Unsteady Convection Diffusion Problems Research Articles

A Rotated Similarity Reduction Approach with Half-Sweep Successive Over-Relaxation Iteration for Solving Two-Dimensional Unsteady Convection-Diffusion Problems

A Rotated Similarity Reduction Approach with Half-Sweep Successive Over-Relaxation Iteration for Solving Two-Dimensional Unsteady Convection-Diffusion Problems

Numerical Methods for Unsteady Convection-diffusion Problems Based on Combining Compact Difference Schemes with Runge-Kutta Methods

The convection-diffusion equation is of primary importance in understanding transport phenomena within a physical system. However, the currently available methods for solving unsteady convection-diffusion problems are generally not able to offer excellent accuracy in space and time. The one-dimensional unsteady convection-diffusion equation was solved by combing a compact difference scheme with the Runge-Kutta method. The combined method has fourth-order accuracy in space and time. To check the accuracy of the combined method, numerical experiments were carried out and comparisons were performed with the Crank-Nicolson method. The analysis results indicated that the combined method is numerically stable at low wave numbers and small Courant-Friedrichs-Lewy numbers. The combined method has higher accuracy than the Crank-Nicolson method.

A half boundary method for two dimensional unsteady convection–diffusion equations

Convection diffusion equation is widely applied in many fields of science and technology. Many practical engineering problems can be expressed by this equation in unsteady state. However, it is usually difficult and time-consuming to find its solution. In this paper, a boundary type method named half boundary method (HBM) is proposed for two dimensional unsteady convection–diffusion equations. The main idea of HBM is introducing new variables to reduce the order of the equation and build relations of variables between nodes inside the area and nodes at half of the boundaries, namely the unknown variables. Since only variables at half of the boundaries are chosen as unknown variables, using HBM can realize dimensionality reduction and the maximum matrix order is less than that in finite volume method when considering large division number, which makes HBM fast and efficient. After the unknown variables are obtained by the boundary conditions, all variables at nodes can be obtained simultaneously according to the relations mentioned above. Numerical studies are carried out for convection dominated problems, problems with variable or discontinuous coefficient and problems with mixed boundary conditions, which show the validity of HBM for two-dimensional unsteady convection-diffusion problems and high accuracy for convection-domination problems.

Numerical Solution to Unsteady One-Dimensional Convection-Diffusion Problems Using Compact Difference Schemes Combined with Runge-Kutta Methods

The convection-diffusion equation is of primary importance in understanding transport phenomena within a physical system. However, the currently available methods for solving unsteady convection-diffusion problems are generally not able to offer excellent accuracy in both time and space variables. A procedure was given in detail to solve the unsteady one-dimensional convection-diffusion equation through a combination of Runge-Kutta methods and compact difference schemes. The combination method has fourth-order accuracy in both time and space variables. Numerical experiments were conducted and the results were compared with those obtained by the Crank-Nicolson method in order to check the accuracy of the combination method. The analysis results indicated that the combination method is numerically stable at low wave numbers and small CFL numbers. The combination method has higher accuracy than the Crank-Nicolson method.

Compact Difference Schemes Combined with Runge-Kutta Methods for Solving Unsteady Convection-Diffusion Problems

The convection-diffusion equation is of primary importance in understanding transport phenomena within a physical system. However, the currently available methods for solving unsteady convection-diffusion problems are generally not able to offer excellent accuracy in both time and space variables. A procedure was given in detail to solve the one-dimensional unsteady convection-diffusion equation through a combination of Runge-Kutta methods and compact difference schemes. The combination method has fourth-order accuracy in both time and space variables. Numerical experiments were conducted and the results were compared with those obtained by the Crank-Nicolson method in order to check the accuracy of the combination method. The analysis results indicated that the combination method is numerically stable at low wave numbers and small CFL numbers. The combination method has higher accuracy than the Crank-Nicolson method.

Local and parallel finite element methods based on two‐grid discretizations for unsteady convection–diffusion problem

AbstractIn this article, some local and parallel finite element methods are proposed and investigated for the time‐dependent convection–diffusion problem. With backward Euler scheme for the temporal discretization, the basic idea of the present methods is that for a solution to the considered equations, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel procedure at each time step. The partition of unity is used to collect the local high frequency components to assemble a global continuous approximation. Theoretical results are obtained and numerical tests are reported to support the theoretical findings.

