Equations Of Convection-diffusion Type Research Articles

Fitted mesh method for singularly perturbed fourth order differential equation of convection diffusion type with integral boundary condition

<abstract><p>This article focuses on a class of fourth-order singularly perturbed convection diffusion equations (SPCDE) with integral boundary conditions (IBC). A numerical method based on a finite difference scheme using Shishkin mesh is presented. The proposed method is close to the first-order convergent. The discrete norm yields an error estimate and theoretical estimations are tested by numerical experiments.</p></abstract>

ROBUST NUMERICAL METHOD FOR SINGULARLY PERTURBED CONVECTION-DIFFUSION TYPE PROBLEMS WITH NON-LOCAL BOUNDARY CONDITION

This paper presents the study of singularly perturbed differential equations of convection diffusion type with non-local boundary condition. The proposed numerical scheme is a combination of classical finite difference method for the initial boundary condition and nonstandard finite difference method for the differential equations at the interior points. Maximum absolute errors and rates of convergence for different values of perturbation parameter and mesh sizes are tabulated for the numerical examples considered. The method is shown to be first-order convergence independent of the perturbation parameter ε.

Comparative Study on Numerical Methods for Singularly Perturbed Advanced-Delay Differential Equations

In this paper, we consider a class of singularly perturbed advanced-delay differential equations of convection-diffusion type. We use finite and hybrid difference schemes to solve the problem on piecewise Shishkin mesh. We have established almost first- and second-order convergence with respect to finite difference and hybrid difference methods. An error estimate is derived with the discrete norm. In the end, numerical examples are given to show the advantages of the proposed results (Mathematics Subject Classification: 65L11, 65L12, and 65L20).

A second order convergent initial value method for singularly perturbed system of differential-difference equations of convection diffusion type

In this article, a system of second order singularly perturbed delay differential equations of convection diffusion type problem is considered. An asymptotic expansion approximation of the solution is constructed. Further the asymptotic expansion approximation is numerically approximated using the Runge Kutta methods and hybrid finite difference methods. The error estimate is obtained and it is of almost second order. Numerical examples are given to illustrate the present method.

Exponential Fitted Operator Method for Singularly Perturbed Convection-Diffusion Type Problems with Nonlocal Boundary Condition

This paper presents the study of singularly perturbed differential equations of convection-diffusion type with nonlocal boundary condition. The proposed numerical scheme is a combination of the classical finite difference method for the boundary conditions and exponential fitted operator method for the differential equations at the interior points. Maximum absolute errors and rates of convergence for different values of perturbation parameter and mesh sizes are tabulated for the numerical examples considered. The method is shown to be first-order accuracy independent of the perturbation parameter ε .

Computational technique for two parameter singularly perturbed parabolic convection-diffusion problem

We study the singularly perturbed parabolic differential equation of convection-diffusion type with two small parameters affecting the derivatives. Using backward Euler process, time discretization is achieved. Problem is discretized in space using two fitting factors on uniform mesh where these factors take care of the two small parameters. Numerical scheme is constructed using two parameter fitting method. Tridiagonal solver is used to solve the resulting system of equations. Numerical results justify the parameter-uniform convergence of the scheme. We also mull numerical examples in comparing with remaining methods in the literature to uphold the method.

Asymptotic numerical method for third-order singularly perturbed convection diffusion delay differential equations

In this paper, an asymptotic numerical method based on a fitted finite difference scheme and the fourth-order Runge–Kutta method with piecewise cubic Hermite interpolation on Shishkin mesh is suggested to solve singularly perturbed boundary value problems for third-order ordinary differential equations of convection diffusion type with a delay. An error estimate is derived using the supremum norm and it is of almost first-order convergence. A nonlinear problem is also solved using the Newton’s quasi linearization technique and the present asymptotic numerical method. Numerical results are provided to illustrate the theoretical results.

Accelerated Exponentially Fitted Operator Method for Singularly Perturbed Problems with Integral Boundary Condition

In this paper, we consider a class of singularly perturbed differential equations of convection diffusion type with integral boundary condition. An accelerated uniformly convergent numerical method is constructed via exponentially fitted operator method using Richardson extrapolation techniques and numerical integration methods to solve the problem. The integral boundary condition is treated using numerical integration techniques. Maximum absolute errors and rates of convergence for different values of perturbation parameter and mesh sizes are tabulated for the numerical example considered. The method is shown to be ε-uniformly convergent.

