Research

15 pages, 4088 KiB

Open AccessArticle

Least-Squares Finite Element Method for Solving Stokes Flow under Point Source Magnetic Field

by Alia Rafiza Che Ayob, Zuhaila Ismail and Erwan Hafizi Kasiman

Symmetry 2022, 14(3), 514; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14030514 - 02 Mar 2022

Cited by 1 | Viewed by 1896

The least-squares finite element method (LSFEM) is successfully employed for the discretization of the Stokes equations and the numerical computation of the behaviour of two-dimensional Stokes flow in a straight rectangular channel under the effect of a point-source magnetic field. LSFEM has several [...] Read more.

The least-squares finite element method (LSFEM) is successfully employed for the discretization of the Stokes equations and the numerical computation of the behaviour of two-dimensional Stokes flow in a straight rectangular channel under the effect of a point-source magnetic field. LSFEM has several advantages in terms of theory and computing, where it can always create a symmetric, positive-definite algebraic system of equations. It also allows for using an equal order shape function for both velocity and pressure, and it is not required to satisfy the Ladyzhenskaya–Babuška–Brezzi (LBB) condition. Despite this, LSFEM has an issue where low-order nodal expansions tend to lock. Thus, the present study proposes the discretization of the problem domain using higher-order nodes elements with full numerical integration. Results concerning velocity contour and streamlines pattern are shown. On the basis of current findings, it can be concluded that the LSFEM can be used to solve Stokes flow problem under the point source magnetic field. Full article

(This article belongs to the Special Issue Symmetry in Numerical Analysis and Numerical Methods)

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17 pages, 369 KiB

Open AccessArticle

A Composite Initialization Method for Phase Retrieval

by Qi Luo, Shijian Lin and Hongxia Wang

Symmetry 2021, 13(11), 2006; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13112006 - 23 Oct 2021

Cited by 1 | Viewed by 1011

Abstract

Phase retrieval is a classical inverse problem with respect to recovering a signal from a system of phaseless constraints. Many recently proposed methods for phase retrieval such as PhaseMax and gradient-descent algorithms enjoy benign theoretical guarantees on the condition that an elaborate estimate [...] Read more.

Phase retrieval is a classical inverse problem with respect to recovering a signal from a system of phaseless constraints. Many recently proposed methods for phase retrieval such as PhaseMax and gradient-descent algorithms enjoy benign theoretical guarantees on the condition that an elaborate estimate of true solution is provided. Current initialization methods do not perform well when number of measurements are low, which deteriorates the success rate of current phase retrieval methods. We propose a new initialization method that can obtain an estimate of the original signal with uniformly higher accuracy which combines the advantages of the null vector method and maximal correlation method. The constructed spectral matrix for the proposed initialization method has a simple and symmetrical form. A lower error bound is proved theoretically as well as verified numerically. Full article

(This article belongs to the Special Issue Symmetry in Numerical Analysis and Numerical Methods)

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12 pages, 283 KiB

Open AccessArticle

A New Application of Gauss Quadrature Method for Solving Systems of Nonlinear Equations

by Hari M. Srivastava, Javed Iqbal, Muhammad Arif, Alamgir Khan, Yusif S. Gasimov and Ronnason Chinram

Symmetry 2021, 13(3), 432; https://0-doi-org.brum.beds.ac.uk/10.3390/sym13030432 - 07 Mar 2021

Cited by 18 | Viewed by 2120

Abstract

In this paper, we introduce a new three-step Newton method for solving a system of nonlinear equations. This new method based on Gauss quadrature rule has sixth order of convergence (with

n = 3

). The proposed method solves nonlinear boundary-value problems and [...] Read more.

In this paper, we introduce a new three-step Newton method for solving a system of nonlinear equations. This new method based on Gauss quadrature rule has sixth order of convergence (with

n = 3

). The proposed method solves nonlinear boundary-value problems and integral equations in few iterations with good accuracy. Numerical comparison shows that the new method is remarkably effective for solving systems of nonlinear equations. Full article

(This article belongs to the Special Issue Symmetry in Numerical Analysis and Numerical Methods)

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10 pages, 294 KiB

Open AccessArticle

A Family of Derivative Free Optimal Fourth Order Methods for Computing Multiple Roots

by Sunil Kumar, Deepak Kumar, Janak Raj Sharma and Lorentz Jäntschi

Symmetry 2020, 12(12), 1969; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12121969 - 28 Nov 2020

Cited by 8 | Viewed by 1330

Abstract

Many optimal order multiple root techniques, which use derivatives in the algorithm, have been proposed in literature. But contrarily, derivative free optimal order techniques for multiple root are almost nonexistent. By this as an inspirational factor, here we present a family of optimal [...] Read more.