Performance of similarity explicit group iteration for solving 2D unsteady convection-diffusion equation

In this paper, a similarity finite difference (SFD) solution is addressed for thetwo-dimensional (2D) parabolic partial differential equation (PDE), specifically on the unsteady convection-diffusion problem. Structuring the similarity transformation using wave variables, we reduce the parabolic PDE into elliptic PDE. The numerical solution of the corresponding similarity equation is obtained using a second-order central SFD discretization schemeto get the second-order SFD approximation equation. We propose a four-point similarity explicit group (4-point SEG) iterative methodasa numericalsolution of the large-scale and sparse linear systems derived from SFD discretization of 2D unsteady convection-diffusion equation (CDE). To showthe 4-point SEG iteration efficiency, two iterative methods, such as Jacobiand Gauss-Seidel (GS) iterations, are also considered. The numerical experiments are carried out using three different problems to illustrate our proposed iterative method's performance. Finally, the numerical results showed that our proposed iterative method is more efficient than the Jacobiand GS iterations in terms of iteration number and execution time.

A Compact Higher-Order Scheme for Two-Dimensional Unsteady Convection–Diffusion Equations

In this paper, we study a compact higher-order scheme for the two-dimensional unsteady convection–diffusion problems using the nearly analytic discrete method (NADM), especially, focusing on the convection dominated-diffusion problems. The numerical scheme is constructed and the stability analysis is implemented. We find the order of accuracy of scheme is [Formula: see text], where [Formula: see text] is the size of time steps and [Formula: see text] the size of spacial steps, especially, making clear the relation between [Formula: see text] and [Formula: see text] is according to the different values of diffusion parameter [Formula: see text] through von Neumann stability analysis. The obtained numerical results for several benchmark problems show that our method makes progress in the numerical study of NADM for convection–diffusion equation and is to be effective and helpful particularly in computations for the convection dominated-diffusion equations and, furthermore, valuable in the numerical treatment of many real-world problems such as MHD natural convection flow.

Radial integration boundary element method for solving two-dimensional unsteady convection–diffusion problem

Numerical solution of two-dimensional unsteady convection–diffusion problem is carried out by using the radial integration boundary element method in the paper. A boundary domain integral equation is established for the purpose by using the fundamental solution of Laplace equation. The convective and time-dependent terms of governing equation lead to the appearance of two domain integrals including the unknown quantities in the integral equation. The radial integration method is used to transform the two domain integrals into the equivalent boundary integrals over the global boundary by approximating the unknown quantities through the augmented fourth-order spline radial basis function. Thus, a pure boundary element algorithm with the requirement of boundary-only discretization and some internal points instead of internal cells is developed. The finite difference scheme for discretizing the time-dependent term is utilized to assemble the final system of equations. Several numerical examples are given to illustrate the accuracy and efficiency of the proposed method.

Adaptive stopping criterion for iterative linear solvers combined with anisotropic mesh adaptation, application to convection-dominated problems

In this work we propose an automatic adaptive stopping criterion integrated with the anisotropic mesh adaptation procedure, thus requiring no additional computational cost. Using information from this procedure we provide control for the linear solver convergence, stopping the iterative solution when the algebraic error is lower than the estimated discretization error. We apply this framework to steady and unsteady convection–diffusion problems, using a stabilized finite element formulation. The proposed method proves to be an effective cost-free strategy to reduce the number of iterations needed, without spoiling the accuracy of the solution.

A High Order Compact ADI Method for Solving 3D Unsteady Convection Diffusion Problems

In this paper, we develop a rational high order compact alternating direction implicit (RHOC ADI) method for solving the three dimensional (3D) unsteady convection diffusion equation. The present scheme, based on the idea of the fourth order rational compact finite difference operator for the spatial discretization and the Crank-Nicolson method for the time discretization, is fourth order accurate in space and second order accurate in time. The solution procedure consists of a number of tridiagonal matrix operations, which makes the computation to be cost-effective. It is shown by means of the discrete Fourier analysis that this method is unconditionally stable. Three test problems are given to demonstrate the performance of the present method. The numerical results show that the present RHOC ADI scheme has higher accuracy and better phase and amplitude error characteristics than the classical second order Douglas-Gunn ADI method [16] and some high order compact ADI methods including the Karaa’s HOC ADI method [26], Cao and Ge’s HOC ADI method [27], and our previous exponential HOC ADI method [28].

