Two-dimensional Convection-diffusion Problems Research Articles

A modified cell-centered nodal integral scheme for the convection-diffusion equation

The nodal integral methods (NIMs) are very efficient and accurate coarse-mesh methods for solving partial differential equations. The cell-centered NIM (CCNIM) is a simplified variant of the NIMs that has recently shown its efficiency in solving fluid flow problems but has been hampered by issues such as inapplicability to one-dimensional problems, complexities in handling Neumann boundary conditions and the formulation of a system of differential-algebraic equations (DAEs) for discrete unknowns. Here, we present a modified version of the CCNIM designed to overcome the challenges associated with its previous version. Our novel development retains the essence of CCNIM while resolving these issues. The proposed scheme, grounded in the nodal framework, achieves second-order accuracy in both spatial and temporal dimensions. Unlike its precursor, the proposed method formulates algebraic equations for discrete variables per node, eliminating the cumbersome DAE system. Neumann boundary conditions are seamlessly incorporated through a straightforward flux definition, and applicability to one-dimensional problems is now feasible. We successfully apply our approach to one and two-dimensional convection-diffusion problems with known analytical solutions to validate our approach. The simplicity and robustness of the approach lay the foundation for its seamless extension to more complex fluid flow problems.

A higher order hybrid-numerical approximation for a class of singularly perturbed two-dimensional convection-diffusion elliptic problem with non-smooth convection and source terms

A higher order hybrid-numerical approximation for a class of singularly perturbed two-dimensional convection-diffusion elliptic problem with non-smooth convection and source terms

The Bakhvalov mesh: a complete finite-difference analysis of two-dimensional singularly perturbed convection-diffusion problems

A linear two-dimensional singularly perturbed convection-diffusion boundary-value problem is considered. The problem is discretized by the upwind finite-difference method. The analysis of this method on Shishkin-type meshes has been well-established, but the discretization mesh in this paper is the original Bakhvalov mesh, introduced in 1969 as the first layer-adapted mesh. We analyze the error of the numerical solution in the maximum norm and prove first-order pointwise accuracy, uniform in the perturbation parameter. This is the first complete analysis of this kind for two-dimensional convection-diffusion problems discretized on the Bakhvalov mesh. Our numerical experiments validate the theoretical analysis.

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 Hybrid Variational Multiscale Element-Free Galerkin Method for Convection-Diffusion Problems

By coupling the dimension splitting method (DSM) and the variational multiscale element-free Galerkin (VMEFG) method, a hybrid variational multiscale element-free Galerkin (HVMEFG) method is developed for the two-dimensional convection-diffusion problems. In the HVMEFG method, the two-dimensional problem is converted into a battery of one-dimensional problems by the DSM. Combining the non-singular improved interpolating moving least-squares (IIMLS) method, the VMEFG method is used to obtain the discrete equations of the one-dimensional problems on the splitting plane. Then, final discretized equations of the entire convection-diffusion problems are assembled by the IIMLS method. The HVMEFG method has high accuracy and efficiency. Numerical examples show that the HVMEFG method can obtain non-oscillating solutions and has higher efficiency and accuracy than the EFG and VMEFG methods for convection-diffusion problems.

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.

The difference scheme for the two-dimensional convection-diffusion problem for large peclet numbers

The purpose of this work is the development of a difference scheme for the solution of convection-diffusion problem at high Peclet numbers (Pe>2). In accordance with this purpose the following problems were solved: difference scheme for convection is built, comparison with the existing schemes is carried out; conditions for stability of the proposed difference scheme are obtained. Solutions of the convection-diffusion equation on the basis of the proposed difference scheme at various Peclet numbers are obtained.

Two-dimensional convection—diffusion problem solved using method of localized particular solutions

A meshless local method of approximated particular solutions (LMAPS) is used to analyze problem described by the convection-diffusion equation. The method solves the steady convection-diffusion equation with reaction term. The discretized system of equations is derived via interpolation procedure using integrated multiquadrics (MQ) radial basis functions (RBF). The solution of the equation is performed over simple geometry with non-uniform velocity field and results are presented in the article. The LMAPS method is capable to produce stable solutions with results comparable to the analytical solutions.

A fast and stabilized meshless method for the convection-dominated convection-diffusion problems

ABSTRACTThis paper presents a fast and stabilized meshless method that combines variational multi-scale element free Galerkin (VMEFG) method and proper orthogonal decomposition (POD) method, namely VMEFG&POD, to solve convection-diffusion problems. Variational multi-scale method is applied to overcome the numerical oscillation for the convection-dominated problems and the POD method is used to improve the computational efficiency of the VMEFG method. This method is validated by considering the simulation of two-dimensional convection-diffusion problems with a small diffusion coefficient. It is demonstrated that the proposed method can largely improve the computational efficiency without a significant loss in accuracy.

