Convection-diffusion Boundary Research Articles

Lattice Boltzmann simulations for the drying of porous media with gas–side convection–diffusion boundary

One of the most significant challenges in describing the drying of porous materials is the realistic integration of internal transport phenomena into mathematical models, coupled with the external free flow. However, the intricacies of pore-scale geometry make experimentation and observation very difficult in practice, thereby making modeling techniques a useful tool for the analysis of the drying process. Among the many modeling techniques developed for drying, the lattice Boltzmann method (LBM)-based modeling approach has gained favor in recent years due to its ability to incorporate realistic geometry and transport at the pore scale. Our previous works on Shan–Chen LBM for drying of capillary porous media were based on the Bhatnagar–Gross–Krook collision operator and diffusion interface boundary conditions. This study elucidates the drying of a capillary porous medium under the influence of convection–diffusion boundary conditions at the gas side, using Shan–Chen LBM. The pore-scale effects of convection–diffusion conditions during the drying process are presented in relation to the macroscale drying kinetics. Moreover, the differences between the convection–diffusion kinetics and purely diffusion–dominated kinetics of the drying process are also presented here. This work also aims to incorporate the convection–diffusion transport phenomena into the drying process of a porous medium under the influence of an imposed thermal gradient, establishing and studying the phenomena of stabilization and destabilization of the drying front under the influence of a temperature gradient, thereby extending the lattice Boltzmann method of modeling for the simulation of convection–diffusion drying, both for the isothermal case and the imposition of a thermal gradient.

Numerical Solution of Two-Parameter Singularly Perturbed Convection-Diffusion Boundary Value Problems via Fourth Order Compact Finite Difference Method

In this study, we have developed a fourth order compact finite difference method for finding the numerical solution of two-parameter singularly perturbed convection-diffusion boundary value problems. We have used fourth order compact finite difference method on uniform mesh which provides a tridiagonal linear system of equations. The convergence analysis of the proposed method is established through a matrix analysis approach and it is proved that present method gives fourth order convergence results. Present method is implemented on two numerical examples for checking the efficiency and precision of the method. Numerical outcomes are exhibited which supports the theoretical outcomes. Numerical outcomes are compared with other existing methods and found that present method gives more accurate approximate solution as compare to the other existing methods.

Superconvergence analysis of FEM and SDFEM on graded meshes for a problem with characteristic layers

We consider a singularly perturbed convection-diffusion boundary value problem whose solution contains exponential and characteristic boundary layers. The problem is numerically solved by the FEM and SDFEM method with bilinear elements on a graded mesh. For the FEM we prove almost uniform convergence and superconvergence. The use of a graded mesh allows for the SDFEM to yield almost uniform estimates in the SD norm, which is not possible for Shishkin type meshes. Numerical results are presented to support theoretical bounds.

Application of Coupled Map Lattice as an Alternative to Classical Finite Difference Method for Solving the Convection-Diffusion Boundary Value Problem

This paper presents a mathematical model for a piston flow reactor based on the material balance law using partial differential equations. A more general, nondimensional variant of the model is also derived. The finite difference method and coupled map lattice are used to create numerical algorithms to simulate spatio-temporal behavior in the studied system. The paper also includes a stability analysis of the proposed algorithms and results of numerous numerical simulations, done in order to compare both methods and to visualize the spatio-temporal behavior of the reactor and the effects of different model parameters. Discussion of the obtained results and comparison of both algorithms is also provided.

A parameter robust computational method for a weakly coupled system of singularly perturbed convection-diffusion boundary value problem with discontinuous source terms

A parameter robust computational method for a weakly coupled system of singularly perturbed convection-diffusion boundary value problem with discontinuous source terms

A parameter robust computational method for a weakly coupled system of singularly perturbed convection-diffusion boundary value problem with discontinuous source terms

In this paper, a parameter robust computational method using an iterative procedure for a weakly coupled system of singularly perturbed convection-diffusion boundary value problem with discontinuou...

The difference schemes for solving singularly perturbed three-point boundary value problem

In this paper, we propose and analyze numerical treatment for a singularly perturbed convection–diffusion boundary value problem with nonlocal condition. First, the boundary layer behavior of the exact solution and its first derivative have been estimated. Then we construct a finite difference scheme on a uniform mesh. We prove the uniform convergence of the proposed difference scheme and give an error estimate. We also present numerical examples, which demonstrate computational efficiency of the proposed method.

Comparative study on difference schemes for singularly perturbed boundary turning point problems with Robin boundary conditions

In this paper, singularly perturbed convection diffusion boundary turning point problem with Robin boundary conditions is considered. Upwind difference and hybrid difference schemes on Shishkin mesh are proposed to solve this problem. The error bounds for the numerical approximation are established for both the schemes. Numerical results are compared for both the schemes to illustrate the corresponding error bounds established.

