Direction Implicit Method Research Articles

HLI, A Direct Method Suitable for Partial and Fully Implicit Time Integration of Primitive Equation Meteorological Models

HLI, the homogeneously linearized implicit method is a direct implicit time integration method for nonlinear prognostic equations. For homogeneous linear test problems it is identical to the known implicit method, when applied to the full model area. As most theoretical results concerning the implicit method use linear homogeneous test problems, these apply immediately to HLI. The implicit method, when applied to the whole model area, is called the fully implicit method. In order to reduce the numerical expense for large integration areas, a partial implicit method is defined by posing artificial numerical boundary conditions, which result in a smaller set of equations, corresponding to the part of the integration area chosen. The application of either full or partial methods to nonlinear and inhomogeneous problems is achieved by defining for each grid point the local homogeneously linearized problem. Partial schemes are convenient when using multiprocessor computers, as only local and neighbouring data have to be used at each grid point. Only grid point discretisations of finite order are admitted. The proposed method would not be suitable for the spectral method. Computational examples are presented to show the suitability of this method for inhomogeneous situations and the ease of the treatment of internal boundary conditions. A semi-implicit treatment was also tested, which treats only the terms responsible for the fast waves implicit. This requires only the Fourier transformation of one field and in the generalised Fourier back transformation only one linear equation has to be solved, rather that a system of equations. In this paper only the trapezoidal implicit method was used. HLI could, however, be applied in the same way with other implicit methods.

A high accuracy variant of the iterative alternating decomposition explicit method for solving the heat equation

We consider three level difference replacements of parabolic equations focusing on the heat equation in two space dimensions. Through a judicious splitting of the approximation, the scheme qualifies as an alternating direction implicit (ADI) method. Using the well known fact of the parabolic-elliptic correspondence, we shall derive a two stage iterative procedure employing a fractional splitting strategy applied alternately at each intermediate time step to the one dimensional heat equation. As the basis of derivation is the unconditionally stable (4,2) accurate ADI scheme, this method is convergent, computationally stable and highly accurate.

Forced Convection in Micropolar Fluid Flow through a Wavy-Wall Channel

ABSTRACT Forced convection of micropolar fluids through a periodic array of wavy-wall channels has been analyzed by using a simple coordinate transformation method and the spline alternating- direction implicit method. The effects of the wavy amplitude, the micropolar parameter, and the Reynolds number on skin friction coefficient and Nusselt number have been examined in detail. Results show that the flow through a sinusoidally curved converging-diverging channel forms a strong forward flow and a reticular vortex within each wave for larger Reynolds number and larger wavy amplitudes. For the micropolar fluids, increasing the vortex viscosity causes an increase in the total viscosity of the fluid, thus the skin friction coefficient increases while the Nusselt number decreases. Also, the influence of vortex viscosity on the minimum cross section of the wavy-wall channel and on a tiny change of the maximum cross section is manifest. Moreover, both Reynolds number and wavy amplitude tend to enhance the total heat transfer rate, regardless of whether the fluids are Newtonian or micropolar fluids.

On coupling the Reynolds‐averaged Navier–Stokes equations with two‐equation turbulence model equations

AbstractTwo methods for coupling the Reynolds‐averaged Navier–Stokes equations with the q–ω turbulence model equations on structured grid systems have been studied; namely a loosely coupled method and a strongly coupled method. The loosely coupled method first solves the Navier–Stokes equations with the turbulent viscosity fixed. In a subsequent step, the turbulence model equations are solved with all flow quantities fixed. On the other hand, the strongly coupled method solves the Reynolds‐averaged Navier–Stokes equations and the turbulence model equations simultaneously. In this paper, numerical stabilities of both methods in conjunction with the approximated factorization‐alternative direction implicit method are analysed. The effect of the turbulent kinetic energy terms in the governing equations on the convergence characteristics is also studied. The performance of the two methods is compared for several two‐ and three‐dimensional problems. Copyright © 2005 John Wiley & Sons, Ltd.

On the solution of Poisson equation by alternating group explicit iterative method

In this paper the alternating group explicit (AGE) iterative methods which are modified analogues of the classical alternating direction implicit (ADI) methods, are introduced. Poisson equation is transformed a linear system using suitably finite difference equations and also the solution of this system is done with AGE methods. Numerical extraordinary results are obtained by using Maple V computer programming.

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.

On the numerical solution of Helmholtz equation by alternating group explicit (AGE) methods

In this paper the alternating group explicit (AGE) iterative methods which is modified analogues of the classical alternating direction implicit (ADI) methods, are introduced. Helmholtz equation is transformed a linear system using by suitably finite difference equations and also the solution of this system is done with AGE methods. Numerical striking results are obtained by using Maple V computer programming.

