Rachford Alternating Direction Implicit Research Articles

Parameter-uniform fractional step hybrid numerical scheme for 2D singularly perturbed parabolic convection–diffusion problems

This article focuses on developing and analyzing an efficient numerical scheme for solving two-dimensional singularly perturbed parabolic convection–diffusion initial-boundary value problems exhibiting a regular boundary layer. For approximating the time derivative, we use the Peaceman–Rachford alternating direction implicit method on uniform mesh and for the spatial discretization, a hybrid finite difference scheme is proposed on a special rectangular mesh which is tensor-product of piecewise-uniform Shishkin meshes in the spatial directions. We prove that the numerical scheme converges uniformly with respect to the perturbation parameter $$\varepsilon $$ and also attains almost second-order spatial accuracy in the discrete supremum norm. Finally, numerical results are presented to validate the theoretical results. In addition to this, numerical experiments are conducted to demonstrate the effect of the time-dependent boundary conditions in the order of convergence numerically by introducing the classical evaluation of the boundary data; and also the improvement in the spatial order of accuracy of the present method by considering the Bakhvalov–Shishkin mesh in the spatial directions.

Computational modeling of MHD flow of blood and heat transfer enhancement in a slowly varying arterial segment

We have developed a computational model to investigate the unsteady flow of blood and heat transfer characteristics through a sinusoidally varying arterial segment under magnetic environment. Direct numerical simulation has been performed by employing stream function-vorticity formulation followed by a coordinate transformation. The transformed governing equations have been solved using the finite difference scheme by developing Peaceman–Rachford Alternating Direction Implicit (P–R ADI) method. The results show that the rate of heat transfer diminishes with a rise in the magnetic field strength, whereas the trait is reversed in the case of Reynolds number and amplitude of wavy vessel. The skin-friction is high for greater values of amplitude of oscillation of arterial wall. An interesting result can be documented that the Nusselt number has decreasing effect with magnetic field strength at higher Reynolds number, while it increases when Reynolds number is low.

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

A new matched alternating direction implicit (ADI) method is proposed in this paper for solving three-dimensional (3D) parabolic interface problems with discontinuous jumps and complex interfaces. This scheme inherits the merits of its ancestor for two-dimensional problems, while possesses several novel features, such as a non-orthogonal local coordinate system for decoupling the jump conditions, two-side estimation of tangential derivatives at an interface point, and a new Douglas–Rachford ADI formulation that minimizes the number of perturbation terms, to attack more challenging 3D problems. In time discretization, this new ADI method is found to be of first order and stable in various experiments. In space discretization, the matched ADI method achieves the second order accuracy based on simple Cartesian grids for various irregularly-shaped surfaces and spatial–temporal dependent jumps. Computationally, the matched ADI method is as efficient as the fastest implicit scheme based on the geometrical multigrid for solving 3D parabolic equations, in the sense that its complexity in each time step scales linearly with respect to the spatial degree of freedom N, i.e., O(N). Furthermore, unlike iterative methods, the ADI method is an exact or non-iterative algebraic solver which guarantees to stop after a certain number of computations for a fixed N. Therefore, the proposed matched ADI method provides a very promising tool for solving 3D parabolic interface problems.

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

A new Peaceman–Rachford alternating direction implicit (PR-ADI) method is proposed in this work for solving two-dimensional (2D) parabolic interface problems with discontinuous solutions. The classical ADI schemes are known to be inaccurate for handling interfaces. This motivates the development of a matched Douglas ADI (D-ADI) method in the literature, in which the finite difference is locally corrected according to the jump conditions. However, the unconditional stability of the matched ADI method cannot be maintained if the D-ADI is simply replaced by the PR-ADI. To stabilize the computation, the tangential derivative approximations in the jump conditions decomposition are substantially improved in this paper. Moreover, a new temporal discretization is adopted for formulating the PR-ADI method, which involves less perturbation terms. Stability analysis is conducted through eigenvalue spectrum analysis, which demonstrates the unconditional stability of the proposed method. The matched PR-ADI method achieves second order of accuracy in space in all tested problems with complex geometries and jumps, while maintaining the efficiency of the ADI. The proposed PR-ADI method is found to be more accurate than the D-ADI method in time integration, even though its formal temporal order is limited in the matched ADI framework.

