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Article
Publication date: 5 May 2015

Martin Joseph Guillot and Steve C McCool

The purpose of this paper is to investigate the effect of numerical boundary condition implementation on local error and convergence in L2-norm of a finite volume discretization…

Abstract

Purpose

The purpose of this paper is to investigate the effect of numerical boundary condition implementation on local error and convergence in L2-norm of a finite volume discretization of the transient heat conduction equation subject to several boundary conditions, and for cases with volumetric heat generation, using both fully implicit and Crank-Nicolson time discretizations. The goal is to determine which combination of numerical boundary condition implementation and time discretization produces the most accurate solutions with the least computational effort.

Design/methodology/approach

The paper studies several benchmark cases including constant temperature, convective heating, constant heat flux, time-varying heat flux, and volumetric heating, and compares the convergence rates and local to analytical or semi-analytical solutions.

Findings

The Crank-Nicolson method coupled with second-order expression for the boundary derivatives produces the most accurate solutions on the coarsest meshes with the least computation times. The Crank-Nicolson method allows up to 16X larger time step for similar accuracy, with nearly negligible additional computational effort compared with the implicit method.

Practical implications

The findings can be used by researchers writing similar codes for quantitative guidance concerning the effect of various numerical boundary condition approximations for a large class of boundary condition types for two common time discretization methods.

Originality/value

The paper provides a comprehensive study of accuracy and convergence of the finite volume discretization for a wide range of benchmark cases and common time discretization methods.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 25 no. 4
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 June 2002

Patrick Dular and Patrick Kuo‐Peng

An efficient and robust time discretization procedure of theta type is proposed in the frame of the finite element‐circuit equation coupling for electronic circuits with switches…

Abstract

An efficient and robust time discretization procedure of theta type is proposed in the frame of the finite element‐circuit equation coupling for electronic circuits with switches, i.e. the use of diodes, thyristors and transistors. This procedure enables the use of the Crank‐Nicolson scheme whatever the circuit and its working conditions by eliminating the undesirable oscillations of the solution peculiar to this scheme. It is based on the accurate determination of the switching instants and on a local modification of the theta parameter.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol. 21 no. 2
Type: Research Article
ISSN: 0332-1649

Keywords

Article
Publication date: 29 July 2021

A. A. Alanazi, Sultan Z. Alamri, S. Shafie and Shazirawati Mohd Puzi

The purpose of this paper is to obtain the nonlinear Schrodinger equation (NLSE) numerical solutions in the presence of the first-order chromatic dispersion using a second-order…

Abstract

Purpose

The purpose of this paper is to obtain the nonlinear Schrodinger equation (NLSE) numerical solutions in the presence of the first-order chromatic dispersion using a second-order, unconditionally stable, implicit finite difference method. In addition, stability and accuracy are proved for the resulting scheme.

Design/methodology/approach

The conserved quantities such as mass, momentum and energy are calculated for the system governed by the NLSE. Moreover, the robustness of the scheme is confirmed by conducting various numerical tests using the Crank-Nicolson method on different cases of solitons to discuss the effects of the factor considered on solitons properties and on conserved quantities.

Findings

The Crank-Nicolson scheme has been derived to solve the NLSE for optical fibers in the presence of the wave packet drift effects. It has been founded that the numerical scheme is second-order in time and space and unconditionally stable by using von-Neumann stability analysis. The effect of the parameters considered in the study is displayed in the case of one, two and three solitons. It was noted that the reliance of NLSE numeric solutions properties on coefficients of wave packets drift, dispersions and Kerr nonlinearity play an important control not only the stable and unstable regime but also the energy, momentum conservation laws. Accordingly, by comparing our numerical results in this study with the previous work, it was recognized that the obtained results are the generalized formularization of these work. Also, it was distinguished that our new data are regarding to the new communications modes that depend on the dispersion, wave packets drift and nonlinearity coefficients.

Originality/value

The present study uses the first-order chromatic. Also, it highlights the relationship between the parameters of dispersion, nonlinearity and optical wave properties. The study further reports the effect of wave packet drift, dispersions and Kerr nonlinearity play an important control not only the stable and unstable regime but also the energy, momentum conservation laws.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 31 no. 8
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 25 February 2022

Yazhou Wang, Ningning Xie, Likun Yin, Tong Zhang, Xuelin Zhang, Shengwei Mei, Xiaodai Xue and Kumar Tamma

The purpose of this paper is to describe a novel universal error estimator and the adaptive time-stepping process in the generalized single-step single-solve (GS4-1) computational…

Abstract

Purpose

The purpose of this paper is to describe a novel universal error estimator and the adaptive time-stepping process in the generalized single-step single-solve (GS4-1) computational framework, applied for the fluid dynamics with illustrations to incompressible Navier–Stokes equations.

