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Article
Publication date: 19 June 2007

Mehdi Dehghan

The diffusion‐advection phenomena occur in many physical situations such as, the transport of heat in fluids, flow through porous media, the spread of contaminants in…

Abstract

Purpose

The diffusion‐advection phenomena occur in many physical situations such as, the transport of heat in fluids, flow through porous media, the spread of contaminants in fluids and as well as in many other branches of science and engineering. So it is essential to approximate the solution of these kinds of partial differential equations numerically in order to investigate the prediction of the mathematical models, as the exact solutions are usually unavailable.

Design/methodology/approach

The difficulties arising in numerical solutions of the transport equation are well known. Hence, the study of transport equation continues to be an active field of research. A number of mathematicians have developed the method of time‐splitting to divide complicated time‐dependent partial differential equations into sets of simpler equations which could then be solved separately by numerical means over fractions of a time‐step. For example, they split large multi‐dimensional equations into a number of simpler one‐dimensional equations each solved separately over a fraction of the time‐step in the so‐called locally one‐dimensional (LOD) method. In the same way, the time‐splitting process can be used to subdivide an equation incorporating several physical processes into a number of simpler equations involving individual physical processes. Thus, instead of applying the one‐dimensional advection‐diffusion equation over one time‐step, it may be split into the pure advection equation and the pure diffusion equation each to be applied over half a time‐step. Known accurate computational procedures of solving the simpler diffusion and advection equations may then be used to solve the advection‐diffusion problem.

Findings

In this paper, several different computational LOD procedures were developed and discussed for solving the two‐dimensional transport equation. These schemes are based on the time‐splitting finite difference approximations.

Practical implications

The new approach is simple and effective. The results of a numerical experiment are given, and the accuracy are discussed and compared.

Originality/value

A comparison of calculations with the results of the conventional finite difference techniques demonstrates the good accuracy of the proposed approach.

Details

Kybernetes, vol. 36 no. 5/6
Type: Research Article
ISSN: 0368-492X

Keywords

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Article
Publication date: 18 May 2010

K.B. Dada and E. Momoniat

The purpose of this paper is to derive a dynamic equation for modelling the behaviour of smectic‐C liquid crystals under the effect of an electric field.

Abstract

Purpose

The purpose of this paper is to derive a dynamic equation for modelling the behaviour of smectic‐C liquid crystals under the effect of an electric field.

Design/methodology/approach

The model equation is solved using a finite difference approximation, method of lines and pseudo‐spectral methods. The solutions are compared for accuracy and efficiency. Comparison is made of the efficiency of finite differences, method of lines and pseudo‐spectral methods.

Findings

The Fourier pseudo‐spectral method is shown to be the most efficient approach.

Originality/value

This work is original; a computational comparison of numerical schemes applied to liquid crystals has not been found in the literature.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 20 no. 4
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 21 October 2019

Cem Civelek

The purpose of this study is the application of the following concepts to the time discrete form. Variational Calculus, potential and kinetic energies, velocity…

Abstract

Purpose

The purpose of this study is the application of the following concepts to the time discrete form. Variational Calculus, potential and kinetic energies, velocity proportional Rayleigh dissipation function, the Lagrange and Hamilton formalisms, extended Hamiltonians and Poisson brackets are all defined and applied for time-continuous physical processes. Such processes are not always time-continuously observable; they are also sometimes time-discrete.

Design/methodology/approach

The classical approach is developed with the benefit of giving only a short table on charge and flux formulation, as they are similar to the classical case just like all other formulation types. Moreover, an electromechanical example is represented as well.

Findings

Lagrange and Hamilton formalisms together with the velocity proportional (Rayleigh) dissipation function can also be used in the discrete time case, and as a result, dissipative equations of generalized motion and dissipative canonical equations in the discrete time case are obtained. The discrete formalisms are optimal approaches especially to analyze a coupled physical system which cannot be observed continuously. In addition, the method makes it unnecessary to convert the quantities to the other. The numerical solutions of equations of dissipative generalized motion of an electromechanical (coupled) system in continuous and discrete time cases are presented.

Originality/value

The formalisms and the velocity proportional (Rayleigh) dissipation function aforementioned are used and applied to a coupled physical system in time-discrete case for the first time to the best of the author’s knowledge, and systems of difference equations are obtained depending on formulation type.

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Article
Publication date: 17 June 2008

Mehdi Dehghan and Reza Mazrooei‐Sebdani

The aim of this paper is to study the global behavior of a higher order nonlinear difference equation.

Abstract

Purpose

The aim of this paper is to study the global behavior of a higher order nonlinear difference equation.

Design/methodology/approach

The character of semicycles, periodicity, global stability and boundedness of positive solutions of the mentioned difference equation are investigated.

Findings

It is deduced that, when q<1, then the positive equilibrium of this equation is globally asymptotically stable and, when q>1, then this difference equation possesses unbounded solutions.

Originality/value

The paper investigates some of the characteristics of nonlinear difference equations.