Multiscale model reduction for transport and flow problems in perforated domains

Convection-dominated transport phenomenon is important for many applications. In these applications, the transport velocity is often a solution of heterogeneous flow problems, which results to a coupled flow and transport phenomena. In this paper, we consider a coupled flow (Stokes problem) and transport (unsteady convection–diffusion problem) in perforated domains. Perforated domains (see Fig. 1) represent void space outside hard inclusions as in porous media, filters, and so on. We construct a coarse-scale solver based on Generalized Multiscale Finite Element Method (GMsFEM) for a coupled flow and transport. The main idea of the GMsFEM is to develop a systematic approach for computing multiscale basis functions. We use a mixed formulation and appropriate multiscale basis functions for both flow and transport to guarantee a mass conservation. For the transport problem, we use Petrov–Galerkin mixed formulation, which provides a stability. As a first approach, we use the multiscale flow solution in constructing the basis functions for the transport equation. In the second approach, we construct multiscale basis functions for coupled flow and transport without solving global flow problem. The novelty of this approach is to construct a coupled multiscale basis function. Numerical results are presented for computations using offline basis. We also present an algorithm for adaptively adding online multiscale basis functions, which are computed using the residual information. Numerical examples using online GMsFEM show the speed up of convergence.

Numerical solutions of two-dimensional unsteady convection–diffusion problems using modified bi-cubic B-spline finite elements

ABSTRACTThis research study presents a numerical scheme to compute approximate solutions of two dimensional unsteady convection–diffusion equation. We used collocation of modified bi-cubic B-spline functions for dependent variable u and for its derivatives w.r.t. space variables x and y. Strong stability preserving Runge–Kutta method (SSP-RK54) has been used for solving system of first-order ordinary differential equations obtained from the collocation form of the partial differential equation. We did not linearize the nonlinear terms by using any transformation or linearization method. The number of computations and the required storage space is very less for the proposed scheme. Four examples have been taken as described in available literature to demonstrate the effect and utility of the proposed scheme. These numerical experiments show that the obtained results are not only quite satisfactory w.r.t. the exact solutions but also competent with the solutions available in earlier research studies. Computational complexity of the proposed scheme has been discussed and shown that it is , where p is total number of nodes. The proposed scheme is easy to implement and the size of required computational work is very small. Moreover, using this scheme, we can compute approximate solutions not only at the mesh points but at any other point of the solution domain as well.

Variational Multiscale Element-Free Galerkin Method and Precise Time Step Integration Method for Convection–Diffusion Problems

To simulate the convection–diffusion problems and improve the computational efficiency of Galerkin meshless methods, an explicit algorithm that combines variational multiscale element-free Gakerkin method with precise time step integration is presented. Variational multiscale element-free Gakerkin method is used to overcome the numerical spurious oscillations for convection-dominated problems; meanwhile in order to save the computational time, the precise time step integration method is adopted for the solution of the ordinary differential equations. Two numerical examples are considered to demonstrate the stability, accuracy, and computational efficiency of the method. The numerical results illustrate that the presented method not only has high calculation accuracy, but also brings a significant computational time saving for solving the unsteady convection–diffusion problems.

A New High-Order Compact ADI Method for 3-D Unsteady Convection-Diffusion Problems with Discontinuous Coefficients

In this article, a fourth-order compact alternating direction implicit (ADI) method, based on the clasical Douglas-Gunn ADI method combined with Richardson's extrapolation technique, is proposed for solving 3-D unsteady convection-diffusion equations with discontinuous coefficients. To achieve fourth-order temporal accuracy, a correction term that ensures the realization of extrapolation and reduces the splitting error is introduced. The scheme has unconditional stability and can be solved by the Thomas algorithm. Numerical experiments, including continuous/discontinuous coefficients cases, are conducted to verify the robustness and the high accuracy of this new method.

Unconditionally Monotone Schemes for Unsteady Convection-Diffusion Problems

Abstract. This paper deals with constructing monotone schemes of the second-order accuracy in space for transient convection-diffusion problems. They are based on a reformulation of the convective and diffusive transport terms using the convective terms in the divergent and nondivergent forms. The stability of the difference schemes is established in the uniform and L1 norm. For 2D problems, unconditionally monotone schemes of splitting with respect to spatial variables are developed. Unconditionally stable schemes for problems of convection-diffusion-reaction are proposed, too.

