ADI Scheme Research Articles

Peaceman‐Rachford ADI scheme for the two dimensional flow of a second‐grade fluid

PurposeThe purpose of this paper is to obtain numerical solutions of a two‐dimensional mixed space‐time PDE modelling the flow of a second‐grade.Design/methodology/approachThe paper derives conditionally stable Crank‐Nicolson schemes to solve both the one and two dimensional mixed‐space time PDE. For the two‐dimensional case we implement the Crank‐Nicolson scheme using a Peaceman‐Rachford ADI scheme.FindingsFor zero‐shear boundaries the Cattanneo representation of the model equation blows up whilst the representation derived by Rajagopal is stable and produces solutions which decay over time.Originality/valueThe use of a Peaceman‐Rachford ADI scheme to solve a mixed space‐time PDE is both novel and new.

A finite difference approach for the initial-boundary value problem of the fractional Klein-Kramers equation in phase space

Abstract Considering the features of the fractional Klein-Kramers equation (FKKE) in phase space, only the unilateral boundary condition in position direction is needed, which is different from the bilateral boundary conditions in [Cartling B., Kinetics of activated processes from nonstationary solutions of the Fokker-Planck equation for a bistable potential, J. Chem. Phys., 1987, 87(5), 2638–2648] and [Deng W., Li C., Finite difference methods and their physical constrains for the fractional Klein-Kramers equation, Numer. Methods Partial Differential Equations, 2011, 27(6), 1561–1583]. In the paper, a finite difference scheme is constructed, where temporal fractional derivatives are approximated using L1 discretization. The advantages of the scheme are: for every temporal level it can be dealt with from one side to the other one in position direction, and for any fixed position only a tri-diagonal system of linear algebraic equations needs to be solved. The computational amount reduces compared with the ADI scheme in [Cartling B., Kinetics of activated processes from nonstationary solutions of the Fokker-Planck equation for a bistable potential, J. Chem. Phys., 1987, 87(5), 2638–2648] and the five-point scheme in [Deng W., Li C., Finite difference methods and their physical constrains for the fractional Klein-Kramers equation, Numer. Methods Partial Differential Equations, 2011, 27(6), 1561–1583]. The stability and convergence are proved and two examples are included to show the accuracy and effectiveness of the method.

An Eulerian MHD model for the analysis of magnetic flux compression by expanding diamagnetic fusion plasma sphere

The process of magnetic flux compression (MFC) inside a solenoid by expanding diamagnetic plasma sphere produced by an inertial fusion micro-explosions and its application as a direct energy conversion scheme to convert a part of plasma kinetic energy into pulsed electrical energy has been recently reported [1]. For a detailed analysis of this concept, an Eulerian multi-material MHD model is developed using magnetic vector potential formulation for electro-magnetic field calculations and classical volume-of-fluid method for material interface tracking. The diffusion term in the magnetic induction equation is solved implicitly while the advection terms are computed using a second-order MUSCL scheme. An iteration procedure using ADI scheme is used for the free space field calculation. In this paper, we describe the details of the new MHD model, its validation against the semi-analytical solutions (for magnetic Reynolds number ≫1) of magnetic convective-diffusion equations and application to explore the concept of MFC by expanding plasma sphere. The simulation results show that the algorithm is capable of handling complex plasma dynamics inside the MFC system. Also, the results indicate the development and the evolution of MRT like instability near the stagnation point. The magnetic field diffusion into the plasma during the expansion phase is found to be negligible.

PARALLEL NUMERICAL METHODS FOR THE SOLUTION OF NONLINEAR PARABOLIC SYSTEMS

Due to the considerable complexity involved in the numerical solution of problems of phase change materials, the development of fast parallel algorithms to solve them is very useful. This large complexity can be reduced by developing faster methods for solving, and suitably modifying programming techniques. We consider the phase field model of the system of p.d.e's [Formula: see text] where ϕ = ϕ(x, y, t) is the phase indicator function, θ = arctan (ϕy/ϕx), u = u(x, y, t) is the temperature, q, p and f are given scalar functions. This system describes the evolution of phase and temperature in a two-phase medium, and is posed for t≥0 on a rectangle in the x, y plane with appropriate boundary and initial conditions. We solve the system using two finite difference methods. The first method is based on the explicit Euler scheme for the first equation and the Crank–Nicolson–ADI method for the second. The other method uses for both equations the Crank–Nicolson–ADI scheme. We show results of relevant numerical experiments, compare the errors of the two methods, and compare their speed-up when we implement them using parallel processors. Comparisons between the methods and for each one separately are given, and furthermore useful conclusions concerning the number of nodes and the speed of executing the methods in one, two or four processors are also developed.

