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Numerical Analysis and Scientific Computing II
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Numerical Analysis and Scientific Computing II

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Computational and Applied Mathematics".

Deadline for manuscript submissions: closed (31 December 2022) | Viewed by 61516

Special Issue Editors

Prof. Dr. Theodore E. Simos

grade E-Mail Website
Guest Editor
1. Laboratory of Applied Mathematics for Solving Interdisciplinary Problems of Energy Production, Ulyanovsk State Technical University, Severny Venetz Street 32, 432027 Ulyanovsk, Russia
2. Digital Industry REC, South Ural State University, 76, Lenin Avenue, 454080 Chelyabinsk, Russia
3. Section of Mathematics, Department of Civil Engineering, Democritus University of Thrace, 67100 Xanthi, Greece
Interests: numerical analysis; scientific computing; applied numerical analysis; computational chemistry; computational material sciences; computational physics; parallel algorithm and expert systems
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Charampos Tsitouras

E-Mail Website
Guest Editor
Mathematics and Applied Mathematics, National and Kapodistrian University of Athens, 10679 Athens, Greece
Interests: numerical analysis; scientific computing
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

In the last few decades, the role of numerical analysis and scientific computing has been increasing constantly, especially for the solution of real-world problems.

This Special Issue will present recent research results in numerical analysis and scientific computing.

Papers on the production, analysis, and computational performance of new and original methods of all areas of numerical analysis and scientific computing are welcome. More specifically, we welcome papers on topics including but not limited to the following:

Numerical analysis of ODEs; numerical analysis of PDEs (including BVPs); scientific computing and algorithms; stochastic differential equations; approximation theory; numerical linear algebra; numerical integral equations; error analysis and interval analysis; difference equations and recurrence relations; numerical problems in dynamical systems; applications to the sciences (computational physics, computational statistics, computational chemistry, computational engineering, etc.); differential algebraic equations, numerical methods in Fourier analysis; mathematical physics; mathematical chemistry; mathematical biology and mathematical medicine; optimization and operational research; theoretical mechanics; discrete applied mathematics; statistics; probability; dynamical systems; algorithms; experimental mathematics; theoretical computer science; applied analysis; mathematical modeling (including but not limited to the mathematical modeling of engineering and environmental processes manufacturing, industrial systems, heat transfer, fluid mechanics, CFD, and transport phenomena solid mechanics and mechanics of metals, electromagnets and MHD, reliability modeling and system optimization, decision sciences in an industrial and manufacturing context, civil engineering systems and structures, mineral and energy resources, relevant software engineering issues associated with CAD and CAE, materials and metallurgical engineering, and the mathematical modelling of social, behavioral and other sciences); decomposition and reconstruction algorithms; subdivision algorithms; continuous and discrete wavelet transform; time-frequency localization; phase-space analysis; sub-band coding; image compression; real-time filtering; radar and sonar applications; transient analysis; medical imaging; multigrid methods; frames; bifurcation and singularity theory; deterministic chaos and fractals; soliton and coherent phenomena; pattern formation; evolution; complexity theory and neural networks; analytical approaches and simulations for more accurate descriptions; predictions; experimental observations and applications of nonlinear phenomena in science and engineering; theoretical and applied aspects of computational geometry; control theory and automation; fuzzy sets and systems and fuzzy logic; applied algebra; quality theory of differential equations; neural networks.

We also welcome papers exploring applications of numerical and mathematical methods to real-world problems in sciences, engineering, and technology.

Prof. Dr. Theodore E. Simos
Prof. Dr. Charampos Tsitouras
Guest Editors

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • numerical analysis
  • computational mathematics
  • scientific computing
  • computational methods
  • algorithms
  • applied and industrial mathematics
  • mathematical methods
  • optimization
  • applications in sciences
  • engineering and technology

Published Papers (34 papers)