Many optimal order multiple root techniques, which use derivatives in the algorithm, have been proposed in literature. But contrarily, derivative free optimal order techniques for multiple root are almost nonexistent. By this as an inspirational factor, here we present a family of optimal fourth order derivative-free techniques for computing multiple roots of nonlinear equations. At the beginning the convergence analysis is executed for particular values of multiplicity afterwards it concludes in general form. Behl et. al derivative-free method is seen as special case of the family. Moreover, the applicability and comparison is demonstrated on different nonlinear problems that certifies the efficient convergent nature of the new methods. Finally, we conclude that our new methods consume the lowest CPU time as compared to the existing ones. This illuminates the theoretical outcomes to a great extent of this study. Full article

(This article belongs to the Special Issue Symmetry in Numerical Analysis and Numerical Methods)

12 pages, 855 KiB

Open AccessArticle

Generating Optimal Eighth Order Methods for Computing Multiple Roots

by Deepak Kumar, Sunil Kumar, Janak Raj Sharma and Matteo d’Amore

Symmetry 2020, 12(12), 1947; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12121947 - 25 Nov 2020

Cited by 1 | Viewed by 1258

Abstract

There are a few optimal eighth order methods in literature for computing multiple zeros of a nonlinear function. Therefore, in this work our main focus is on developing a new family of optimal eighth order iterative methods for multiple zeros. The applicability of [...] Read more.

There are a few optimal eighth order methods in literature for computing multiple zeros of a nonlinear function. Therefore, in this work our main focus is on developing a new family of optimal eighth order iterative methods for multiple zeros. The applicability of proposed methods is demonstrated on some real life and academic problems that illustrate the efficient convergence behavior. It is shown that the newly developed schemes are able to compete with other methods in terms of numerical error, convergence and computational time. Stability is also demonstrated by means of a pictorial tool, namely, basins of attraction that have the fractal-like shapes along the borders through which basins are symmetric. Full article

(This article belongs to the Special Issue Symmetry in Numerical Analysis and Numerical Methods)

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13 pages, 1052 KiB

Open AccessArticle

A Novel Family of Efficient Weighted-Newton Multiple Root Iterations

by Deepak Kumar, Janak Raj Sharma and Lorentz Jăntschi

Symmetry 2020, 12(9), 1494; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12091494 - 10 Sep 2020

Cited by 9 | Viewed by 1454

Abstract

We propose a novel family of seventh-order iterative methods for computing multiple zeros of a nonlinear function. The algorithm consists of three steps, of which the first two are the steps of recently developed Liu–Zhou fourth-order method, whereas the third step is based [...] Read more.

We propose a novel family of seventh-order iterative methods for computing multiple zeros of a nonlinear function. The algorithm consists of three steps, of which the first two are the steps of recently developed Liu–Zhou fourth-order method, whereas the third step is based on a Newton-like step. The efficiency index of the proposed scheme is 1.627, which is better than the efficiency index 1.587 of the basic Liu–Zhou fourth-order method. In this sense, the proposed iteration is the modification over the Liu–Zhou iteration. Theoretical results are fully studied including the main theorem of local convergence analysis. Moreover, convergence domains are also assessed using the graphical tool, namely, basins of attraction which are symmetrical through the fractal like boundaries. Accuracy and computational efficiency are demonstrated by implementing the algorithms on different numerical problems. Comparison of numerical experiments shows that the new methods have an edge over the existing methods. Full article

(This article belongs to the Special Issue Symmetry in Numerical Analysis and Numerical Methods)

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11 pages, 761 KiB

Open AccessArticle

Quadratic Stability of Non-Linear Systems Modeled with Norm Bounded Linear Differential Inclusions

by Mutti-Ur Rehman, Jehad Alzabut and Arfan Hyder

Symmetry 2020, 12(9), 1432; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12091432 - 29 Aug 2020

Cited by 1 | Viewed by 1512

Abstract

In this article we present an ordinary differential equation based technique to study the quadratic stability of non-linear dynamical systems. The non-linear dynamical systems are modeled with norm bounded linear differential inclusions. The proposed methodology reformulate non-linear differential inclusion to an equivalent non-linear [...] Read more.