Uniformly convergent hybrid schemes for solutions and derivatives in quasilinear singularly perturbed BVPs

In this paper, a class of hybrid difference schemes with variable weights on Bakhvalov–Shishkin mesh is proposed to compute both the solution and the derivative in quasilinear singularly perturbed convection–diffusion boundary value problems. The parameter-uniform second-order convergence of approximating to the solution and the derivative on Bakhvalov–Shishkin mesh and that of nearly second-order on Shishkin mesh are proved clearly by use of an (l∞,l1)-stability property, where the former sufficient conditions for uniform convergence are modestly relaxed on Bakhvalov–Shishkin mesh and are clarified on Shishkin mesh. The numerical examples support the proposed schemes with new sufficient conditions and their error estimates.

Uniform Second-Order Hybrid Schemes on Bakhvalov-Shishkin Mesh for Quasi-Linear Convection-Diffusion Problems

In this paper, we propose a class of hybrid difference schemes combining the central difference scheme and the midpoint upwind scheme on the Bakhvalov-Shishkin mesh for solving quasi-linear singularly perturbed convection-diffusion boundary value problems. Point-wise second-order convergence uniform in the perturbation is proved clearly by using the-stability. The numerical experiments support the schemes and the uniform second-order estimate.

A ROBUST COMPUTATIONAL TECHNIQUE FOR A SYSTEM OF SINGULARLY PERTURBED CONVECTION-DIFFUSION EQUATIONS

In this paper, a singularly perturbed system of convection-diffusion boundary value problem (BVP) is examined. To solve such type of problem, a modified initial value technique (MIVT) is proposed on an appropriate piecewise uniform Shishkin mesh. The MIVT is shown to be uniformly convergent with respect to the perturbation parameter. Numerical results are presented which are in agreement with the theoretical results.

Parameter-uniform numerical methods for some linear and nonlinear singularly perturbed convection diffusion boundary turning point problems

Both linear and nonlinear singularly perturbed two point boundary value problems are examined in this paper. In both cases, the problems have a boundary turning point and are of convection-diffusion type. Parameter-uniform numerical methods composed of monotone finite difference operators and piecewise-uniform Shishkin meshes, are constructed and analyzed for both the linear and the nonlinear class of problems. Numerical results are presented to illustrate the theoretical parameter-uniform error bounds established.

A parameter robust Petrov–Galerkin scheme for advection–diffusion–reaction equations

In this paper, a singularly perturbed convection diffusion boundary value problem, with discontinuous diffusion coefficient is examined. In addition to the presence of boundary layers, strong and weak interior layers can also be present due to the discontinuities in the diffusion coefficient. A priori layer adapted piecewise uniform meshes are used to resolve any layers present in the solution. Using a Petrov---Galerkin finite element formulation, a fitted finite difference operator is shown to produce numerical approximations on this fitted mesh, which are uniformly second order (up to logarithmic terms) globally convergent in the pointwise maximum norm.

A higher order richardson scheme for a singularly perturbed semilinear elliptic convection-diffusion equation

The Dirichlet problem on a vertical strip is examined for a singularly perturbed semilinear elliptic convection-diffusion equation. For this problem, the basic nonlinear difference scheme based on the classical approximations on piecewise uniform grids condensing in the vicinity of boundary layers converges ɛ-uniformly with an order at most almost one. The Richardson technique is used to construct a nonlinear scheme that converges ɛ-uniformly with an improved order, namely, at the rate O(N 1 −2 ln2 N 1 + N 2 −2 ), where N 1 + 1 and N 2 + 1 are the number of grid nodes along the x 1-axis and per unit interval of the x 2-axis, respectively. This nonlinear basic scheme underlies the linearized iterative scheme, in which the nonlinear term is calculated using the values of the sought function found at the preceding iteration step. The latter scheme is used to construct a linearized iterative Richardson scheme converging ɛ-uniformly with an improved order. Both the basic and improved iterative schemes converge ɛ-uniformly at the rate of a geometric progression as the number of iteration steps grows. The upper and lower solutions to the iterative Richardson schemes are used as indicators, which makes it possible to determine the iteration step at which the same ɛ-uniform accuracy is attained as that of the non-iterative nonlinear Richardson scheme. It is shown that no Richardson schemes exist for the convection-diffusion boundary value problem converging ɛ-uniformly with an order greater than two. Principles are discussed on which the construction of schemes of order greater than two can be based.

B-Spline collocation method for a two-parameter singularly perturbed convection–diffusion boundary value problems

In this paper, we have presented a B-spline collocation method for solving a class of two-parameter singularly perturbed boundary value problems. We use B-spline collocation method on piecewise-uniform Shishkin mesh, which leads to a tridiagonal linear system. The convergence analysis is given and the method is shown to have uniform convergence of second order. Numerical results are presented in the end which support the theoretical results.