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13 pages, 401 KiB  
Article
Runge–Kutta–Nyström Pairs of Orders 8(6) for Use in Quadruple Precision Computations
by Vladislav N. Kovalnogov, Alexander F. Matveev, Dmitry A. Generalov, Tamara V. Karpukhina, Theodore E. Simos and Charalampos Tsitouras
Mathematics 2023, 11(4), 891; https://0-doi-org.brum.beds.ac.uk/10.3390/math11040891 - 09 Feb 2023
Cited by 2 | Viewed by 996
Abstract
The second-order system of non-stiff Initial Value Problems (IVP) is considered and, in particular, the case where the first derivatives are absent. This kind of problem is interesting since since it arises in many significant problems, e.g., in Celestial mechanics. Runge–Kutta–Nyström (RKN) pairs [...] Read more.
The second-order system of non-stiff Initial Value Problems (IVP) is considered and, in particular, the case where the first derivatives are absent. This kind of problem is interesting since since it arises in many significant problems, e.g., in Celestial mechanics. Runge–Kutta–Nyström (RKN) pairs are perhaps the most successful approaches for solving such type of IVPs. To achieve a pair attaining orders eight and six, we have to solve a well-defined set of equations with respect to the coefficients. Here, we provide a simplified form of these equations in a robust algorithm. When creating such pairings for use in double precision arithmetic, numerous conditions are often satisfied. First and foremost, we strive to keep the coefficients’ magnitudes small to prevent accuracy loss. We may, however, allow greater coefficients when working with quadruple precision. Then, we may build pairs of orders eight and six with significantly smaller truncation errors. In this paper, a novel pair is generated that, as predicted, outperforms state-of-the-art pairs of the same orders in a collection of important problems. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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25 pages, 1223 KiB  
Article
Algorithmic Aspects of Simulation of Magnetic Field Generation by Thermal Convection in a Plane Layer of Fluid
by Daniil Tolmachev, Roman Chertovskih and Vladislav Zheligovsky
Mathematics 2023, 11(4), 808; https://0-doi-org.brum.beds.ac.uk/10.3390/math11040808 - 05 Feb 2023
Cited by 3 | Viewed by 983 | Correction
Abstract
We present an algorithm for numerical solution of the equations of magnetohydrodynamics describing the convective dynamo in a plane horizontal layer rotating about an arbitrary axis under geophysically sound boundary conditions. While in many respects we pursue the general approach that was followed [...] Read more.
We present an algorithm for numerical solution of the equations of magnetohydrodynamics describing the convective dynamo in a plane horizontal layer rotating about an arbitrary axis under geophysically sound boundary conditions. While in many respects we pursue the general approach that was followed by other authors, our main focus is on the accuracy of simulations, especially in the small scales. We employ the Galerkin method. We use products of linear combinations (each involving two to five terms) of Chebyshev polynomials in the vertical Cartesian space variable and Fourier harmonics in the horizontal variables for space discretisation of the toroidal and poloidal potentials of the flow (satisfying the no-slip conditions on the horizontal boundaries) and magnetic field (for which the boundary conditions mimick the presence of a dielectric over the fluid layer and an electrically conducting bottom boundary), and of the deviation of temperature from the steady-state linear profile. For the chosen coefficients in the linear combinations, the products satisfy the respective boundary conditions and constitute non-orthogonal bases in the weighted Lebesgue space. Determining coefficients in the expansion of a given function in such a basis (for instance, for computing the time derivatives of these coefficients) requires solving linear systems of equations for band matrices. Several algorithms for determining the coefficients, which are exploiting algebraically precise relations, have been developed, and their efficiency and accuracy have been numerically investigated for exponentially decaying solutions (encountered when simulating convective regimes which are spatially resolved sufficiently well). For the boundary conditions satisfied by the toroidal component of the flow, our algorithm outperforms the shuttle method, but the latter proves superior when solving the problem for the conditions characterising the poloidal component. While the conditions for the magnetic field on the horizontal boundaries are quite specific, our algorithms for the no-slip boundary conditions are general-purpose and can be applied for treating other boundary-value problems in which the zero value must be admitted on the boundary. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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14 pages, 1120 KiB  
Article
Time-Varying Pseudoinversion Based on Full-Rank Decomposition and Zeroing Neural Networks
by Hadeel Alharbi, Houssem Jerbi, Mourad Kchaou, Rabeh Abbassi, Theodore E. Simos, Spyridon D. Mourtas and Vasilios N. Katsikis
Mathematics 2023, 11(3), 600; https://0-doi-org.brum.beds.ac.uk/10.3390/math11030600 - 24 Jan 2023
Cited by 6 | Viewed by 1215
Abstract