Numerical investigation of MHD flow of blood and heat transfer in a stenosed arterial segment

A numerical investigation of unsteady flow of blood and heat transfer has been performed with an aim to provide better understanding of blood flow through arteries under stenotic condition. The blood is treated as Newtonian fluid and the arterial wall is considered to be rigid having deposition of plaque in its lumen. The heat transfer characteristic has been analyzed by taking into consideration of the dissipation of energy due to applied magnetic field and the viscosity of blood. The vorticity-stream function formulation has been adopted to solve the problem using implicit finite difference method by developing well known Peaceman–Rachford Alternating Direction Implicit (ADI) scheme. The quantitative profile analysis of velocity, temperature and wall shear stress as well as Nusselt number is carried out over the entire arterial segment. The streamline and temperature contours have been plotted to understand the flow pattern in the diseased artery, which alters significantly in the downstream of the stenosis in the presence of magnetic field. Both the wall shear stress and Nusselt number increases with increasing magnetic field strength. However, wall shear stress decreases and Nusselt number enhances with Reynolds number. The results show that with an increase in the magnetic field strength upto 8T, does not causes any damage to the arterial wall, but the study is significant for assessing temperature rise during hyperthermic treatment.

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.

SECOND-ORDER METHOD FOR SOLVING 2D NONLINEAR PARABOLIC DIFFERENTIAL EQUATIONS BASED ON ADI METHOD

In this article, a new unconditionally stable method based on alternating direction implicit (ADI) method for solving nonlinear unsteady 2D parabolic problems is presented. The order of this method in both time and space is 2. For this development, the Peaceman–Rachford ADI method has been modified suitably to take care of nonlinear forcing term of the equation appropriately. The unconditional stability of the method is shown under discrete L2 norm on bounded sets. The method of solution consists of a number of strictly diagonally dominant tridiagonal matrices, which make the method computationally efficient. The accuracy of the proposed method is also demonstrated by some numerical examples.

Strategies for damping the oscillations of the alternating direction implicit method of simulation of diffusion-limited chronoamperometry at disk electrodes

Five methods are described for reducing or eliminating the error oscillations resulting from the application of the Peaceman–Rachford alternating direction implicit algorithm to disk electrode simulation problems involving an initial discontinuity, such as results from a potential jump at an electrode. The methods are: (a) the straight-forward application of the Peaceman–Rachford ADI algorithm, using sufficiently small time intervals so that the oscillations are damped at times for which accurate current values are needed; (b) using the first-order alternating direction implicit algorithm by Douglas and Rachford; (c) subdivision of the first simulation time interval into subintervals; (d) beginning the simulation with a few steps of the Douglas–Rachford algorithm followed by the Peaceman–Rachford method; and (e) presetting a reasonable approximation to the concentration profile at the end of the first simulation step and simulating from there on. Methods (d) and (e) are found clearly to be the most efficient at damping the oscillations, but method (b) also eliminates oscillations and leads to reasonable computation times, about one third of those needed for a sparse matrix solution of a two-dimensional system.

A high-order ADI method for parabolic problems with variable coefficients

A high-order compact alternating direction implicit (ADI) method is proposed for solving two-dimensional (2D) parabolic problems with variable coefficients. The computational problem is reduced to sequence one-dimensional problems which makes the computation cost-effective. The method is easily extendable to multi-dimensional problems. Various numerical tests are performed to test its high-order accuracy and efficiency, and to compare it with the standard second-order Peaceman–Rachford ADI method. The method has been applied to obtain the numerical solutions of the lid-driven cavity flow problem governed by the 2D incompressible Navier–Stokes equations using the stream function-vorticity formulation. The solutions obtained agree well with other results in the literature.

A generalized alternating-direction implicit scheme for incompressible magnetohydrodynamic viscous flows at low magnetic Reynolds number

Abstract This paper presents numerical simulations of incompressible fluid flows in the presence of a magnetic field at low magnetic Reynolds number. The equations governing the flow are the Navier–Stokes equations of fluid motion coupled with Maxwell’s equations of electromagnetics. The study of fluid flows under the influence of a magnetic field and with no free electric charges or electric fields is known as magnetohydrodynamics. The magnetohydrodynamics approximation is considered for the formulation of the non-dimensional problem and for the characterization of similarity parameters. A finite-difference technique is used to discretize the equations. In particular, an extension of the generalized Peaceman and Rachford alternating-direction implicit (ADI) scheme for simulating two-dimensional fluid flows is presented. The discretized conservation equations are solved in stream function–vorticity formulation. We compare the ADI and generalized ADI schemes, and show that the latter is more efficient in simulating low Reynolds number and magnetic Reynolds number problems. Numerical results demonstrating the applicability of this technique are also presented. The simulation of incompressible magnetohydrodynamic fluid flows is illustrated by numerical solution for two-dimensional cases.