Fitted Finite Difference Method for Third Order Singularly Perturbed Delay Differential Equations of Convection Diffusion Type

In this paper, a fitted finite difference method on Shishkin mesh is suggested to solve a class of third order singularly perturbed boundary value problems for ordinary delay differential equations of convection-diffusion type. Numerical solution converges uniformly to the exact solution. The order of convergence of the numerical method is almost first order. Numerical results are provided to illustrate the theoretical results.

Finite difference scheme for third order singularly perturbed delay differential equation of convection diffusion type with integral boundary condition

A class of third order singularly perturbed delay differential equations of convection diffusion type with integral boundary condition is considered. A numerical method based on a finite difference scheme on a Shishkin mesh is presented. The method suggested is almost first order convergent. An error estimate is derived in the discrete norm. Numerical examples are presented, which validate the theoretical estimates.

Uniformly Convergent Finite Difference Schemes for Singularly Perturbed Convection Diffusion Type Delay Differential Equations

In this paper, uniformly convergent finite difference schemes with piecewise linear interpolation on Shishkin meshes are suggested to solve singularly perturbed boundary value problems for second order ordinary delay differential equations of convection-diffusion type. Error estimates are derived and are found to be of almost first order. Numerical results are provided to illustrate the theoretical results.

A NUMERICAL SCHEME FOR SINGULARLY PERTURBED DELAY DIFFERENTIAL EQUATIONS OF CONVECTION-DIFFUSION TYPE ON AN ADAPTIVE GRID

In this paper, an adaptive mesh strategy is presented for solving singularly perturbed delay differential equation of convection-diffusion type using second order central finite difference scheme. Layer adaptive meshes are generated via an entropy production operator. The details of the location and width of the layer is not required in the proposed method unlike the popular layer adaptive meshes mainly by Bakhvalov and Shishkin. An extensive amount of computational work has been carried out to demonstrate the applicability of the proposed method.

Stabilized FEM Solutions of MHD Equations Around a Solid and Inside a Conducting Medium

In this study, the numerical solution of the magnetohydrodynamic (MHD) flow is considered in a circular pipe around a conducting solid and in an insulating or conducting medium. An external magnetic field is applied through axis of the pipe with an angle α with through the x-axis. The mathematical model of the considered physical problem can be defined in terms of coupled MHD equations in the pipe domain and the Laplace equations on the solid and external mediums. The coupled equations are transformed into decoupled inhomogeneous convection-diffusion type equations in order to apply stabilization in the finite element method solution procedure. Obtained stabled solutions for the high values of the problem parameters display the well-known characteristics of the MHD pipe flow.

Exponential difference schemes for solving boundary-value problems for convection-diffusion type equations

The paper considers the numerical solution of boundary-value problems for multidimensional convection-diffusion type equations (CDEs). Such equations are useful for various physical processes in solids, liquids and gases. A new approach to the spatial approximation for such equations is proposed. This approach is based on an integral transformation of second-order one-dimensional differential operators. A linear version of CDE was chosen for simplicity of the analysis. In this setting, exponential difference schemes were constructed, algorithms for their implementation were developed, a brief analysis of the stability and convergence was made. This approach was numerically tested for a two-dimensional problem of motion of metallic particles in water flow subject to a constant magnetic field.

Computational fluid dynamics for nematic liquid crystals

Due to recent advances in fast iterative solvers in the field of computational fluid dynamics, more complex problems which were previously beyond the scope of standard techniques can be tackled. In this paper, we describe one such situation, namely, modelling the interaction of flow and molecular orientation in a complex fluid such as a liquid crystal. Specifically, we consider a nematic liquid crystal in a spatially inhomogeneous flow situation where the orientational order is described by a second rank alignment tensor. The evolution is determined by two coupled equations: a generalised Navier–Stokes equation for flow in which the divergence of the stress tensor also depends on the alignment tensor and its time derivative, and a convection-diffusion type equation with non-linear terms that stem from a Landau-Ginzburg-DeGennes potential for the alignment. In this paper, we use a specific model with three viscosity coefficients that allows the contribution of the orientation to the viscous stress to be cast in the form of an orientation-dependent force. This effectively decouples the flow and orientation, with each appearing only on the right-hand side of the other equation. In this way, difficulties associated with solving the fully coupled problem are circumvented and a stand-alone fast solver, such as the state-of-the-art preconditioned iterative solver implemented here, can be used for the flow equation. A time-discretised strategy for solving the flow-orientation problem is illustrated using the example of Stokes flow in a lid-driven cavity.