Design/methodology/approach

The proposed error estimator is universal and versatile that it works for the entire subsets of the GS4-1 framework, encompassing the nondissipative Crank–Nicolson method, the most dissipative backward differential formula and anything in between. It is new and novel that the cumbersome design work of error estimation for specific time integration algorithms can be avoided. Regarding the numerical implementation, the local error estimation has a compact representation that it is determined by the time derivative variables at four successive time levels and only involves vector operations, which is simple for numerical implementation. Additionally, the adaptive time-stepping is further illustrated by the proposed error estimator and is used to solve the benchmark problems of lid-driven cavity and flow past a cylinder.

Findings

The proposed computational procedure is capable of eliminating the nonphysical oscillations in GS4-1(1,1)/Crank–Nicolson method; being CPU-efficient in both dissipative and nondissipative schemes with better solution accuracy; and detecting the complex physics and hence selecting a suitable time step according to the user-defined error threshold.

Originality/value

To the best of the authors’ knowledge, for the first time, this study applies the general purpose GS4-1 family of time integration algorithms for transient simulations of incompressible Navier–Stokes equations in fluid dynamics with constant and adaptive time steps via a novel and universal error estimator. The proposed computational framework is simple for numerical implementation and the time step selection based on the proposed error estimation is efficient, benefiting to the computational expense for transient simulations.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 32 no. 10
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 16 June 2010

M.H. Hojjati and H. Tari

The purpose of this paper is to show how a system of differential equations of one‐dimensional transient cooling heat conduction of different multi‐layer slabs has been solved…

Abstract

Purpose

The purpose of this paper is to show how a system of differential equations of one‐dimensional transient cooling heat conduction of different multi‐layer slabs has been solved numerically. A simple deterministic filtering matrix has been developed to remove errors involved in the experimental temperature measurements.

Design/methodology/approach

The system of differential equations is solved through Crank‐Nicolson method using a developed computer code. The developed matrix is based on the available information about the system and is strong enough to detect and remove errors from the measured temperatures.

Findings

The filtering algorithm is very straightforward and easy to implement and needs to be developed once for a given system.

Originality/value

This paper shows how the governing equations of transient heat conduction of multi‐layer slabs have been solved through Crank‐Nicolson method using a developed computer code. The code is user friendly and solves a large system of simultaneous differential equations for any given composite slab configurations. Furthermore, a matrix filter has been suggested to remove experimental errors. The filter is based on the available information about the considered system. The developed matrix filter is shown to be a powerful technique to detect the noisy data and correct them while intelligent enough not to harm good data. The filtering algorithm is very straightforward and easy to implement and needs to be developed once for a given system.

Details

Multidiscipline Modeling in Materials and Structures, vol. 6 no. 1
Type: Research Article
ISSN: 1573-6105

Keywords

Article
Publication date: 23 March 2012

E. Momoniat and C. Harley

The purpose of this paper is to obtain numerical solutions of a two‐dimensional mixed space‐time PDE modelling the flow of a second‐grade.

Abstract

Purpose

The 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/approach

The 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.

Findings

For 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/value

The use of a Peaceman‐Rachford ADI scheme to solve a mixed space‐time PDE is both novel and new.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 22 no. 2
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 June 2006

Mile R. Vujičić

To provide an analysis of transient heat conduction, which is solved using different iterative solvers for graduate and postgraduate students (researchers) which can help them…

1679

Abstract

Purpose

To provide an analysis of transient heat conduction, which is solved using different iterative solvers for graduate and postgraduate students (researchers) which can help them develop their own research.

Design/methodology/approach

Three‐dimensional transient heat conduction in homogeneous materials using different time‐stepping methods such as finite difference (Θ explicit, implicit and Crank‐Nicolson) and finite element (weighted residual and least squared) methods. Iterative solvers used in the paper are conjugate gradient (CG), preconditioned gradient, least square CG, conjugate gradient squared (CGS), preconditioned CGS, bi‐conjugate gradient (BCG), preconditioned BCG, bi‐conjugate gradient stabilized (BCGSTAB), reconditioned BCGSTAB and Gaussian elimination with incomplete Cholesky factorization.

Findings

Provides information on which time‐stepping method is the most accurate, which solver is the fastest to solve a symmetric and positive system of linear matrix equations of all those considered.

Practical implications

Fortran 90 code given as an abstract can be very useful for graduate and postgraduate students to develop their own code.

Originality/value

This paper offers practical help to an individual starting his/her research in the finite element technique and numerical methods.