THERMOSOLUTAL TRANSFER WITHIN TRAPEZOIDAL CAVITY

The heat and mass transfer problem in a trapezoidal cavity is treated in this article. The lower part of the cavity is heated and the top inclined part is cooled. Phenomenological equations are solved using the alternating direction implicit (ADI) method combined with a fourth-order compact Hermitian method. The results are compared to those obtained experimentally and numerically by other authors in the triangular and trapezoidal cavity cases. The thermoconvective instabilities obtained are similar to those obtained in rectangular cavities. The influence of geometric parameters, global solicitations, and Lewis numbers on fluid flow configurations and on heat and mass transfer ratios is also studied.

COMPUTATION OF TRANSIENT NATURAL CONVECTION IN A SQUARE CAVITY BY AN IMPLICIT FINITE-DIFFERENCE SCHEME IN TERMS OF THE STREAM FUNCTION AND TEMPERATURE

In this article we discuss the application of an implicit scheme to the solution by finite differences of transient natural convection in terms of the stream function and temperature. The second-order energy differential and fourth-order momentum equations are discretized according to the well-known alternate direction implicit (ADI) method. Then the temperature field solution is built based on the classic tridiagonal matrix algorithm (TDMA), and the stream function solution is built based on the two original hypotheses proposed here together with the penta diagonal matrix algorithm (PDMA). The time step in the algorithm is obtained analytically as a function of the Prandtl number and the size of the grid imposing the diagonal dominance condition in the pentadiagonal matrix generated by the hypothesis. We verify the hypothesis and the algorithm using the transient natural convection solution in the following parameter ranges: 10 3 r Ra r 10 6 , 10 -2 r Pr r 10 2 , 21 2 21 r grid r 61 2 61. The transient solution to the problem is presented using the average Nusselt number and local Nusselt numbers on the hot wall, CPU time, the temperature and velocity profiles in the y = 0.5 section, and the temperature in the central point of the cavity. The results of the permanent solution are presented using the maximum value of the stream function within the domain and the average Nusselt number on the cold wall as a function of the Rayleigh number, the relaxation parameter, Prandtl number, and the size of the grid. Finally, both the transient and permanent solution results are compared with the results of five other published studies.

ON THE APPLICATION OF ADI METHODS TO PREDICT CONJUGATE PHASE CHANGE AND DIFFUSION HEAT TRANSFER

In this article, the process of phase-change heat transfer is investigated. To simplify the representation of the interface between phases, an enthalpy method is implemented to describe the energy conservation process. This equation describes an equivalence of the time rate of change of the continuous-phase quality x to the diffusive energy transport via a Laplacian of the continuous temperature field T. Because of the mixture of terms (x , T) , relaxation techniques are usually employed to solve the discrete energy equation. In the present study these equations are extended for implementation of an alternating direction implicit (ADI) method with the main goal of decreasing computational wall clock time. The phase quality formulation allows a discretized model to possess both phase-change nodes and solid diffusion nodes, the former defaulting to basic diffusion above and below the melt point of the material in question. In the present study, one set of equations is used to represent all thermal nodes. By relating the temperature to the phase quality as a linear function of x, the discretized equations are converted to the matrix form Ax = f, where x is the phase quality vector and is of a form suitable for ADI implementation. This produces the sought-after conversion from a temperature solution to a quality-based solution. The sweep equations are developed for both the 3-D splitting and 3-D Brian ADI methods. Because the overall heat transfer process involves transient solidification or melting, the time-space variations in the quality field cause the diagonals of A to change in time, forcing an additional LDU decomposition at each time step. These incremental expenses of an ADI implementation can be significant; however, for the case study presented, the overall ADI technique is nearly 30 times faster than conventional relaxation methods. The current method is validated against analytical solutions of 2-D flat and cylindrical solidification front propagation. A pseudo-code is presented to assist the reader in algorithmic implementation.

Numerical Solution of 2D Free Surface Flow to Ditch Drains in Anisotropic Soils

A numerical solution of two-dimensional free surface flow to ditch drains in homogeneous and anisotropic soils is presented. The differential equation governing two dimensional ground-water flow in anisotropic soils is solved by the alternating direction implicit (ADI) method of finite difference. In the approach the method is modified to make it applicable for curved boundaries. This simplified approach does not involve any kind of smoothing or linearization technique. The model solutions for ditch drainage with constant replenishment and instantaneous drawdown are found to be in close agreement with the finite-difference and finite-element solutions proposed by other researchers.