An exponentially fitted tridiagonal finite difference method for singularly perturbed differential-difference equations with small shift

This paper deals with the singularly perturbed boundary value problem for a linear second order differential-difference equation of convection-diffusion type. In the numerical treatment of such type of problems, first we use Taylor’s approximation to tackle the term containing the small shift. A fitting parameter has been introduced in a tridiagonal finite difference method and is obtained from the theory of singular perturbations. Thomas algorithm is used to solve the tridiagonal system. The method is analysed for convergence. Several numerical examples are solved to demonstrate the applicability of the method. Graphs are plotted for the solutions of these problems to illustrate the effect of small shift on the boundary layer solution.

Analytic approximation to delayed convection dominated systems through transforms

In this paper we consider a family of singularly perturbed delay differential equation of convection diffusion type. When the perturbation parameter is very small, the solution of the problem exhibits layer behavior. In the layer region the solution changes rapidly, while away from this region the change in the solution is moderate. This simultaneous presence of two different scales phenomena makes the problem stiff. In this work, the problem is solved by applying a new Liouville–Green transform and the asymptotic solutions are obtained. Application to multi-point boundary value problem is also illustrated. Several test examples are taken into account so as to test the efficiency of the proposed method. The method presented is compared with other existing numerical or asymptotic methods. It is observed that the method presented is very easy to implement and is capable of reducing the size of calculations significantly while still maintaining high accuracy of the solution.

Numerical Method for Singularly Perturbed Third Order Ordinary Differential Equations of Convection-Diffusion Type

Numerical Method for Singularly Perturbed Third Order Ordinary Differential Equations of Convection-Diffusion Type

A finite volume formulation for transient convection and diffusion equations with unstructured distorted grids and its applications in fluid flow simulations with a collocated variable arrangement

This paper describes a finite volume method for simulating transport processes governed by convection–diffusion type equations. The formulation is based on a cell-centred, unstructured grid. With an edge-based data structure, discretisation is independent of control volume (or cell) shape. By using a surface vector decomposition at the midpoint of the interface between cells, along with a deferred-correction approach, any cross-diffusion due to grid skewness can be readily accounted for when discretising the diffusive flux. For modelling fluid flow processes, a collocated arrangement of variables is employed so that a single coefficient matrix applies for the momentum equations of each velocity component. To avoid ‘checkerboard oscillation’ (arising from pressure–velocity decoupling) occurring under the collocated variable arrangement when a pressure-based solution algorithm is employed, a novel pseudo-flux interpolation method is proposed for unstructured grids, ensuring that the solution is both under-relaxation factor and time-step (for transient calculation) independent. The methodology can be formulated within a framework whereby either a coupled or a decoupled solution algorithm can be employed. The features and advantages of the method are demonstrated by solution of the Navier–Stokes equations for two benchmark flow problems.

A numerical solution of the steady MHD flow through infinite strips with BEM

The magnetohydrodynamic (MHD) flow of an incompressible, viscous, electrically conducting fluid in infinite channels in the presence of a magnetic field is investigated. The fluid is driven either by a pressure gradient or by the currents produced by electrodes placed parallel in the middle of the walls. The applied magnetic field is perpendicular to the infinite walls which are combined from conducting and insulated parts. A boundary element method (BEM) solution has been obtained by using a fundamental solution which enables to threat the convection-diffusion type equations in coupled form with general wall conductivities. Constant elements are used for the discretization of the walls by keeping them as finite since the boundary integrals are restricted to these boundaries due to the regularity conditions as x , y → ± ∞ . The solutions are presented in terms of equivelocity and induced magnetic field contours for several values of Hartmann number and conducting lengths. The effect of the parameters on the solution is visualized.