Peaceman Rachford Alternating Direction Implicit Research Articles

ALTERNATING DIRECTION IMPLICIT METHOD FOR POISSON EQUATION WITH INTEGRAL CONDITIONS

In this paper, we investigate the convergence of the Peaceman-Rachford Alternating Direction Implicit method for the system of difference equations, approximating the two-dimensional elliptic equations in rectangular domain with nonlocal integral conditions. The main goal of the paper is the analysis of spectrum structure of difference eigenvalue problem with nonlocal conditions. The convergence of iterative method is proved in the case when the system of eigenvectors is complete. The main results are generalized for the system of difference equations, approximating the differential problem with truncation error O(h4).

Analysis of a Peaceman-Rachford ADI scheme for Maxwell equations in heterogeneous media

The Peaceman-Rachford alternating direction implicit (ADI) scheme for linear time-dependent Maxwell equations is analyzed on a heterogeneous cuboid. Due to discontinuities of the material parameters, the solution of the Maxwell equations is less than H2-regular in space. For the ADI scheme, we prove a rigorous time-discrete error bound with a convergence rate that is half an order lower than the classical one. Our statement imposes only assumptions on the initial data and the material parameters, but not on the solution. To establish this result, we analyze the regularity of the Maxwell equations in detail in an appropriate functional analytical framework. The theoretical findings are complemented by a numerical experiment indicating that the proven convergence rate is indeed observable and optimal.

Spatio-temporal evolution of magnetohydrodynamic blood flow and heat dynamics through a porous medium in a wavy-walled artery

Background and objectiveIn a healthy body, the elastic wall of the arteries forms wave-like structures resulting from the continuous pumping of the heart. The systolic and diastolic phases generate a contraction and expansion pattern, which is mimicked in this study by considering a wavy-walled arterial structure. A numerical investigation of the spatio-temporal flow of blood and heat transfer through a porous medium under the action of magnetic field strength is conducted. MethodThe governing equations of the blood flow in the Darcy model are simulated by applying a vorticity-stream function formulation approach. The transformed dimensionless equations are further discretized using the finite difference method by developing the Peaceman-Rachford alternating direction implicit (P-R ADI) scheme. ResultsThe computational results for the axial velocity, temperature distribution, flow visualization using the streamlines and vorticity contours, isotherms, wall shear stress and the average Nusselt number are presented graphically for different values of the physical parameters. ConclusionsThe study shows that the axial velocity increases with an increase in the Darcy number, and a similar phenomenon is observed because of an amplitude variation in the wavy wall. Both temperature and wall shear stress decreases with an increase in the Darcy number. The average Nusselt number increases with the magnetic field strength, while it has a reducing tendency due to the permeability of the porous medium.

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 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.

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.

Fast numerical solution of nonlinear diffusion equation for the simulation of ion-exchanged micro-optics components in glass

Modulation of the refractive index of glass using ion exchange is a common fabrication method for various micro-optics components, such as waveguide devices, microlenses, and diffractive elements. The design of elements with complicated optical functions has increased the demand for fast numerical simulation of the nonlinear ion-exchange process of two monovalent ions. In this paper, a linear implicit finite-difference Crank-Nicholson-type method is implemented for thermal two- and three-dimensional ion-exchange processes. The resulting system of linear equations is solved by using various iterative methods. Grid spacing, length of the time step, and the number of iterations are chosen so that desired accuracy is obtained within minimum computing time. The implemented methods are compared with the Peaceman-Rachford Alternating Direction Implicit (PR-ADI) and the explicit Du Fort-Frankel method in modeling a computer-synthesized diffractive optical element and a waveguide channel system.

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.

Stability and convergence of the Peaceman-Rachford ADI method for initial-boundary value problems

In this paper an analysis will be presented for the ADI (alternating direction implicit) method of Peaceman and Rachford applied to initial-boundary value problems for partial differential equations in two space dimensions. We shall use the method of lines approach. Motivated by developments in the field of stiff nonlinear ordinary differential equations, our analysis will focus on problems where the semidiscrete system, obtained after discretization in space, satisfies a one-sided Lipschitz condition with a constant independent of the grid spacing. For such problems, unconditional stability and convergence results will be derived.

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.