be arranged in a rectangular shape to improve the chloride ion intrusion resistance efficiency. These results have important implications for the further application of the new anti-chloride erosion technique in engineering.

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.

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.

Investigation of the influence of the parameters of the mathematical model of two-dimensional stationary convective diffusion on the calculations of the concentration of depositing solid particles in the channel

Investigation of the influence of the parameters of the mathematical model of two-dimensional stationary convective diffusion on the calculations of the concentration of depositing solid particles in the channel

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.

A Computational Approach for One and Two Dimensional Fisher’s Equation Using Quadrature Technique

In this paper, a refined form of the differential quadrature method is proposed to compute the numerical solution of one and two-dimensional convection-diffusion Fisher’s equation. The cubic trigonometric B-spline basis functions are applied in the differential quadrature method in a modified form to obtain the weighting coefficients. The application of the method reduces nonlinear Fisher’s partial differential equation into a system of ordinary differential equations which can be solved by applying the Runge-Kutta method. Six numerical test problems of Fisher’s equation are analyzed numerically to establish the efficiency of the proposed method. The stability of the method is also discussed using the matrix method.

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.

Two-dimensional convection–diffusion in multipolar flows with applications in microfluidics and groundwater flow

Advection–diffusion in two-dimensional plane flows plays a key role in numerous transport problems in physics, including groundwater flow, micro-scale sensing, heat dissipation, and, in general, microfluidics. However, transport profiles are usually only known in a purely convective approximation or for the simplest geometries, such as for quasi one-dimensional planar microchannels. This situation greatly limits the use of these models as design tools for fully 2D planar flows. We present a complete analysis of the problem of convection–diffusion in low Reynolds number 2D flows with distributions of singularities, such as those found in open-space microfluidics and in groundwater flows. Using Boussinesq transformations and solving the problem in streamline coordinates, we obtain concentration profiles in flows with complex arrangements of sources and sinks for both high and low Peclet numbers. These yield the complete analytical concentration profile at every point in applications such as microfluidic probes, groundwater heat pumps, or diffusive flows in porous media, which previously relied on material surface tracking, local lump models, or numerical analysis. Using conformal transforms, we generate families of symmetrical solutions from simple ones and provide a general methodology that can be used to analyze any arrangement of source and sinks. The solutions obtained include explicit dependence on the various parameters of the problems, such as Pe, the spacing of the apertures, and their relative injection and aspiration rates. We then show how these same models can be used to model diffusion in confined geometries, such as channel junctions and chambers, and give examples for classic microfluidic devices such as T-mixers and hydrodynamic focusing. The high Pe models can model problems with Pe as low as 1 with a maximum error committed of under 10%, and this error decreases approximately as Pe−1.5.

Systematic formulation of a general numerical framework for solving the two-dimensional convection–diffusion–reaction system

Abstract A general numerical framework is designed for the two-dimensional convection–diffusion–reaction (CDR) system. The compatibility of differential quadrature and finite difference methods (FDM) are utilized for the formulation. The idea is to switch one numerical scheme to another numerical scheme without changing the formulation. The only requirement is to input the weighting coefficients associated with the derivative discretizations to the general algorithm. Three numerical schemes comprising combinations of differential quadrature and FDMs are studied using the general algorithm. Properties of numerical schemes and the algorithm are analyzed by using the simulations of two-dimensional linear CDR system, Burgers’ equation, and Brusselator model.

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.

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.

The streamline-diffusion finite element method on graded meshes for a convection–diffusion problem

In this paper, the streamline-diffusion finite element method is applied to a two-dimensional convection–diffusion problem posed on the unit square, using a graded mesh of O(N2) points based on standard Lagrange polynomials of degree k≥1. We prove the method is convergent almost uniformly in the perturbation parameter ϵ, and a convergence order O(N−klogk+1⁡(1ϵ)) is obtained in a streamline-diffusion norm under certain assumptions. Numerical experiments support the theoretical results.