Точность численного решение уравнения диффузии-конвекции на основе разностных схем второго и четвертого порядков погрешности аппроксимации

The paper deals with the scheme of the second and fourth order approximation error for solving convection-diffusion problems. To model initial boundary value problem in the case when the functions of the right and the initial condition can be represented by finite sums of Fourier series in the trigonometric basis, we investigated the accuracy of difference schemes. It was found that the accuracy of the numerical solution depends on the number of units attributable to half the wavelength corresponding to the most high frequency harmonics in the final sum of the Fourier series, necessary to describe the behavior of calculated objects. The dependence of the diffusion approximation error terms difference schemes of second and fourth order of accuracy of the number of nodes. The comparison of the calculation results of two-dimensional convectiondiffusion problems and tasks of the Poisson-based schemes of the second and fourth order accuracy. In the expediency of transition to a scheme of high accuracy for solving applied problems of the estimates and is easy to obtain the numerical values of the gain in computation time by using schemes of higher order accuracy.

A Posteriori Local Discontinuous Galerkin Error Estimation for Two-Dimensional Convection–Diffusion Problems

We present a simple, efficient, and asymptotically correct a posteriori error estimates for a minimal dissipation local discontinuous Galerkin method applied to two-dimensional diffusion and convection---diffusion problems on rectangular meshes. The finite element spaces are obtained by performing a local error analysis and a posteriori error estimates are computed by solving local problems on each element. We present computational results for several problems to show the efficiency and accuracy of our error estimates. It is shown that even in the presence of boundary layers our error estimates converge to the true error under mesh refinement when Shishkin meshes are used.

Skew-Hermitian based iterations for nine-point approximations of convection–diffusion problems

We study skew-Hermitian based splitting methods for the iterative solution of nonsymmetric linear systems arising from high-order compact (HOC) approximations of two-dimensional convection–diffusion problems. Such discretisations lead to nine-point system matrices with block-tridiagonal structures. The nonsymmetric discretisation matrix for a constant-coefficient grid-aligned flow problem is shown to be positive definite and thus Hermitian and skew-Hermitian splitting (HSS) methods can be considered for the solution of the corresponding linear system. For the solution of this linear system using the HSS iteration, we derive an analytical expression for the optimal value for the upper bound of the contraction factor. Comparison of the HSS iteration with triangular and skew-Hermitian splitting (TSS) methods and their corresponding block variants (BTSS) are carried out for various test problems. Cpu timings for the different splitting methods using experimentally determined optimal acceleration parameters are also given. Our results show that the combination of high-order compact discretisations and skew-Hermitian based iterative methods for solving the corresponding linear systems provide efficient procedures for approximating the solutions of two-dimensional convection–diffusion equations.

An unconditionally stable second-order accurate ALE-FEM scheme for two-dimensional convection-diffusion problems

The aim of this paper is to investigate the stability of time integration schemes for the solution of a finite element semi-discretization of a scalar convection–diffusion equation defined on a moving domain. An arbitrary Lagrangian–Eulerian formulation is used to reformulate the governing equation with respect to a moving reference frame. We devise an adaptive θ-method time integrator that is shown to be unconditionally stable and asymptotically second-order accurate for smoothly evolving meshes. An essential feature of the method is that it satisfies a discrete equivalent of the well-known geometric conservation law. Numerical experiments are presented to confirm the findings of the analysis.

A meshless solution to two-dimensional convection–diffusion problems

An integral equation domain decomposition method has been implemented in a meshless fashion. The method exploits the advantage of placing the source point always in the centre of circular sub-domains in order to avoid singular or near-singular integrals. Three equations for two-dimensional (2D) or four for three-dimensional (3D) potential problems are required at each node. The first equation is the integral equation arising from the application of the Green's identities and the remaining equations are the derivatives of the first equation in respect to space coordinates. Radial basis function interpolation is applied in order to obtain the values of the field variable and partial derivatives at the boundary of the circular sub-domains, providing this way the boundary conditions for solution of the integral equations at the nodes (centres of circles). Dual reciprocity method (DRM) has been applied to convert the domain integrals into boundary integrals, though the approach is general and can be applied without the DRM. The accuracy and robustness of the method has been tested on a convection–diffusion problem. The results obtained using the current approach have been compared with previously reported results obtained using the finite element method (FEM), and the DRM multi-domain approach (DRM-MD) showing similar level of accuracy.