The computation of the time-varying matrix pseudoinverse has become crucial in recent years for solving time-varying problems in engineering and science domains. This paper investigates the issue of calculating the time-varying pseudoinverse based on full-rank decomposition (FRD) using the zeroing neural network (ZNN) [...] Read more.
The computation of the time-varying matrix pseudoinverse has become crucial in recent years for solving time-varying problems in engineering and science domains. This paper investigates the issue of calculating the time-varying pseudoinverse based on full-rank decomposition (FRD) using the zeroing neural network (ZNN) method, which is currently considered to be a cutting edge method for calculating the time-varying matrix pseudoinverse. As a consequence, for the first time in the literature, a new ZNN model called ZNNFRDP is introduced for time-varying pseudoinversion and it is based on FRD. Five numerical experiments investigate and confirm that the ZNNFRDP model performs as well as, if not better than, other well-performing ZNN models in the calculation of the time-varying pseudoinverse. Additionally, theoretical analysis and numerical findings have both supported the effectiveness of the proposed model. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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25 pages, 1948 KiB  
Article
Generation of Boxes and Permutations Using a Bijective Function and the Lorenz Equations: An Application to Color Image Encryption
by Víctor Manuel Silva-García, Rolando Flores-Carapia, Manuel Alejandro Cardona-López and Miguel Gabriel Villarreal-Cervantes
Mathematics 2023, 11(3), 599; https://0-doi-org.brum.beds.ac.uk/10.3390/math11030599 - 24 Jan 2023
Cited by 3 | Viewed by 1283
Abstract
Some images that contain sensitive information and travel through the network require security. Therefore, a symmetric cryptosystem that encrypts images and resists known attacks is developed. Subsequently, in this work, an encryption algorithm known as Image Cipher utilizing Lorenz equation and a Bijective [...] Read more.
Some images that contain sensitive information and travel through the network require security. Therefore, a symmetric cryptosystem that encrypts images and resists known attacks is developed. Subsequently, in this work, an encryption algorithm known as Image Cipher utilizing Lorenz equation and a Bijective Function—ICLEBF are proposed. In the proposal, the Lorenz equations and the Bijective function are used to generate boxes, the permutation, and schedule keys, considering that all these elements are different in each encryption process. The encryption procedure consists of 14 rounds, where a different box is applied in each round. In this type of algorithm, the impact of quantum computers will be less forceful and can be useful for that epoch. On the other hand, the quality of the encrypted images and the loss of sharpness in decoded images with damage are measured. In addition, an attack from five types of noise (one of which is a developed proposal) is carried out by applying it to encrypted images. Finally, the results of the proposed ICLEBF are compared with other recent image encryption algorithms, including the Advanced Encryption Standard. As a result, this proposal resists known attacks and others that the current standard does not support. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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19 pages, 4407 KiB  
Communication
Implementation and Performance Analysis of Kalman Filters with Consistency Validation
by Dah-Jing Jwo and Amita Biswal
Mathematics 2023, 11(3), 521; https://0-doi-org.brum.beds.ac.uk/10.3390/math11030521 - 18 Jan 2023
Cited by 6 | Viewed by 2710
Abstract
This paper provides a useful supplement note for implementing the Kalman filters. The material presented in this work points out several significant highlights with emphasis on performance evaluation and consistency validation between the discrete Kalman filter (DKF) and the continuous Kalman filter (CKF). [...] Read more.
This paper provides a useful supplement note for implementing the Kalman filters. The material presented in this work points out several significant highlights with emphasis on performance evaluation and consistency validation between the discrete Kalman filter (DKF) and the continuous Kalman filter (CKF). Several important issues are delivered through comprehensive exposition accompanied by supporting examples, both qualitatively and quantitatively for implementing the Kalman filter algorithms. The lesson learned assists the readers to capture the basic principles of the topic and enables the readers to better interpret the theory, understand the algorithms, and correctly implement the computer codes for further study on the theory and applications of the topic. A wide spectrum of content is covered from theoretical to implementation aspects, where the DKF and CKF along with the theoretical error covariance check based on Riccati and Lyapunov equations are involved. Consistency check of performance between discrete and continuous Kalman filters enables readers to assure correctness on implementing and coding for the algorithm. The tutorial-based exposition presented in this article involves the materials from a practical usage perspective that can provide profound insights into the topic with an appropriate understanding of the stochastic process and system theory. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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15 pages, 2713 KiB  