Details

Kybernetes, vol. 37 no. 6
Type: Research Article
ISSN: 0368-492X

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Article
Publication date: 14 June 2011

Allaberen Ashyralyev and Betul Hicdurmaz

The purpose of this paper is to introduce stability analysis for the initial value problem for the fractional Schrödinger differential equation: Equation 1 in a Hilbert…

Abstract

Purpose

The purpose of this paper is to introduce stability analysis for the initial value problem for the fractional Schrödinger differential equation: Equation 1 in a Hilbert space H with a self‐adjoint positive definite operator A under the condition |α(s)|<M1/s1/2 and the first order of accuracy difference scheme for the approximate solution of this initial value problem.

Design/methodology/approach

The paper considers the stability estimates for the solution of this problem and the stability estimate for the approximate solution of first order of accuracy difference scheme of this problem.

Findings

The paper finds the stability for the fractional Schrödinger differential equation and the first order of accuracy difference scheme of that equation by applications to one‐dimensional fractional Schrödinger differential equation with nonlocal boundary conditions and multidimensional fractional Schrödinger differential equation with Dirichlet conditions.

Originality/value

The paper is a significant work on stability of fractional Schrödinger differential equation and its difference scheme presenting some numerical experiments which resulted from applying obtained theorems on several mixed fractional Schrödinger differential equations.

Details

Kybernetes, vol. 40 no. 5/6
Type: Research Article
ISSN: 0368-492X

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Article
Publication date: 18 September 2009

Yuri N. Skiba and Denis M. Filatov

The purpose of this paper is to suggest a new approach to the numerical simulation of shallow‐water flows both in plane domains and on the sphere.

Abstract

Purpose

The purpose of this paper is to suggest a new approach to the numerical simulation of shallow‐water flows both in plane domains and on the sphere.

Design/methodology/approach

The approach involves the technique of splitting of the model operator by geometric coordinates and by physical processes. Specially chosen temporal and spatial approximations result in one‐dimensional finite difference schemes that conserve the mass and the total energy. Therefore, the mass and the total energy of the whole two‐dimensional split scheme are kept constant too.

Findings

Explicit expressions for the schemes of arbitrary approximation orders in space are given. The schemes are shown to be mass‐ and energy‐conserving, and hence absolutely stable because the square root of the total energy is the norm of the solution. The schemes of the first four approximation orders are then tested by simulating nonlinear solitary waves generated by a model topography. In the analysis, the primary attention is given to the study of the time‐space structure of the numerical solutions.

Originality/value

The approach can be used for the numerical simulation of shallow‐water flows in domains of both Cartesian and spherical geometries, providing the solution adequate from the physical and mathematical standpoints in the sense of keeping its mass and total energy constant even when fully discrete shallow‐water models are applied.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 19 no. 8
Type: Research Article
ISSN: 0961-5539

Keywords

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Book part
Publication date: 11 July 2019

Daniel Dench and Michael Grossman

In this chapter, we investigate two-way causality between health and the hourly wage. We employ insights from the human capital and compensating wage differential models…

Abstract

In this chapter, we investigate two-way causality between health and the hourly wage. We employ insights from the human capital and compensating wage differential models, a panel formed from the National Longitudinal Survey of Youth 1997, and dynamic panel estimation methods in this investigation. We adopt plausible specifications in which a change in health induces a change in the wage with a lag and a change in the wage induces a change in health, also with a lag. We uncover a causal relationship between two of the five measures of health and the wage in which a reduction in health leads to an increase in the wage rate in a panel of US young adults who had completed their formal schooling by 2006 and were continuously employed from that year through 2011. There is no evidence of a causal relationship running from the wage rate to health in this panel. The former result highlights the multidimensional nature of health. It is consistent with an extension of the compensating wage differential model in which a large amount of effort in one period is required to obtain promotions and the wage increases that accompany them in subsequent periods. That effort may cause reductions in health and to a negative effect of health in the previous period on the current period wage. In this framework, employees have imperfect information about the effort requirements of a particular job when they are hired, and employers have imperfect information about the amount of effort new hirers are willing to exert. The result is also consistent with a model in which investments in career advancement compete with investments in health for time – the ultimate scarce resource. The lack of a causal effect of the wage on health may suggest that forces that go in opposite directions in the human capital and compensating wage differential models offset each other.

Details

Health and Labor Markets
Type: Book
ISBN: 978-1-78973-861-2

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Article
Publication date: 17 September 2008

Mehdi Dehghan and Akbar Mohebbi

The purpose of this paper is to introduce efficient methods for solving the 2D biharmonic equation with Dirichlet boundary conditions of second kind. This equation

Abstract

Purpose

The purpose of this paper is to introduce efficient methods for solving the 2D biharmonic equation with Dirichlet boundary conditions of second kind. This equation describes the deflection of loaded plate with boundary conditions of simply supported plate kind. Also it can be derived from the calculus of variations combined with the variational principle of minimum potential energy. Because of existing fourth derivatives in this equation, introducing high‐order accurate methods need to use artificial points. Also solving the resulted linear system of equations suffers from slow convergence when iterative methods are used. This paper aims to introduce efficient methods to overcome these problems.