Electrohydrodynamics in hierarchically structured monolithic and particulate fixed beds

We have investigated the basic dependence of electroosmotic flow (EOF) velocity and hydrodynamic dispersion in capillary electrochromatography (CEC) on the variation of applied field and mobile phase ionic strengths employing silica-based particulate and monolithic fixed beds. These porous media have a hierarchical structure characterized by discrete intraparticle (intraskeleton) mesoporous and interparticle (interskeleton) macroporous spatial domains. While the macroporous domains contain quasi-electroneutral electrolyte solution, the ion-permselectivity (charge-selectivity) of the mesoporous domains determines the co-ion exclusion and counter-ion enrichment at electrochemical equilibrium (without superimposed electrical field) which depends on mesopore-scale electrical double layer (EDL) overlap and surface charge density. This adjustable, locally charge-selective transport realized under most general conditions forms the basis for concentration polarization (CP) induced by electrical fields superimposed in CEC. CP characterizes the formation of convective diffusion boundary layers with reduced (depleted CP zone) and increased (enriched CP zone) electrolyte concentration, respectively, at the anodic and cathodic interfaces in fixed beds containing the cation-selective, silica-based particles (or monolith skeleton). CP originates in the electrical field-induced coupled mass and charge transport normal to the charge-selective interfaces and has consequences for the EOF dynamics, hydrodynamic dispersion, and analyte retention in CEC. A secondary EDL with mobile counter-ionic space charge can be induced in the depleted CP zone leading to induced-charge EOF in the macroporous domains. It is characterized by a nonlinear dependence of the average EOF velocities on applied field strength and strong local velocity components tangential to the surface which enhance lateral pore-scale dispersion, thereby decreasing (axial) zone spreading. Differences in the pore space morphology of random-close sphere packings and monoliths criticially affect the intensity of CP and induced-charge EOF in these materials. CP is identified as a key phenomenon in CEC which also influences effective migration and the retention of charged analytes because the local intensity of CP inherently depends on applied field and mobile phase ionic strengths.

Uniform Pointwise Convergence of Difference Schemes for Convection-Diffusion Problems on Layer-Adapted Meshes

We consider two convection-diffusion boundary value problems in conservative form: for an ordinary differential equation and for a parabolic equation. Both the problems are discretized using a four-point second-order upwind space difference operator on arbitrary and layer-adapted space meshes. We give ɛ-uniform maximum norm error estimates O(N−2ln2N(+τ)) and O(N−2(+τ)), respectively, for the Shishkin and Bakhvalov space meshes, where N is the space meshnodes number, τ is the time meshinterval. The smoothness condition for the Bakhvalov mesh is replaced by a weaker condition.

Analysis of the streamline-diffusion finite element method on a piecewise uniform mesh for a convection-diffusion problem with exponential layers

On the unit square, we consider a singularly perturbed convection-diffusion boundary value problem whose solution has two exponential boundary layers. We apply the streamline-diffusion finite element method with piecewise bilinear trial functions on a Shishkin mesh of O ( N 2 ) points and show that the error in the discrete space between the computed solution and the interpolant of the true solution is, uniformly in the diffusion parameter ɛ , of order ɛ 1/2 N –1 ln N + N – 3/2 in the usual streamline-diffusion norm. This includes an L 2 -norm error estimate of order O ( N – 3/2 ) in the convection–dominated case ɛ ⩽ N – 1 ln –2 N . As a corollary we prove that the method is convergent of order N –1/2 ln 3/2 N (again uniformly in ɛ ) in the local L ∞ norm on the fine part of the mesh (i.e., inside the boundary layers). This local L ∞ estimate within the layers can be improved to order ɛ 1/2 N –1/2 ln 3/2 N + N –1 ln 1/2 N , uniformly in ɛ , away from the corner layer.

Uniform convergence analysis of an upwind finite-difference approximation of a convection-diffusion boundary value problem on an adaptive grid

We derive an ∈-uniform error estimate for the first-order upwind discretization of a model convection-diffusion boundary value problem on a non-uniform grid. Here, ∈ is the small parameter multiplying the highest derivative term. The grid is constructed by the equidistribution of the power of the gradient of the solution of a constant coefficient problem. The use of equidistribution principles appears in many grid adaption schemes and our analysis indicates the convergence behaviour on such grids. We also show that the condition number of the resulting linear algebraic systems increases at the same rate as the singular perturbation parameter is decreased. A simple preconditioning of the original system is shown to lead to systems with condition numbers that are independent of the singular perturbation parameter. Numerical results are given that confirm the ∈-uniform convergence rate of the nodal errors and the effect of preconditioning on the linear algebraic systems.

On the magnetohydrodynamics of natural convective diffusion boundary layers in coupled horizontal electric and magnetic fields

An approximate method for the estimation of the mean velocity and the root mean square velocity in the natural-convective diffusion boundary layer under the influence of weak magnetic fields is presented. The method applies to vertical solid surfaces when the electric and magnetic field are horizontal and mutually transverse; the magnetic field may support or oppose the gravity field. Extension to axisymmetric surfaces is feasible by appropriate modifications of the constitutive equations.