On the finite differences schemes for the numerical solution of two-dimensional moving boundary problem

In this paper, two different finite differences schemes are presented for numerical solution of two-dimensional moving boundary problem. The methods are illustrated by solving sample problem in two-dimensional solidification of the square prism. The finite difference schemes developed for this purpose are based on the method of lines using interpolation polynomials. The fully explicit method developed here which has reasonable accuracy, and Peaceman and Rachford alternating direction implicit (ADI) formula illustrated by solving sample problem.The accuracies of the resulting methods are verified by numerical testing. The numerical results are obtained by present methods are compared with earlier authors.

A Formulation of Peaceman and Rachford ADI Method for the Three-Dimensional Heat Diffusion Equation

A formulation of an alternating direction implicit (ADI) method is given by extending peaceman and Rachford Scheme to three dimensions. The scheme becomes conditionally stable. The von Neumann stability analysis is performed. Numerical results for solving heat diffusion equation have been obtained for different specified boundary value problems to obtain a simple explicit stability.

High order ADI method for solving unsteady convection–diffusion problems

We propose a high order alternating direction implicit (ADI) solution method for solving unsteady convection–diffusion problems. The method is fourth order in space and second order in time. It permits multiple use of the one-dimensional tridiagonal algorithm with a considerable saving in computing time, and produces a very efficient solver. It is shown through a discrete Fourier analysis that the method is unconditionally stable for 2D problems. Numerical experiments are conducted to test its high accuracy and to compare it with the standard second-order Peaceman–Rachford ADI method and the spatial third-order compact scheme of Noye and Tan.

A NUMERICAL SCHEME FOR SOLVING CREEPING FLOWS

This work shows an extension of the generalized Peaceman and
Rachford alternating-direction implicit (ADI) scheme for simulating
two-dimensional fluid flows at low Reynolds numbers. The
conservation equations are solved in stream function - vorticity
formulation. We compare the ADI and generalized ADI schemes, and
show that the latter is more efficient to simulate a creeping flow.
Numerical results demonstrating the applicability of this technique are
also presented.

A NUMERICAL SCHEME FOR SOLVING CREEPING FLOWS

This work shows an extension of the generalized Peaceman and Rachford alternating-direction implicit (ADI) scheme for simulating two-dimensional fluid flows at low Reynolds numbers. The conservation equations are solved in stream function - vorticity formulation. We compare the ADI and generalized ADI schemes, and show that the latter is more efficient to simulate a creeping flow. Numerical results demonstrating the applicability of this technique are also presented.

A Generalized Peaceman–Rachford ADI Scheme for Solving Two-Dimensional Parabolic Differential Equations

A generalized Peaceman–Rachford alternating-direction implicit (ADI) scheme for solving two-dimensional parabolic differential equations has been developed based on the idea of regularized difference scheme. It is to be very well to simulate fast transient phenomena and to efficiently capture steady state solutions of parabolic differential equations. Numerical example is illustrated.

Accuracy of the general peaceman and Rachford alternating direction implicit method

In this paper, the truncation error of the general Peaceman and Rachford alternating direction implicit (ADI) method in a N-dimensional space was derived by using the Taylor-series-expansion approach. The formulation of this truncation error is TE= 2−N 2N 2 ∑ i=l N ∑ j=l N ∂ 4θ ∂η 2 iη 2 j δτ+O(δτ 2)+ ∑ i=l N O(δη 2 i which indicates that the method is second order accurate in both time and space for two dimensions but reduces to first order accuracy in time for higher dimensions. However the second order time accuracy is recovered when N → ∞. The worst case corresponding to the maximum value of (2-N) 2N 2 occurs at N = 4.

Two-dimensional Hartmann flows of fluids with cross-stream dependent electrical conductivity

The equations for two‐dimensional Hartmann flow through ducts of rectangular cross section are solved numerically by the Peaceman–Rachford alternating direction implicit method for boundary conditions appropriate to those encountered in a Faraday magnetohydrodynamic generator. Cross‐stream variation in fluid electrical condictivity as well as variable conductivity of the electrode walls are considered and quantities such as velocity distribution, current streamline distribution, volume flow rate, and conversion efficiency are obtained for a range of Hartmann numbers up to M=100. Results show that decreasing the electrical conductivity in a continuous manner from a high value near the walls to a minimum along the duct axis leads to a high velocity region near the duct axis, a result differing from the essential slug flow character found for Hartmann flows of constant conductivity fluids at high M. The known one‐dimensional Hartmann flow solutions are found to give excellent agreement with the two‐dimensional results for constant conductivity fluids whenever the product of the duct aspect ratio and the square root of the Hartmann number is greater than about ten.