Design/methodology/approach

The paper considers several compact finite difference approximations that are derived on a nine‐point compact stencil using the values of the solution and its second derivatives as the unknowns. In these approximations there is no need to define special formulas near the boundaries and boundary conditions can be incorporated with these techniques. Several iterative linear systems solvers such as Krylov subspace and multigrid methods and their combination (with suitable preconditioner) have been developed to compare the efficiency of each method and to design powerful solvers.

Findings

The paper finds that the combination of compact finite difference schemes with multigrid method and Krylov iteration methods preconditioned by multigrid have excellent results for the second biharmonic equation, and that Krylov iteration methods preconditioned by multigrid are the most efficient methods.

Originality/value

The paper is of value in presenting, via some tables and figures, some numerical experiments which resulted from applying new methods on several test problems, and making comparison with conventional methods.

Details

Kybernetes, vol. 37 no. 8
Type: Research Article
ISSN: 0368-492X

Keywords

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Article
Publication date: 2 November 2010

Mohamed Rady, Eric Arquis, Dominique Gobin and Benoît Goyeau

This paper aims to tackle the problem of thermo‐solutal convection and macrosegregation during ingot solidification of metal alloys. Complex flow structures associated…

Abstract

Purpose

This paper aims to tackle the problem of thermo‐solutal convection and macrosegregation during ingot solidification of metal alloys. Complex flow structures associated with the development of channels segregate and sharp gradients in the solutal field call for the implementation of accurate methods for numerical modeling of alloy solidification. In particular, the solute transport equation is convection dominated and requires special non‐oscillarity type high‐order schemes to handle the regions of channels segregates.

Design/methodology/approach

In the present study, a time‐splitting approach has been adopted to separately handle solute advection and diffusion. This splitting technique allows the application of accurate total variation dimensioning (TVD) schemes for solution of solute advection. Applications of second‐order Lax‐Wendroff TVD SUPERBEE and fifth‐order weighted essentially non‐oscillatory (WENO) schemes are described in the present article. Classical numerical solution of solute transport using hybrid and central‐difference schemes are also employed for the purpose of comparisons. Numerical simulations for solidification of Pb‐18%Sn in a two‐dimensional rectangular cavity have been carried out using different numerical schemes.

Findings

Numerical results show the difficulty of obtaining grid‐independent solutions with respect to local details in the region of channels. Grid convergence patterns and numerical uncertainty are found to be dependent on the applied scheme. In general, the first‐order hybrid scheme is diffusive and under predicts the formation of channels. The second‐order central‐difference scheme brings about oscillations with possible non‐physical extremes of solute composition in the region of channel segregates due to sharp gradients in the solutal field. The results obtained using TVD and WENO schemes contain no oscillations and show an excellent capture of channels formation and resolution of the interface between solute‐rich and depleted bands. Different stages of channels formation are followed by analyzing thermo‐solutal convection and macrosegregation at different times during solidification.

Research limitations/implications

Accurate prediction of local variation in the solutal and flow fields in the channels regions requires grid refinement up to scales in the order of microscopic dendrite arm spacing. This imposes limitations in terms of large computational time and applicability of available macroscopic models based on classical volume‐averaging techniques.

Practical implications

The present study is very useful for numerical simulation of macrosegregation during ingot casting of metal alloys.

Originality/value

The paper provides the methodology and application of TVD schemes to predict channel segregates during columnar solidification of metal alloys. It also demonstrates the limitations of classical schemes for simulation of alloy solidification.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 20 no. 8
Type: Research Article
ISSN: 0961-5539

Keywords

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Article
Publication date: 1 August 2006

Weizhong Dai, Hertong Song, Shengjun Su and Raja Nassar

To develop a numerical method for solving hyperbolic two‐step micro heat transport equations, which have attracted attention in thermal analysis of thin metal films…

Abstract

Purpose

To develop a numerical method for solving hyperbolic two‐step micro heat transport equations, which have attracted attention in thermal analysis of thin metal films exposed to ultrashort‐pulsed lasers.

Design/methodology/approach

An energy estimation for the hyperbolic two‐step model in a three‐dimensional (3D) micro sphere irradiated by ultrashort‐pulsed lasers is first derived, and then a finite difference scheme for solving the hyperbolic two‐step model based on the energy estimation is developed. The scheme is shown to be unconditionally stable and satisfies a discrete analogue of the energy estimation. The method is illustrated by investigating the heat transfer in a micro gold sphere exposed to ultrashort‐pulsed lasers.

Findings

Provides information on normalized electron temperature change with time on the surface of the sphere, and shows the changes in electron and lattice temperatures.

Research limitations/implications

The hyperbolic two‐step model is considered under the assumption of constant thermal properties.

Practical implications

A useful tool to investigate the temperature change in a micro sphere irradiated by ultrashort‐pulsed lasers.

Originality/value

Provides a new unconditionally stable finite difference scheme for solving the hyperbolic two‐step model in a 3D micro sphere irradiated by ultrashort‐pulsed lasers.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 16 no. 6
Type: Research Article
ISSN: 0961-5539

Keywords

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