An exponential high‐order compact ADI method for 3D unsteady convection–diffusion problems

AbstractIn this article, we develop an exponential high order compact alternating direction implicit (EHOC ADI) method for solving three dimensional (3D) unsteady convection–diffusion equations. The method, which requires only a regular seven‐point 3D stencil similar to that in the standard second‐order methods, is second order accurate in time and fourth‐order accurate in space and unconditionally stable. The resulting EHOC ADI scheme in each alternating direction implicit (ADI) solution step corresponding to a strictly diagonally dominant matrix equation can be solved by the application of the one‐dimensional tridiagonal Thomas algorithm with a considerable saving in computing time. Numerical experiments for three test problems are carried out to demonstrate the performance of the present method and to compare it with the classical Douglas–Gunn ADI method and the Karaa's high‐order compact ADI method. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013

A multi-block ADI finite-volume method for incompressible Navier–Stokes equations in complex geometries

An efficient second-order accurate finite-volume method is developed for a solution of the incompressible Navier–Stokes equations on complex multi-block structured curvilinear grids. Unlike in the finite-volume or finite-difference-based alternating-direction-implicit (ADI) methods, where factorization of the coordinate transformed governing equations is performed along generalized coordinate directions, in the proposed method, the discretized Cartesian form Navier–Stokes equations are factored along curvilinear grid lines. The new ADI finite-volume method is also extended for simulations on multi-block structured curvilinear grids with which complex geometries can be efficiently resolved. The numerical method is first developed for an unsteady convection–diffusion equation, then is extended for the incompressible Navier–Stokes equations. The order of accuracy and stability characteristics of the present method are analyzed in simulations of an unsteady convection–diffusion problem, decaying vortices, flow in a lid-driven cavity, flow over a circular cylinder, and turbulent flow through a planar channel. Numerical solutions predicted by the proposed ADI finite-volume method are found to be in good agreement with experimental and other numerical data, while the solutions are obtained at much lower computational cost than those required by other iterative methods without factorization. For a simulation on a grid with O(10 5) cells, the computational time required by the present ADI-based method for a solution of momentum equations is found to be less than 20% of that required by a method employing a biconjugate-gradient-stabilized scheme.

A rational high-order compact ADI method for unsteady convection–diffusion equations

Based on a fourth-order compact difference formula for the spatial discretization, which is currently proposed for the one-dimensional (1D) steady convection–diffusion problem, and the Crank–Nicolson scheme for the time discretization, a rational high-order compact alternating direction implicit (ADI) method is developed for solving two-dimensional (2D) unsteady convection–diffusion problems. The method is unconditionally stable and second-order accurate in time and fourth-order accurate in space. The resulting scheme in each ADI computation step corresponds to a tridiagonal matrix equation which can be solved by the application of the 1D tridiagonal Thomas algorithm with a considerable saving in computing time. Three examples supporting our theoretical analysis are numerically solved. The present method not only shows higher accuracy and better phase and amplitude error properties than the standard second-order Peaceman–Rachford ADI method in Peaceman and Rachford (1959) [4], the fourth-order ADI method of Karaa and Zhang (2004) [5] and the fourth-order ADI method of Tian and Ge (2007) [23], but also proves more effective than the fourth-order Padé ADI method of You (2006) [6], in the aspect of computational cost. The method proposed for the diffusion–convection problems is easy to implement and can also be used to solve pure diffusion or pure convection problems.

A hybrid Padé ADI scheme of higher‐order for convection–diffusion problems

AbstractA high‐order Padé alternating direction implicit (ADI) scheme is proposed for solving unsteady convection–diffusion problems. The scheme employs standard high‐order Padé approximations for spatial first and second derivatives in the convection‐diffusion equation. Linear multistep (LM) methods combined with the approximate factorization introduced by Beam and Warming (J. Comput. Phys. 1976; 22: 87–110) are applied for the time integration. The approximate factorization imposes a second‐order temporal accuracy limitation on the ADI scheme independent of the accuracy of the LM method chosen for the time integration. To achieve a higher‐order temporal accuracy, we introduce a correction term that reduces the splitting error. The resulting scheme is carried out by repeatedly solving a series of pentadiagonal linear systems producing a computationally cost effective solver. The effects of the approximate factorization and the correction term on the stability of the scheme are examined. A modified wave number analysis is performed to examine the dispersive and dissipative properties of the scheme. In contrast to the HOC‐based schemes in which the phase and amplitude characteristics of a solution are altered by the variation of cell Reynolds number, the present scheme retains the characteristics of the modified wave numbers for spatial derivatives regardless of the magnitude of cell Reynolds number. The superiority of the proposed scheme compared with other high‐order ADI schemes for solving unsteady convection‐diffusion problems is discussed. A comparison of different time discretizations based on LM methods is given. Copyright © 2009 John Wiley & Sons, Ltd.