Snake pedals: compact and versatile geometric models with physics-based control

We introduce a geometric shape modeling scheme which allows for representation of global and local shape characteristics of an object. Geometric models are well-suited for representing global shapes without local detail, but we propose a scheme which represents global shapes with local detail and permits model shaping as well as topological changes via physics-based control. The scheme represents shapes by pedal curves and surfaces, i.e. the loci of the foot of perpendiculars to the tangents of a fixed curve/surface from a fixed point called the pedal point. By varying the location of the pedal point, one can synthesize a large class of shapes which exhibit both local and global deformations. We introduce physics-based control for shaping these geometric models by letting the pedal point vary and use a snake to represent the position of this varying point. The model, a "snake pedal", allows for interactive manipulation via forces applied to the snake. We develop a fast numerical iterative algorithm for shape recovery from image data using this scheme. The algorithm involves the Levenberg-Marquardt (LM) method in the outer loop for solving the global parameters and the alternating direction implicit (ADI) method in the inner loop for solving the local parameters of the model. The combination of the global and local scheme leads to an efficient numerical solution to the model fitting problem. We demonstrate the applicability of this modeling scheme via examples of shape synthesis and shape estimation from real image data.

Numerical methods for conjunctive two-dimensional surface and three-dimensional sub-surface flows

SUMMARY Sophisticated catchment runoff problems necessitate conjunctive modeling of overland flow and subsurface flow. In this paper, finite difference numerical methods are studied for simulation of catchment runoff of two-dimensional surface flow interacting with three-dimensional unsaturated and saturated sub-surface flows. The equations representing the flows are mathematically classified as a type of heat diffusion equation. Therefore, two- and three-dimensional numerical methods for heat diffusion equations were investigated for applications to the surface and sub-surface flow sub-models in terms of accuracy, stability, and calculation time. The methods are the purely explicit method, Saul’yev’s methods, the alternating direction explicit (ADE) methods, and the alternating direction implicit (ADI) methods. The methods are first examined on surface and sub-surface flows separately; subsequently, 12 selected combinations of methods were investigated for modeling the conjunctive flows. Saul’yev’s downstream (S-d) method was found to be the preferred method for two-dimensional surface flow modeling, whereas the ADE method of Barakat and Clark is a less accurate, stable alternative. For the three-dimensional sub-surface flow model, the ADE method of Larkin (ADE-L) and Brian’s ADI method are unconditionally stable and more accurate than the other methods. The calculations of the conjunctive models utilizing the S-d surface flow sub-model give excellent results and confirm the expectation that the errors of the surface and sub-surface sub-models interact. The surface sub-model dominates the accuracy and stability of the conjunctive model, whereas the sub-surface sub-model dominates the calculation time, suggesting the desirability of using a smaller time increment for the surface sub-model. Copyright © 2000 John Wiley & Sons, Ltd.

Comments on "Investigation of 3D semivectorial finite-difference beam propagation method for bent waveguides"

The technique used to model curved waveguides described in the above paper is analyzed. Both the model and alternating direction implicit (ADI) method used to implement it are discussed.

Numerical simulation of sheath plasma by the particle method taking account of coulomb collisions

We study the problem of the behavior of a plasma bounded longitudinally by an absorbing sheath. This model contains charged particles (electrons and ions) moving subject to a self-consistent electrostatic field. New particle pairs are generated in the region of a distributed source. As a numerical model we used the electrostatic “particle-in-cell” method supplemented by the Emmert model for a bulk source and the algorithm of binary Coulomb collisions using the Monte Carlo method. We give a mathematical statement of the problem. The computations were carried out using the direct implicit method with the “explicit limit” time step. The results of numerical simulation of this system are given. We consider the formation and evoluiton of potential structures (multiple weak nonmonotonic double layers). Five figures. Bibliography: 35 titles.

Characterization of the Darwin Direct Implicit Particle-in-Cell Method and Resulting Guidelines for Operation

We investigate the linear dispersion and other properties of the Darwin Direct Implicit Particle-in-cell (DADIPIC) method in order to deduce guidelines for its use in the simulation of long time-scale, kinetic phenomena in plasmas. The Darwin part of this algorithm eliminates the Courant constraint for light propagation across a grid cell in a time step and divides the field solution into several elliptic equations. The direct implicit method is only applied to the electrostatic field relieving the need to resolve plasma oscillations. Linear theory and simulations verifying the theory are used to generate the desired guidelines as well as show the utility of DADIPIC for a wide range of low frequency, electromagnetic phenomena. We find that separation of the fields has made the task of predicting algorithm behavior easier and produced a robust method without restrictive constraints.

