Solution Of Convection-diffusion Equation Research Articles

Formulation of Low Peclet Number Based Grid Expansion Factor for the Solution of the Convection Diffusion Equation

Convection-diffusion problems, due to its fundamental nature, are found in various science and engineering applications. In this research, the importance of the relationship between grid structure and flow parameters in such problems is emphasized. In particular, we propose a systematic technique in the selection of the grid expansion factor based on its logarithmic relationship with low Peclet number. Such linear mathematical connection between the two non-dimensional parameters serves as a guideline for more structured decision-making and improves the heuristic process in the determination of the computational domain grid for the numerical solution of convection-diffusion equations especially in the prediction of the concentration of the scalar. Results confirm the effectiveness of the new approach.

Decay rate for degenerate convection diffusion equations in both one and several space dimensions

We consider degenerate convection-diffusion equations in both one space dimension and several space dimensions. In the first part of this article, we are concerned with the decay rate of solutions of one dimension convection diffusion equation. On the other hand, in the second part of this article, we are concerned with a decay rate of derivatives of solution of convection diffusion equation in several space dimensions.

基于牛顿插值的高分辨率有限体积格式

有限体积方法在计算流体和热传导的数值计算中占据重要的地位。在数值格式构造的过程中，如何消除对流扩散方程选取间断波为初始条件时数值解所产生的非物理震荡，是有限体积方法主要的研究内容。本文基于对流有界准则(CBC)原理，通过Newton插值多项式的方法构造新的高分辨率有限体积格式。经典算例表明，本文构造的数值格式对于大梯度或间断解有较好的分辨率和数值稳定性。 Finite volume method plays an important role in fluid flow and heat transfer numerical calculation. How to eliminate unphysical oscillations caused by numerical solution of convection diffusion equation selecting discontinuity wave as the initial condition is a key task for studying finite volume method. New high-resolution schemes were constructed by Newton interpolation polynomial based on convection boundness criterion (CBC). Classic test cases demonstrated that the present numerical scheme possesses high resolution and good stability for high gradient and discontinuous solution.

Studying the Effect of Preconditioner Scheme on the Solution of Convection Diffusion Equations

Studying the Effect of Preconditioner Scheme on the Solution of Convection Diffusion Equations

Cubic Spline Iterative Method for Poisson’s Equation in Cylindrical Polar Coordinates

Using nonpolynomial cubic spline approximation in x- and finite difference in y-direction, we discuss a numerical approximation of O(k2+h4) for the solutions of diffusion-convection equation, where k>0 and h>0 are grid sizes in y- and x-coordinates, respectively. We also extend our technique to polar coordinate system and obtain high-order numerical scheme for Poisson’s equation in cylindrical polar coordinates. Iterative method of the proposed method is discussed, and numerical examples are given in support of the theoretical results.

Methods for Determining Agent Concentration Profiles in Agarose Gel During Convection-Enhanced Delivery

Convection-enhanced delivery (CED) is a promising technique to deliver large molecular weight drugs to the human brain for treatment of Parkinson's, Alzheimer's, or brain tumors. Researchers have used agarose gels to study mechanisms of agent transport in soft tissues like brain due to its similar mechanical and transport properties. However, inexpensive quantitative techniques to precisely measure achieved agent distribution in agarose gel phantoms during CED are missing. Such precise measurements of concentration distribution are needed to optimize drug delivery. An optical experimental method to accurately quantify agent concentration in agarose is presented. A novel geometry correction algorithm is used to determine real concentrations from observable light intensities captured by a digital camera. We demonstrate the technique in dye infusion experiments that provide cylindrical and spherical distributions when infusing with porous membrane and conventional single-port catheters, respectively. This optical method incorporates important parameters, such as optimum camera exposure, captured camera intensity calibration, and use of collimated light source for maximum precision. We compare experimental results with numerical solutions to the convection diffusion equation. The solutions of convection-diffusion equations in the cylindrical and spherical domains were found to match the experimental data obtained by geometry correction algorithm.

Error Analysis for hp-FEM Semi-Lagrangian Second Order BDF Method for Convection-Dominated Diffusion Problems

We present in this paper an analysis of a semi-Lagrangian second order Backward Difference Formula combined with hp-finite element method to calculate the numerical solution of convection diffusion equations in ?2. Using mesh dependent norms, we prove that the a priori error estimate has two components: one corresponds to the approximation of the exact solution along the characteristic curves, which is $O(\Delta t^{2}+h^{m+1}(1+\frac{\mathopen{|}\log h|}{\Delta t}))$ ; and the second, which is $O(\Delta t^{p}+\| \vec{u}-\vec{u}_{h}\|_{L^{\infty}})$ , represents the error committed in the calculation of the characteristic curves. Here, m is the degree of the polynomials in the finite element space, $\vec{u}$ is the velocity vector, $\vec{u}_{h}$ is the finite element approximation of $\vec{u}$ and p denotes the order of the method employed to calculate the characteristics curves. Numerical examples support the validity of our estimates.

