EXPONENTIAL COMPACT HIGHER-ORDER SCHEMES AND THEIR STABILITY ANALYSIS FOR UNSTEADY CONVECTION-DIFFUSION EQUATIONS

In practical engineering applications, convection diffusion equations are generally used to describe the transport processes involving fluid motion, heat transfer, astrophysics, oceanography, meteorology, semiconductors, hydraulics, pollutant & sediment transport and chemical engineering. In this paper, a high order compact difference scheme based on the fourth order compact difference scheme in spatial discretization and the fourth order Runge-Kutta method in time integration is proposed for the numerical simulation of the unsteady convection-diffusion equation. The validity and effectiveness of the proposed method is firstly tested by a two-dimensional convection-diffusion equation with a Gaussian pulse type concentration. The L 2 error norms are used to measure differences between the exact and numerical solutions and compared to those obtained by other methods. It is shown that the results obtained by proposed method agree very well with the analytical solutions and is more accurate than other methods. Then, a two-dimensional non-linear Burgers equation is used to validate the effectiveness of the proposed method used to solve the non-linear convection-diffusion equation, which also models well. Finally, the Taylor’s vortex problem is investigated by the proposed method and good agreement is obtained with the exact solutions. From the three test problems, it is shown that the proposed high order compact difference scheme is an efficient and accurate method to simulate the transport problems and also can be applied to many engineering problems.

A sixth order finite difference scheme for the convection diffusion equation

In 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

In this paper, a novel compact alternating direction implicit (ADI) scheme is proposed for solving the time-fractional subdiffusion equation in two space dimensions. The established scheme is based on the modified L1 method in time and the compact finite difference method in space. The unique solvability, unconditionally stability and convergence of the scheme are proved. The derived compact ADI scheme is coincident with the one for 2D integer order parabolic equation when the , where is the order of the Riemann–Liouville derivative operator. In addition, the novel ADI scheme is used to solve the 2D modified fractional diffusion equation, and the corresponding stability and convergence results are also given. Numerical results are provided to verify the theoretical analysis.

In this work, a numerical scheme based on artificial compressibility formulation of a diffuse-interface model is developed for simulating two-dimensional two-phase incompressible flows. The coupled nonlinear systems composing of the incompressible Navier-Stoke equations(NSeqs) and volume preserving Allen-Cahn type phase equation are recast into conservative forms with source terms, which are suited to implementing high-order schemes. The Boussinesq approximation is used for buoyancy effects in flows with small density densities. The fifth-order weighted essentially non-oscillatory scheme is used for discretizing the convective terms while for obtaining time accuracy at each physical time step dual-time stepping (DTS) approach is used. Beam-Warming approximate factorization scheme is utilized so as to obtain block tridiagonal system of equations in each spatial direction which can be solved efficiently with the alternating direction implicit (ADI) algorithm. The performance of the numerical method is demonstrated by its application to benchmark viscous two-phase flow problems. Numerical experiments shows that the method allows larger time-steps and smaller interface value η than previous projection methods in literature when computing problems with moving interfaces and topological changes.

A spatially second order alternating direction implicit (ADI) method for solving three dimensional parabolic interface problems

ABSTRACTHeat transfer at microscale plays an important role in microtechnology applications. First, this paper is concerned with the numerical solution of a two-dimensional (2D) heat transfer equation at microscale through a second-order alternating direction implicit (ADI) method. By the discrete energy method, it is shown that the ADI solution converges to the exact solution with a convergence order of in L2-norm. Second, the ADI method is generalized to numerically solve the three-dimensional (3D) problem, and provides numerical solutions of order two in both time and space. Finally, numerical results testify the temporal and spatial accuracy of the ADI method.

This paper presents an exponential compact higher order scheme for Convection-Diffusion Equations (CDE) with variable and nonlinear convection coefficients. The scheme is for one-dimensional problems and produces a tri-diagonal system of equations which can be solved efficiently using Thomas algorithm. For two-dimensional problems, the scheme produces an accuracy over a compact nine point stencil which can be solved using any line iterative approach with alternate direction implicit procedure. The convergence of the iterative procedure is guaranteed as the coefficient matrix of the developed scheme satisfies the conditions required to be positive. Wave number analysis has been carried out to establish that the scheme is comparable in accuracy with spectral methods. The higher order accuracy and better rate of convergence of the developed scheme have been demonstrated by solving numerous model problems for one and two-dimensional CDE, where the solutions have the sharp gradient at the solution boundary.

The Advection-Diffusion Reaction (ADR) equation appears in many problems in nature. This constitutes a general model that is useful in various scenarios, from porous media to atmospheric processes. Particularly, it is used at the interface between two fluids where different types of instabilities due to surface mobility may appear. Together with the ADR equation, the Darcy-Brinkman model describes the phenomena known as fingering that appear in different contexts. The study of this type of system gains in complexity when the number of chemical species dissolved in both fluids increases. With more solutes, the increasing complexity of this phenomenon generally requires much computational power. To face the need for more computational resources, we build a solver tool based on an Alternating Direction Implicit (ADI) scheme that can be run in Central Processing Unit (CPU) and Graphic Processing Unit (GPU) architectures on any notebook. The implementation is done using the MATLAB platform to compare both versions. It is shown that using the GPU version strongly saves both resources and calculation times.

A high order finite difference method with Richardsonextrapolation for 3D convection diffusion equation

Contaminant transport in a soil formation is described by advection dispersion equation. In this study, a horizontal and transversal contaminant transport along transient groundwater flow under non-linear sorption is solved numerically to examine the contaminant distribution profiles in finite soil media. The horizontal and transversal pore-water seepage velocities are defined as varying exponentially with time. Dispersion is considered directly proportional to the first power of the pore-water velocity. Axial input sources varying exponentially with time are assigned along the two-dimensional coordinate axes. For approximating the governing solute transport equation into algebraic equations, Crank–Nicolson (CN) and alternating direction implicit (ADI) methods are used. Both the numerical solutions are illustrated graphically with the help of computer software for various hydrological input data. In a special case, the numerical solutions are also compared with an approximate solution obtained by PDEtool. The comparison is performed with the help of contour plots. The CN method gives more accurate result than ADI method for the present model problem.

A matched Peaceman–Rachford ADI method for solving parabolic interface problems

Stability of ADI schemes applied to convection–diffusion equations with mixed derivative terms

Parallelization via domain decomposition techniques of multigrid and adi solvers for navier-stokes equations

The initial-boundary problems of two-dimension and second order partial differential equation, with variable coefficients, is developed with piecewise rectangle domain. Numerical solution process is carried out with Alternating Direction Implicit (ADI) sheme, and stability analysis is conducted by von Neumann method. Results of numerical solution show clearly the advantages of rapid calculation speed, good stability and etc, which provide the basis for solving the equation with computer in this paper.