In this article we present an ordinary differential equation based technique to study the quadratic stability of non-linear dynamical systems. The non-linear dynamical systems are modeled with norm bounded linear differential inclusions. The proposed methodology reformulate non-linear differential inclusion to an equivalent non-linear system. Lyapunov function demonstrate the existence of a symmetric positive definite matrix to analyze the stability of non-linear dynamical systems. The proposed method allows us to construct a system of ordinary differential equations to localize the spectrum of perturbed system which guarantees the stability of non-linear dynamical system. Full article

(This article belongs to the Special Issue Symmetry in Numerical Analysis and Numerical Methods)

16 pages, 1319 KiB

Open AccessArticle

Analysis of Heat and Mass Transfer for Second-Order Slip Flow on a Thin Needle Using a Two-Phase Nanofluid Model

by Siti Nur Alwani Salleh, Norfifah Bachok, Fadzilah Md Ali and Norihan Md Arifin

Symmetry 2020, 12(7), 1176; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12071176 - 16 Jul 2020

Cited by 9 | Viewed by 2351

Abstract

The present paper concentrates on the second-order slip flow over a moving thin needle in a nanofluid. The combined effects of thermophoresis and Brownian motion are considered to describe the heat and mass transfer performance of nanofluid. The resulting system of equations are [...] Read more.

The present paper concentrates on the second-order slip flow over a moving thin needle in a nanofluid. The combined effects of thermophoresis and Brownian motion are considered to describe the heat and mass transfer performance of nanofluid. The resulting system of equations are obtained using similarity transformations and being executed in MATLAB software via bvp4c solver. The physical characteristics of embedded parameters on velocity, temperature, concentration, coefficient of skin friction, heat and mass transfer rates are demonstrated through a graphical approach and are discussed in detail. The obtained outcomes are validated with the existing works and are found to be in good agreement. It is shown that, for a specific domain of moving parameter, dual solutions are likely to exist. The stability analysis is performed to identify the stability of the solutions gained, and it is revealed that only one of them is numerically stable. The analysis indicated that the percentage of increment in the heat and mass transfer rates from no-slip to slip condition for both thin and thick surfaces of the needle (

a = 0.1

and

a = 0.2

) are

10.77 %

and

12.56 %

, respectively. Moreover, the symmetric behavior is noted for the graphs of reduced heat and mass transfer when the parameters

N b

and

N t

are the same. Full article

(This article belongs to the Special Issue Symmetry in Numerical Analysis and Numerical Methods)

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Graphical abstract

19 pages, 6466 KiB

Open AccessArticle

The Motion of a Point Vortex in Multiply-Connected Polygonal Domains

by El Mostafa Kalmoun, Mohamed M. S. Nasser and Khalifa A. Hazaa

Symmetry 2020, 12(7), 1175; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12071175 - 16 Jul 2020

Cited by 1 | Viewed by 2142

Abstract

We study the motion of a single point vortex in simply- and multiply-connected polygonal domains. In the case of multiply-connected domains, the polygonal obstacles can be viewed as the cross-sections of 3D polygonal cylinders. First, we utilize conformal mappings to transfer the polygonal [...] Read more.

We study the motion of a single point vortex in simply- and multiply-connected polygonal domains. In the case of multiply-connected domains, the polygonal obstacles can be viewed as the cross-sections of 3D polygonal cylinders. First, we utilize conformal mappings to transfer the polygonal domains onto circular domains. Then, we employ the Schottky-Klein prime function to compute the Hamiltonian governing the point vortex motion in circular domains. We compare between the topological structures of the contour lines of the Hamiltonian in symmetric and asymmetric domains. Special attention is paid to the interaction of point vortex trajectories with the polygonal obstacles. In this context, we discuss the effect of symmetry breaking, and obstacle location and shape on the behavior of vortex motion. Full article

(This article belongs to the Special Issue Symmetry in Numerical Analysis and Numerical Methods)

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15 pages, 4932 KiB

Open AccessArticle

Application of Symmetry Law in Numerical Modeling of Hydraulic Fracturing by Finite Element Method

by Shanhui Sun, Meihua Zhou, Wei Lu and Afshin Davarpanah

Symmetry 2020, 12(7), 1122; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12071122 - 06 Jul 2020

Cited by 44 | Viewed by 3179

Abstract

In this paper, influential parameters on the hydraulic fracturing processes in porous media were investigated. Besides, the simultaneous stimulation of solids, fluids and fractures geomechanical equations were numerically analyzed as a developed 3D model. To do this, the Abacus software was used as [...] Read more.