MODELING OF PROBE MOTION PROCESS IN MICROANALYTICAL ELECTROCHEMILUMINESCENT SYSTEM OF CAPILLARY ELECTROPHORESIS

The model of electrokinetic probe motion that is based on numerical solution of convective diffusion equation using finite-difference method was developed and implemented. The distribution of monodisperse dissolved substance (amino acid proline) along zone of probe that migrate in buffer solution in straight rectangular separation channel of microanalytical electrochemiluminescent of capillary electrophoresis system (CE) was calculated. It was shown that increase of voltage controlling axial electric field during conduction of CE leads to narrowing of electrophoretic profile elution; passage time of concentration peak has nonlinear dependence versus applied voltage, and dependence between peak position and channel length is practically linear.

Determination of Hydro-chemical Parameters of Salt Transportation in Soil by Using the Solution of Convective Diffusion Equation

Determination of Hydro-chemical Parameters of Salt Transportation in Soil by Using the Solution of Convective Diffusion Equation

Numerical simulation of interphase mass transfer with the level set approach

A level set approach is applied for simulating the interphase mass transfer of single drops in immiscible liquid with resistance in both phases. The control volume formulation with the SIMPLEC (semi-implicit method for pressure-linked equations consistent) algorithm incorporated is used to solve the governing equations of incompressible two-phase flow with deformable free interface on a staggered Eulerian grid. The solution of convective diffusion equation for interphase mass transfer is decoupled with the momentum equations. Different spatial discretization schemes including the fifth-order WENO (weighted essentially nonoscillatory), second-order ENO (essentially nonoscillatory) and power-law schemes, are tested for the solution of mass transfer to or from single drops. The conjugate cases with different equilibrium distribution coefficients are simulated successfully with the transformation of concentrations, molecular diffusivities, mass transfer time and velocities. The predicted drop concentration, overall mass transfer coefficient and flow structure are compared with the reported experimental data of a typical extraction system, i.e., n-butanol–succinic acid–water, and good agreement is observed.

High Accuracy Iterative Solution of Convection Diffusion Equation with Boundary Layers on Nonuniform Grids

A fourth-order compact finite difference scheme and a multigrid method are employed to solve the two-dimensional convection diffusion equations with boundary layers. The computational domain is first discretized on a nonuniform (stretched) grid to resolve the boundary layers. A grid transformation technique is used to map the nonuniform grid to a uniform one. The fourth-order compact scheme is applied to the transformed uniform grid. A multigrid method is used to solve the resulting linear system. Numerical experiments are used to show that a graded mesh and a grid transformation are necessary to compute high accuracy solutions for the convection diffusion problems with boundary layers and dicretized by the fourth-order compact scheme.

Discrete Sobolev inequalities andLperror estimates for finite volume solutions of convection diffusion equations

The topic of this work is to obtain discrete Sobolev inequalities for piecewise constant functions, and to deduce Lp error estimates on the approximate solutions of convection diffusion equations by finite volume schemes.

Error Estimates on the Approximate Finite Volume Solution of Convection Diffusion Equations with General Boundary Conditions

We study here the convergence of a finite volume scheme for a diffusion convection equation on an open bounded set of $\R^\dim$ ($\dim=2$ or 3) for which we consider Dirichlet, Neumann, or Robin boundary conditions. We consider unstructured meshes which include Voronoi or triangular meshes; we use for the diffusion term an "s points" (where s is the number of sides of each cell) finite volume scheme and for the convection term an upstream finite volume scheme. Assuming the exact solution at least in H2 we prove error estimates in a discrete $H^1_0$ norm of order the size of the mesh. Discrete Poincaré inequalities then allow one to prove error estimates in the L2 norm.