Convergence of compact ADI method for solving linear Schrödinger equations

AbstractA compact ADI scheme of second‐order in time and fourth‐order in space is proposed for solving linear Schrödinger equations with periodic boundary conditions. By using the recently suggested discrete energy method, it is shown that the stable compact ADI method is unconditionally convergent in the maximum norm. Numerical experiments, including the comparisons with the second‐order ADI scheme and the time‐splitting Fourier pseudospectral method, are presented to support the theoretical results and show the effectiveness of our method. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012

Stability Analysis and Improvement of the Conformal ADI-FDTD Methods

In order to model curved conducting boundaries with high accuracy, conformal alternating direction implicit finite-difference time-domain (ADI-FDTD) methods have been developed with applications of Dey-Mittra's and Yu-Mittra's conformal techniques. We show that the conformal ADI-FDTD methods developed as such lose the unconditional stability although it is expected with the ADI schemes. We present the theoretical analysis and numerical experiments that proves our findings. We then present a modified conformal ADI-FDTD technique with better stability than those based on Yu-Mittra's and Dey-Mittra's conformal techniques.

ON SOME FUČÍK TYPE PROBLEM WITH CUBIC NONLINEARITY ADI APPROACH TO THE PARTICLE DIFFUSION PROBLEM IN MAGNETIC FLUIDS

The present study is devoted to the development of an ADI approach to simulate two-dimensional time-dependent diffusion process of ferromagnetic particles in magnetic fluids. Specific features of the problem are the Neumann boundary conditions. We construct an ADI scheme of formally second order accuracy approximation in time and space. It is proved that the scheme is absolutely stable and it has the accuracy in the energy norm of the second order in time and the order 3/2 in space. The numerical results of a test problem indicate that the convergence rate in space is of the second order as well.

NUMERICAL SIMULATION OF POLLUTANT TRANSPORT FROM FISH FARMING IN RIVER

This paper presented a 3D model for substance transport in river and its application for simulation of pollutant transport in Mekong river due to floating cages-raising. 3D flow-field was solved by logarithmic distributing 2D flow-field of averaged height. Pollutant transport is calculated by solving its full 3D transport equation. The 2D continuum and momentum equations were solved by finite difference method with ADI scheme of Ponce-Yabusaki. The 3D transport equation was solved by finite volume method with ADI scheme of Douglas-Gunn in sigma transformed co-ordinate. The model was tested over analytical solution. Some preliminary results of simulation for pollutant transport of Myhoahung floating cages area (Angiang province) are also presented.

Maximum norm error estimates of efficient difference schemes for second-order wave equations

The three-level explicit scheme is efficient for numerical approximation of the second-order wave equations. By employing a fourth-order accurate scheme to approximate the solution at first time level, it is shown that the discrete solution is conditionally convergent in the maximum norm with the convergence order of two. Since the asymptotic expansion of the difference solution consists of odd powers of the mesh parameters (time step and spacings), an unusual Richardson extrapolation formula is needed in promoting the second-order solution to fourth-order accuracy. Extensions of our technique to the classical ADI scheme also yield the maximum norm error estimate of the discrete solution and its extrapolation. Numerical experiments are presented to support our theoretical results.

Parallel implementation of the ADI‐FDTD method

AbstractWe present a parallel implementation of the three‐dimensional alternating direction implicit finite‐difference time‐domain (ADI‐FDTD) method in Cartesian coordinates using the message passing interface (MPI) library. Parallel implementations not only speed up computations but also increase the maximum solvable problem size. The application of the ADI scheme results in freeing the time‐step size from the Courant‐Friedrichs‐Lewy stability constraint. We demonstrate a speedup of the ADI‐FDTD method through parallelization, even though the communication overhead between the processors is proportional to the volume of the domain, rather than just the surface area of the subdomains as for standard parallel FDTD. We benchmarked our code on an IBM p690 symmetric multiprocessor and also a distributed memory computer cluster with a high‐bandwidth Infiniband interconnect. In both cases, satisfactory scalability and efficiency was demonstrated for simulation domains comprising up to 8 billion mesh cells. © 2009 Wiley Periodicals, Inc. Microwave Opt Technol Lett 51: 1298–1304, 2009; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.24310

Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations

AbstractAlternating direction implicit (ADI) schemes are computationally efficient and widely utilized for numerical approximation of the multidimensional parabolic equations. By using the discrete energy method, it is shown that the ADI solution is unconditionally convergent with the convergence order of two in the maximum norm. Considering an asymptotic expansion of the difference solution, we obtain a fourth‐order, in both time and space, approximation by one Richardson extrapolation. Extension of our technique to the higher‐order compact ADI schemes also yields the maximum norm error estimate of the discrete solution. And by one extrapolation, we obtain a sixth order accurate approximation when the time step is proportional to the squares of the spatial size. An numerical example is presented to support our theoretical results. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010

Unconditional stability of second-order ADI schemes applied to multi-dimensional diffusion equations with mixed derivative terms

We consider the unconditional stability of second-order ADI schemes in the numerical solution of finite difference discretizations of multi-dimensional diffusion problems containing mixed spatial-derivative terms. We investigate an ADI scheme proposed by Craig and Sneyd, an ADI scheme that is a modified version thereof, and an ADI scheme introduced by Hundsdorfer and Verwer. Both sufficient and necessary conditions are derived on the parameters of each of these schemes for unconditional stability in the presence of mixed derivative terms. Our main result is that, under a mild condition on its parameter θ, the second-order Hundsdorfer and Verwer scheme is unconditionally stable when applied to semi-discretized diffusion problems with mixed derivative terms in arbitrary spatial dimensions k ⩾ 2 .

A combined mixed finite element ADI scheme for solving Richards' equation with mixed derivatives on irregular grids

We propose and analyze an efficient numerical method for solving semilinear parabolic problems with mixed derivative terms on non-rectangular domains. The spatial semidiscretization process is based on an expanded mixed finite element scheme which, combined with suitable quadrature rules, is converted into a cell-centered finite difference scheme. This choice preserves the asymptotic accuracy and local conservation of mass of the method, while substantially reducing the computational cost of the totally discrete scheme. To obtain it, an alternating direction implicit scheme is used for the integration in time. The resulting numerical algorithm involves sets of uncoupled tridiagonal systems which can be solved in parallel. We set out some theoretical results of unconditional convergence (of second order in space and first order in time) for our method. Finally, a numerical experiment is shown in order to illustrate the theoretical results.

A Hybrid ADI and SBTD Scheme for Unconditionally Stable Time-Domain Solutions of Maxwell's Equations

Based upon the sampling bi-orthogonal algorithm and alternating direction implicit scheme, we present an unconditionally stable time-domain method, SB-ADI. Similar to the ADI scheme, this method is a second-order approximation in temporal discretization. However, theoretically, the new method produces no spatial discretization error due to the sampling property and compact support of the SBTD algorithm. Dispersion and stability analysis are presented. In a numerical example of 2-D electromagnetic bandgap structure, the new method has reduced the error by a factor of 3.5 at 16% increase of CPU time with respect to the standard ADI.

The Simplest V-Cycle Fast Adaptive Composite Grid ADI-FDTD Method for Two-Dimensional Electromagnetic Simulations

An alternating-direction-implicit finite-difference time-domain (ADI-FDTD) method based on the fast-adaptive-composite grid ADI (FAC-ADI) scheme is introduced to reduce the truncation error generated when ADI-FDTD is derived from the Crank-Nicolson (CN) ADI scheme, and its efficiency is investigated. The relaxation equation and residual equation for FAC-FDTD are constructed by defining the error vector between the traditional ADI and CN FDTD formulas, and the iterative formulas are defined using the iterative schemes. A comparison of the computational efficiency between the FAC-ADI and ADI-FDTD is presented.

Free Convection Flow of Viscous Dissipative Fluid in a Rectangular Cavity

Free convection flow of incompressible viscous fluid with dissipation in a rectangular cavity has been studied, a finite difference technique based on ADI scheme is adopted in the solution of the problem. The effect of dissipation parameter (e ), which usually appears as a term in the energy equation, has been taken into account.The results indicated that the effect of dissipation number ( 004 . 0 ∓ = e ) was very small which is accepted with the fact of neglecting the dissipation function in the energy equation of most convection problems.