In this paper, influential parameters on the hydraulic fracturing processes in porous media were investigated. Besides, the simultaneous stimulation of solids, fluids and fractures geomechanical equations were numerically analyzed as a developed 3D model. To do this, the Abacus software was used as a multi-objective program to solve the physical-mechanical symmetry law governing equations, according to the finite element method. Two different layers, A (3104–2984 m) and B (4216–4326 m), are considered in the model. According to the result of this study, the maximum fracture opening length in the connection of the wall surface is 10 and 9 mm for layer B and layer A, respectively. Moreover, the internal fracture fluid pressure for layer B and layer A is 65 and 53 Mpa. It is indicated that fracture fluid pressure reduced with the increase in fracture propagation length. Consequently, the results of this study would be of benefit for petroleum industries to consider several crucial geomechanical characteristics in hydraulic fractures simultaneously as a developed numerical model for different formation layers to compare a comprehensive analysis between each layer. Full article

(This article belongs to the Special Issue Symmetry in Numerical Analysis and Numerical Methods)

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22 pages, 1624 KiB

Open AccessArticle

On Two-Derivative Runge–Kutta Type Methods for Solving u‴ = f(x,u(x)) with Application to Thin Film Flow Problem

by Khai Chien Lee, Norazak Senu, Ali Ahmadian and Siti Nur Iqmal Ibrahim

Symmetry 2020, 12(6), 924; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12060924 - 02 Jun 2020

Cited by 5 | Viewed by 2715

Abstract

A class of explicit Runge–Kutta type methods with the involvement of fourth derivative, denoted as two-derivative Runge–Kutta type (TDRKT) methods, are proposed and investigated for solving a special class of third-order ordinary differential equations in the form [...] Read more.

A class of explicit Runge–Kutta type methods with the involvement of fourth derivative, denoted as two-derivative Runge–Kutta type (TDRKT) methods, are proposed and investigated for solving a special class of third-order ordinary differential equations in the form

u^{‴} (x) = f (x, u (x))

. In this paper, two stages with algebraic order four and three stages with algebraic order five are presented. The derivation of TDRKT methods involves single third derivative and multiple evaluations of fourth derivative for every step. Stability property of the methods are analysed. Accuracy and efficiency of the new methods are exhibited through numerical experiments. Full article

(This article belongs to the Special Issue Symmetry in Numerical Analysis and Numerical Methods)

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11 pages, 417 KiB

Open AccessArticle

On the Convergence Rate of Clenshaw–Curtis Quadrature for Jacobi Weight Applied to Functions with Algebraic Endpoint Singularities

by Ahlam Arama, Shuhuang Xiang and Suliman Khan

Symmetry 2020, 12(5), 716; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12050716 - 02 May 2020

Cited by 3 | Viewed by 2232

Abstract

Applying the aliasing asymptotics on the coefficients of the Chebyshev expansions, the convergence rate of Clenshaw–Curtis quadrature for Jacobi weights is presented for functions with algebraic endpoint singularities. Based upon a new constructed symmetric Jacobi weight, the optimal error bound is derived for [...] Read more.

Applying the aliasing asymptotics on the coefficients of the Chebyshev expansions, the convergence rate of Clenshaw–Curtis quadrature for Jacobi weights is presented for functions with algebraic endpoint singularities. Based upon a new constructed symmetric Jacobi weight, the optimal error bound is derived for this kind of function. In particular, in this case, the Clenshaw–Curtis quadrature for a new constructed Jacobi weight is exponentially convergent. Numerical examples illustrate the theoretical results. Full article

(This article belongs to the Special Issue Symmetry in Numerical Analysis and Numerical Methods)

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