Interstage Disk-Cavity/Brush Seal Numerical Flow Visualization Study

This paper presents work-in-progress results and model developments that are directed towards the computational fluid dynamics simulation of the secondary flow system in a gas turbine engine. Numerical flow visualization results of the 2-D axisymmetric rotational fluid flow (Re=105) in a generic cross section of an interstage turbine cavity are displayed. The core flow is driven by an imposed pressure drop along the vane row of the main flow path. Three interconnected disk cavities separated by a brush seal are located along the secondary flow path. A new computational algorithm was developed in order to predict the flow patterns and leakages for different seal and cavity configurations. The code is based on the numerical solution of the transient laminar Navier-Stokes equations, written in primitive variables and approximated on a nonuniform rectangular collocated grid. The program uses a mass and momentum conservative formulation as well as a set of boundary conditions for pressure and conservation of mass. The pressure solution uses the Poisson equation under a direct implementation procedure. A time dependent Alternating Direction Implicit (ADI) method is also used for the solution of all primitive variables. This integrated computational approach allows the simulation of the rotating disk cavity vortex flow periodic features, as well as their impact on leakage flow and sealing effectiveness. In the present study this approach is used to numerically investigate the flow through the brush. The main purpose of this study is to present an integrated brush seal/disk cavity flow model, to demonstrate the effectiveness of this approach, and to emphasize the necessity for further work in this direction.Copyright © 1995 by ASME

The Darwin Direct Implicit Particle-in-Cell (DADIPIC) Method for Simulation of Low Frequency Plasma Phenomena

We describe a new algorithm for simulating low frequency, kinetic phenomena in plasmas. Darwin direct implicit particle-in-cell (DADIPIC), as its name implies, is a combination of the Darwin and direct implicit methods. Through the Darwin method the hyperbolic Maxwell's equations are reformulated into a set of elliptic equations. Propagating light waves do not exist in the formulation so the Courant constraint on the time step is eliminated. The direct implicit method is applied only to the electrostatic field with the result that electrostatic plasma oscillations do not have to be resolved for stability. With the elimination of these constraints spatial and temporal discretization can be much larger than that possible with explicit, electrodynamic PIC. We discuss the algorithms for pushing the particles and solving for the fields in 2D cartesian geometry. We also detail boundary conditions for conductors and dielectrics. Finally, we present two test cases, electron cyclotron waves and collisionless heating in inductively coupled plasmas. For these test cases DADIPIC shows agreement with analytic kinetic theory and good energy conservation characteristics.

The Alternating Direction Multi-zone Implicit Method

Within the structured grid approach, numerical solution of partial differential equations (PDE) in complex regions requires the decomposition of the domain into several zones. Implicit solution methods of the discrete equations are preferred because of their superior numerical properties. Indeed, many existing multi-zone solution methods use implicit techniques, but the zonal boundaries are updated explicitly. The zonal boundaries are created between the zones in the process of domain decomposition, but otherwise they are regular interior field points. If only steady state solutions are sought, the explicit calculation of the zonal boundaries usually effects only the convergence rate but not the accuracy of the solution. However, in time-dependent cases this may degrade the time accuracy of the solution. In the present work, we propose a novel fully implicit method for solving sets of PDE using multi-zones and structured grids. The method combines the zonal approach with the alternating direction implicit (ADI) method, and hence the method is referred to as the alternating direction multi-zone implicit (ADMZI) method. The key idea is the generation of different sets of zones for each stage (factor) of the ADI method, rather than using the same set of zones for all the stages. Consequently, the ADI sweeps are performed between physical boundaries, so that zonal boundaries are avoided and the solution is fully implicit. The ADMZI method can be applied to any set of PDE that employs an ADI (or approximate factorization) method. Typical examples include the time-dependent compressible or incompressible Navier-Stokes equations. Several conceptual examples and actual numerical test cases (that solve the heat conduction equation) confirm the versatility, efficiency, and accuracy of the ADMZI method.

The acceleration of the adi method by sor

In this paper, the alternating direction implicit (ADI) method is considered for solving linear systems of equations. It is shown that the rate of convergence of the ADI method can be accelerated by the successive over relaxation (SOR) method in the commutative case. Theoretical results demonstrate that the accelerated procedure, called the ADI-SOR method, results in an order of magnitude improvement. For the model problem, it is shown that the rate of convergence for the ADI-SOR scheme is asymptotically proportional to h 1/2m (where h is the mesh size and m is the number of parameters for the ADI method), while for the ADI method the rate of convergence is asymptotically proportional to h 1/m. It is also shown that the ADI-SOR method is faster than the ADI scheme with 2 m parameters.