On block-circulant preconditioners for high-order compact approximations of convection–diffusion problems

We study some properties of block-circulant preconditioners for high-order compact approximations of convection–diffusion problems. For two-dimensional problems, the approximation gives rise to a nine-point discretisation matrix and in three dimensions, we obtain a nineteen-point matrix. We derive analytical expressions for the eigenvalues of the block-circulant preconditioner and this allows us to establish the invertibility of the preconditioner in both two and three dimensions. The eigenspectra of the preconditioned matrix in the two-dimensional case is described for different test cases. Our numerical results indicate that the block-circulant preconditioning leads to significant reduction in iteration counts and comparisons between the high-order compact and upwind discretisations are carried out. For the unpreconditioned systems, we observe fewer iteration counts for the HOC discretisation but for the preconditioned systems, we find similar iteration counts for both finite difference approximations of constant-coefficient two-dimensional convection–diffusion problems.

The combination technique for a two-dimensional convection-diffusion problem with exponential layers

Convection-diffusion problems posed on the unit square and with solutions displaying exponential layers are solved using a sparse grid Galerkin finite element method with Shishkin meshes. Writing N for the maximum number of mesh intervals in each coordinate direction, our “combination” method simply adds or subtracts solutions that have been computed by the Galerkin FEM on \( N \times \sqrt N , \sqrt N \times N \) and \( \sqrt N \times \sqrt N \) meshes. It is shown that the combination FEM yields (up to a factor ln N) the same order of accuracy in the associated energy norm as the Galerkin FEM on an N × N mesh, but it requires only \( \mathcal{O} \)(N 3/2) degrees of freedom compared with the \( \mathcal{O} \)(N 2) used by the Galerkin FEM. An analogous result is also proved for the streamline diffusion finite element method.

Domain decomposition boundary element method with overlapping sub-domains

A numerical approach based on the domain decomposition boundary element method (BEM) with overlapping sub-domains has been developed. The approach simplifies the assembly of the equations arising from the BEM sub-domain methods, reduces the size of the system matrix, produces a closed system of equations when continuous elements are used, and reduces any problems arising from near-singular or singular integrals which otherwise may appear in the integral equations. The overlapping numerical approach is tested on three different problems, i.e., the Poisson equation, and a one-dimensional and two-dimensional convection–diffusion problems. The approach is implemented in combination with the dual reciprocity method (DRM) with two different radial basis functions (RBFs), though the approach is general and can be applied with other BEM formulations. The results are compared with the previous results obtained using the dual reciprocity method–multi domain (DRM–MD) approach, showing comparable accuracy and convergence.

Analysis of incomplete factorizations for a nine-point approximation to a convection–diffusion model problem

We study the stability of zero-fill incomplete LU factorizations of a nine-point coefficient matrix arising from a high-order compact discretisation of a two-dimensional constant-coefficient convection–diffusion problem. Nonlinear recurrences for computing entries of the lower and upper triangular matrices are derived and we show that the sequence of diagonal entries of the lower triangular factor is unconditionally convergent. A theoretical estimate of the limiting value is derived and we show that this estimate is a good predictor of the computed value. The unconditional convergence of the diagonal sequence of the lower triangular factor to a positive limit implies that the incomplete factorization process never encounters a zero pivot and that the other diagonal sequences are also convergent. The characteristic polynomials associated with the lower and upper triangular solves that occur during the preconditioning step are studied and conditions for the stability of the triangular solves are derived in terms of the entries of the tridiagonal matrices appearing in the lower and upper subdiagonals of the block triangular system matrix and a triplet of parameters which completely determines the solution of the nonlinear recursions. Results of ILU-preconditioned GMRES iterations and the effects of orderings on their convergence are also described.

Probing non-integer dimensions

We show that a two-dimensional convection–diffusion problem with a radial sink or sourceat the origin may be recast as a pure diffusion problem in a fictitious space in which thespatial dimension is continuously tunable with the Péclet number. This formulationallows us to probe various diffusion-controlled processes in non-integer dimensions.

Domain decomposition for a parabolic convection-diffusion problem

A two-dimensional convection-diffusion problem of parabolic type is considered. A multidomain decomposition algorithm with nonoverlapping subdomains based on a upwind scheme and on a piecewise equidistant mesh is investigated. Uniform in a perturbation parameter convergence properties of the algorithm are established. Numerical experiments complement the theoretical results. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006