A new finite volume method for the solution of convection-diffusion equations: analysis of stability and convergence

A new high order FV method is presented for the solution of convection-diffusion equations, based on a 4-point approximation of the diffusive term and on the definition of a quadratic profile for the approximation of the convective term, in which coefficients are obtained by imposing conditions on the truncation error. The method works on unequal volumes (ratios of consecutive volume sizes in the interval [1/3,3]). The properties of this method are analysed formally. Numerical examples are given

A boundary element method for the solution of convective diffusion and burgers’ equations

The paper gives the description of boundary element method (BEM) with subdomains for the solution of convection—diffusion equations with variable coefficients and Burgers’ equations. At first, the whole domain is discretized into K subdomains, in which linearization of equations by representing convective velocity by the sum of constant and variable parts is carried out. Then using fundamental solutions for convection—diffusion linear equations for each subdomain the boundary integral equation (in which the part of the convective term with the constant convective velocity is not included into the pseudo‐body force) is formulated. Only part of the convective term with the variable velocity, which is, as a rule, more than one order less than convective velocity constant part contribution, is left as the pseudo‐source. On the one hand, this does not disturb the numerical BEM—algorithm stability and, on the other hand, this leads to significant improvement in the accuracy of solution. The global matrix, similar to the case of finite element method, has block band structure whereas its width depends only on the numeration order of nodes and subdomains. It is noted, that in comparison with the direct boundary element method the number of global matrix non‐zero elements is not proportional to the square of the number of nodes, but only to the total number of nodal points. This allows us to use the BEM for the solution of problems with very fine space discretization. The proposed BEM with subdomains technique has been used for the numerical solution of one‐dimensional linear steady‐state convective—diffusion problem with variable coefficients and one‐dimensional non‐linear Burgers’ equation for which exact analytical solutions are available. It made it possible to find out the BEM correctness according to both time and space. High precision of the numerical method is noted. The good point of the BEM is the high iteration convergence, which is disturbed neither by high Reynolds numbers nor by the presence of negative velocity zones.

The stability and convergence of the finite analytic method for the numerical solution of convective diffusion equation

In this paper we make a close study of the finite analytic method by means of the maximum principles in differential equations and give the proof of the stability and convergence of the finite analytic method.

A Finite-Element Singular-Perturbation Technique for Convection-Diffusion Problems—Part 1: The One-Dimensional Case

Approximation procedures for the solution of convection-diffusion equations, occurring in various physical problems, are considered. Several finite-element algorithms based on singular-perturbation methods are proposed for the solution of these equations. A method of variational matched asymptotic expansions is employed to develop shape functions which are particularly useful when convection effects dominate diffusion effects in these problems. When these shape functions are used, in conjunction with the standard Galerkin method, to solve convection-diffusion equations, increased solution accuracy is obtained. Numerical results for various one-dimensional problems are presented to establish the workability of the developed methods.

Quantitative Evaluation of Numerical Diffusion (Truncation Error)

Abstract Numerical diffusion (truncation error) can limit the usefulness of numerical finite-difference approximations to solve partial differential equations. Many reservoir simulation users are aware of these limitations but are not as familiar with actually quantifying the magnitude of the truncation error. This paper illustrates that, over a wide range of block size and time step, the truncation error expressions for convective-diffusion partial differential equations are quantitative. Since miscible, thermal, and immiscible processes can be of the convective-diffusion equation form, the truncation error expressions presented can provide guidelines for choosing block size-time step combinations that minimize the effect of numerical diffusion. Introduction Truncation error limits the use of numerical finite-difference approximations to solve partial differential equations. In the solution of convection-diffusion equations, such as occur in miscible displacement and thermal transport, truncation error results in an artificial dispersion term often denoted as numerical diffusion. The differential equations describing two-phase fluid flow can also be rearranged into a convection-diffusion form. And, in fact, miscible and immiscible differential equations have been shown to be completely analogous. In this form, it is easy to infer that numerical diffusion will result in an additional term resembling flow due to capillarity. Many users of numerical programs, and probably all numerical analysts, recognize that the magnitude of the numerical diffusivity for convection-diffusion equations can depend on both block size and time step. Most expressions developed in the literature have been used primarily to determine the order of the error rather than to quantify it. The primary purpose of this paper is to give the user more than just a qualitative feel for the importance of truncation error. In this paper, insofar as possible, analytical expressions for truncation error are compared by experiment to computed values for the numerical diffusivity. Consequently, the reservoir simulator user can observe that these expressions are quantitative and can use them as guidelines for choosing block sizes and time steps that keep the numerical diffusivity small. DEVELOPMENT OF EXPRESSIONS FOR TRUNCATION ERROR APPLICATION TO CONVECTION-DIFFUSION EQUATION To illustrate the method of quantifying numerical diffusivity, consider a convective-diffusion equation of the form: ..............(1) Symbols are defined in the Nomenclature. The first term on the right-hand side represents the diffusion, and the second term represents convection. Such an equation describes the flow of either a two-component miscible mixture or heat in one dimension with constant diffusivity. EXPLICIT DIFFERENCE FORMS An explicit expression for the truncation error (the space derivatives are approximated at a known time level) can be developed by examining the Taylor's series expansion representing first- and second-order derivatives. For the time derivative: .....(2) SPEJ P. 315