Details

Engineering Computations, vol. 23 no. 4
Type: Research Article
ISSN: 0264-4401

Keywords

Article
Publication date: 23 November 2018

Neeraj Dhiman and Mohammad Tamsir

The purpose of this paper is to present a new method, namely, “Re-modified quintic B-spline collocation method” to solve the Kuramoto–Sivashinsky (KS) type equations. In this…

Abstract

Purpose

The purpose of this paper is to present a new method, namely, “Re-modified quintic B-spline collocation method” to solve the Kuramoto–Sivashinsky (KS) type equations. In this method, re-modified quintic B-spline functions and the Crank–Nicolson formulation is used for space and time integration, respectively. Five examples are considered to test out the efficiency and accuracy of the method. The main objective is to develop a method which gives more accurate results and reduces the computational cost so that the authors require less memory storage.

Design/methodology/approach

A new collocation technique is developed to solve the KS type equations. In this technique, quintic B-spline basis functions are re-modified and used to integrate the space derivatives while time derivative is discretized by using Crank–Nicolson formulation. The discretization yields systems of linear equations, which are solved by using Gauss elimination method with partial pivoting.

Findings

Five examples are considered to test out the efficiency and accuracy of the method. Finally, the present study summarizes the following outcomes: first, the computational cost of the proposed method is the less than quintic B-spline collocation method. Second, the present method produces better results than those obtained by Lattice Boltzmann method (Lai and Ma, 2009), quintic B-spline collocation method (Mittal and Arora, 2010), quintic B-spline differential quadrature method (DQM) (Mittal and Dahiya, 2017), extended modified cubic B-spline DQM (Tamsir et al., 2016) and modified cubic B-splines collocation method (Mittal and Jain, 2012).

Originality/value

The method presented in this paper is new to best of the authors’ knowledge. This work is the original work of authors and the manuscript is not submitted anywhere else for publication.

Details

Multidiscipline Modeling in Materials and Structures, vol. 18 no. 3
Type: Research Article
ISSN: 1573-6105

Keywords

Article
Publication date: 27 May 2014

Sahin Ahmed, Abdul Batin and Ali J. Chamkha

The purpose of this paper is to examine the effects of Darcian drag force and radiation-conduction on unsteady two-dimensional magnetohydrodynamic flow of viscous, electrically…

Abstract

Purpose

The purpose of this paper is to examine the effects of Darcian drag force and radiation-conduction on unsteady two-dimensional magnetohydrodynamic flow of viscous, electrically conducting and Newtonian fluid over a vertical plate adjacent to a Darcian regime in presence of thermal radiation and transversal magnetic field. A well-tested, numerically stable Crank-Nicolson finite-difference procedure is employed for the conservation equations. Excellent agreement is obtained for numerical solutions with previously published work.

Design/methodology/approach

In this investigation, an efficient, accurate, extensively validated and unconditionally stable finite-difference scheme based on the Crank-Nicolson model is developed to solve the governing coupled, non-linear partial differential equations. The accuracy and effectiveness of the method are demonstrated.

Findings

Different numerical results are obtained and presented graphically to explain the effect of various physical parameters on the velocity and temperature profiles, local, as well as average, skin friction and Nusselt number. It is found that, with a rise in Darcian drag force, flow velocity and temperature are reduced, but increased for all times. Both average and local skin frictions are reduced considerably with an increase in Darcian drag force, but reversed behavior is observed for the local Nusselt number. Increasing the thermal radiation effects accelerated the flow velocity as well as the fluid temperature and wall local skin friction in a saturated porous medium, but effectively reduced the local Nusselt number and average Nusselt number at the wall. Comparison with previously published works in the limits shows excellent agreement.

Research limitations/implications

The analysis is valid for unsteady, two-dimensional laminar flow of an optically thick no-gray gas, electrically conducting, and Newtonian fluid past an isothermal vertical surface adjacent to the Darcian regime with variable surface temperature. An extension to three-dimensional flow case is left for future work.

Practical implications

Practical interest of such study includes applications in electromagnetic lubrication, boundary cooling, bio-physical systems and in many branches of engineering and science. It is well known that the effect of thermal radiation is important in space technology and high temperature processes. Thermal radiation also plays an important role in controlling heat transfer process in polymer processing industry.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 24 no. 5
Type: Research Article
ISSN: 0961-5539

Keywords

Article
Publication date: 1 January 1991

Paul WILD

Implicit one‐step methods for the system of differential equations arising from a space discretisation of the semiconductor equations are considered. It is shown that mere…

Abstract

Implicit one‐step methods for the system of differential equations arising from a space discretisation of the semiconductor equations are considered. It is shown that mere spectral conditions like A‐stability or L‐stability do not give a reliable answer to the behaviour of the numerical solution. Rather, positivity arguments for the corresponding rational matrix functions play an important role.

Details

COMPEL - The international journal for computation and mathematics in electrical and electronic engineering, vol. 10 no. 1
Type: Research Article
ISSN: 0332-1649

1 – 10 of 222