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1 – 10 of over 77000The 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…
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.
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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.
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Keywords
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…
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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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.
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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 space H…
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.
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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.
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Hafiz Faiz Rasool, Muhammad Ali Qureshi, Abdul Aziz, Zain Ul Abiden Akhtar and Usman Ali Khan
The purpose of this paper is to provide a brief introduction of the finite difference based parabolic equation (PE) modeling to the advanced engineering students and academic…
Abstract
Purpose
The purpose of this paper is to provide a brief introduction of the finite difference based parabolic equation (PE) modeling to the advanced engineering students and academic researchers.
Design/methodology/approach
A three-dimensional parabolic equation (3DPE) model is developed from the ground up for modeling wave propagation in the tunnel via a rectangular waveguide structure. A discussion of vector wave equations from Maxwell’s equations followed by the paraxial approximations and finite difference implementation is presented for the beginners. The obtained simulation results are compared with the analytical solution.
Findings
It is shown that the alternating direction implicit finite difference method (FDM) is more efficient in terms of accuracy, computational time and memory than the explicit FDM. The reader interested in maximum details of individual contributions such as the latest achievements in PE modeling until 2021, basic PE derivation, PE formulation’s approximations, finite difference discretization and implementation of 3DPE, can learn from this paper.
Research limitations/implications
For the purpose of this paper, a simple 3DPE formulation is presented. For simplicity, a rectangular waveguide structure is discretized with the finite difference approach as a design problem. Future work could use the PE based FDM to study the possibility of utilization of meteorological techniques, including the effects of backward traveling waves as well as making comparisons with the experimental data.
Originality/value
The proposed work is directly applicable to typical problems in the field of tunnel propagation modeling for both national commercial and military applications.
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The purpose of this paper is to analyze the dynamical state of a discrete time engineering/physical dynamic system. The analysis is performed based on observability…
Abstract
Purpose
The purpose of this paper is to analyze the dynamical state of a discrete time engineering/physical dynamic system. The analysis is performed based on observability, controllability and stability first using difference equations of generalized motion obtained through discrete time equations of dissipative generalized motion derived from discrete Lagrange-dissipative model [{L,D}-model] for short of a discrete time observed dynamic system. As a next step, the same system has also been analyzed related to observability, controllability and stability concepts but this time using discrete dissipative canonical equations derived from a discrete Hamiltonian system together with discrete generalized velocity proportional Rayleigh dissipation function. The methods have been applied to a coupled (electromechanical) example in different formulation types.
Design/methodology/approach
An observability, controllability and stability analysis of a discrete time observed dynamic system using discrete equations of generalized motion obtained through discrete {L,D}-model and discrete dissipative canonical equations obtained through discrete Hamiltonian together with discrete generalized velocity proportional Rayleigh dissipation function.
Findings
The related analysis can be carried out easily depending on the values of classical elements.
Originality/value
Discrete equations of generalized motion and discrete dissipative canonical equations obtained by discrete Lagrangian and discrete Hamiltonian, respectively, together with velocity proportional discrete dissipative function are used to analyze a discrete time observed engineering system by means of observability, controllability and stability using state variable theory and in the method proposed, the physical quantities do not need to be converted one to another.
Details
Keywords
- Control systems
- Circuit analysis
- Sensors
- Coupled systems
- Mechatronics
- MEMS modelling
- Observability
- Controllability and stability analysis of discrete time systems
- Discrete physical coupled dynamic systems
- Discrete energy and dissipative systems
- Discrete Lagrangian and Hamiltonian systems
- Discrete time engineering systems
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 describes the…
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.
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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 with the…
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.
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