High‐order ADI method for stream‐function vorticity equations

AbstractMany recent works have demonstrated the efficiency of high‐order compact (HOC) difference schemes on the stream‐function and vorticity formulation of 2‐D incompressible Navier‐Stokes equations. HOC discretizations induce cross spatial derivatives which are treated explicitly in most ADI schemes. Recently, Karaa and Zhang proposed a fourth‐order ADImethod for solving convection‐diffusion problems efficiently. In this work, we extend this method to the solution of incompressible Navier‐Stokes equations. The driven flow in a square cavity is used as a model problem and numerical results are compared with other results. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Split step non-paraxial finite difference method for 3D scalar wave propagation

We describe a simple and efficient wide-angle, split-step finite-difference based explicit transfer matrix method to solve the 3D scalar wave equation. The formulation is completely analytic and does not involve any numerical matrix inversion or diagonalization. It also does not use the ADI scheme. The PML boundary condition can be easily implemented with only a marginal increase in computational effort.

An efficient mode-splitting method for a curvilinear nearshore circulation model

A mode-splitting method is applied to the quasi-3D nearshore circulation equations in generalized curvilinear coordinates. The gravity wave mode and the vorticity wave mode of the equations are derived using the two-step projection method. Using an implicit algorithm for the gravity mode and an explicit algorithm for the vorticity mode, we combine the two modes to derive a mixed difference–differential equation with respect to surface elevation. McKee et al.'s [McKee, S., Wall, D.P., and Wilson, S.K., 1996. An alternating direction implicit scheme for parabolic equations with mixed derivative and convective terms. J. Comput. Phys., 126, 64–76.] ADI scheme is then used to solve the parabolic-type equation in dealing with the mixed derivative and convective terms from the curvilinear coordinate transformation. Good convergence rates are found in two typical cases which represent respectively the motions dominated by the gravity mode and the vorticity mode. Time step limitations imposed by the vorticity convective Courant number in vorticity-mode-dominant cases are discussed. Model efficiency and accuracy are verified in model application to tidal current simulations in San Francisco Bight.

Stability analysis of uniform and non-uniform annular passages conducting incompressible laminar flows for small and large amplitude oscillatory motions of the outer cylinder

A computational method is developed involving the simultaneous integration of the Navier–Stokes and structural equations for the purpose of studying the stability of concentric annular passages conducting incompressible laminar flows. It is assumed that one side of the annulus, i.e. the centre-body, is fixed and the outer cylindrical duct is flexibly supported. The outer cylinder is displaced from its equilibrium position and is then released. In this situation, the fluid part of the problem is solved by an accurate method using a three-point backward implicit scheme, followed by a pseudo-time iteration using an artificial compressibility factor. The fluid equations are discretized in space based on a finite-difference formulation and primitive variables, for which stretched staggered grids are used. The resulting equations are cast in delta form and are solved using an ADI scheme. The fluid forces acting on the vibrating cylinder are calculated from the integration of the unsteady pressure and shear stresses resulting from the unsteady primitive variables calculated. The equations of motion of the structure, subjected to the calculated fluid forces are solved using the Runge–Kutta scheme to obtain the displacement of the moving cylinder. The problem is solved: (a) for small-amplitude motions, by means of the so-called mean position (MP) analysis and (b) for large amplitude oscillations of the outer cylinder for which a time-dependent coordinate transformation (TDCT) is used to fix the computational domain. Both these approaches (MP and TDCT) are applied to uniform and non-uniform (backstep-shaped) annuli for translational motion of the cylinder. The problem is also solved for (i) rocking motion and uniform annuli and (ii) translational motion for diffuser-shaped annuli, with only MP analysis. When the MP analysis is considered, it is shown that, for translational motion of the outer cylinder, the most stable configuration is that of a uniform geometry and the least stable one is the backward step geometry. For rocking motion in uniform annular geometry, for some system parameters and high enough flow velocity, the outer cylinder can develop flutter (limit-cycle oscillation). The comparison between the results of MP and TDCT analyses for uniform and backstep geometries for translational motion of the cylinder indicates that the outer cylinder is less stable when TDCT is used and the coupled frequency of oscillation is also changed considerably.