Article
Criteria-Based Model of Hybrid Photovoltaic–Wind Energy System with Micro-Compressed Air Energy Storage
by Georgios E. Arnaoutakis, Gudrun Kocher-Oberlehner and Dimitris Al. Katsaprakakis
Mathematics 2023, 11(2), 391; https://0-doi-org.brum.beds.ac.uk/10.3390/math11020391 - 11 Jan 2023
Cited by 6 | Viewed by 1340
Abstract
Utilization of solar and wind energy is increasing worldwide. Photovoltaic and wind energy systems are among the major contributing tec4hnologies to the generation capacity from renewable energy sources; however, the generation often does not temporally match the demand. Micro-compressed air energy storage (micro-CAES) [...] Read more.
Utilization of solar and wind energy is increasing worldwide. Photovoltaic and wind energy systems are among the major contributing tec4hnologies to the generation capacity from renewable energy sources; however, the generation often does not temporally match the demand. Micro-compressed air energy storage (micro-CAES) is among the low-cost storage options, and its coupling with the power generated by photovoltaics and wind turbines can provide demand shifting, modeled by efficient algorithms. A model based on criteria that are preset according to the demand is presented. The model decides on the distribution of the generated energy, depending on the state of the energy storage and the preset criteria of each storage technology. The satisfaction of the demand by the energy production and micro-CAES is compared to that of storage batteries. The demand originates in a case study of a household and optimal configurations of photovoltaics and wind turbines, and the storage capacities and costs are compared. An optimal configuration of 30 photovoltaic panels and two wind turbines was found for micro-CAES. The annual stored energy of micro-CAES was 114 kWh higher than that of the system with batteries. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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30 pages, 374 KiB  
Article
Dual Variational Formulations for a Large Class of Non-Convex Models in the Calculus of Variations
by Fabio Silva Botelho
Mathematics 2023, 11(1), 63; https://0-doi-org.brum.beds.ac.uk/10.3390/math11010063 - 24 Dec 2022
Cited by 2 | Viewed by 811
Abstract
This article develops dual variational formulations for a large class of models in variational optimization. The results are established through basic tools of functional analysis, convex analysis and duality theory. The main duality principle is developed as an application to a Ginzburg–Landau-type system [...] Read more.
This article develops dual variational formulations for a large class of models in variational optimization. The results are established through basic tools of functional analysis, convex analysis and duality theory. The main duality principle is developed as an application to a Ginzburg–Landau-type system in superconductivity in the absence of a magnetic field. In the first section, we develop new general dual convex variational formulations, more specifically, dual formulations with a large region of convexity around the critical points, which are suitable for the non-convex optimization for a large class of models in physics and engineering. Finally, in the last section, we present some numerical results concerning the generalized method of lines applied to a Ginzburg–Landau-type equation. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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11 pages, 487 KiB  
Article
Solving Poisson Equations by the MN-Curve Approach
by Lin-Tian Luh
Mathematics 2022, 10(23), 4582; https://0-doi-org.brum.beds.ac.uk/10.3390/math10234582 - 02 Dec 2022
Viewed by 1068
Abstract
In this paper, we adopt the choice theory of the shape parameters contained in the smooth radial basis functions to solve Poisson equations. Luh’s choice theory, based on harmonic analysis, is mathematically complicated and applies only to function interpolation. Here, we aim at [...] Read more.
In this paper, we adopt the choice theory of the shape parameters contained in the smooth radial basis functions to solve Poisson equations. Luh’s choice theory, based on harmonic analysis, is mathematically complicated and applies only to function interpolation. Here, we aim at presenting an easily accessible approach to solving differential equations with the choice theory which proves to be very successful, not only by its easy accessibility but also by its striking accuracy and efficiency. Our emphases are on the highly reliable prediction of the optimal value of the shape parameter and the extremely small approximation errors of the numerical solutions to the differential equations. We hope that our approach can be accepted by both mathematicians and non-mathematicians. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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21 pages, 4057 KiB  
Article
One-Step Suicide Substrate Inactivation Kinetics of a Ping-Pong Reaction with One Substrate Undergoing Disproportionation: A Theoretical Approach with Approximate Solutions
by Ismael Gutiérrez-Fernández, Ouardia Bendou, Nara Bueno-Ramos, Emilio L. Marcos-Barbero, Rosa Morcuende and Juan B. Arellano
Mathematics 2022, 10(22), 4240; https://0-doi-org.brum.beds.ac.uk/10.3390/math10224240 - 13 Nov 2022
Viewed by 1573
Abstract
Understanding the kinetic mechanism of enzyme inactivation by suicide substrate is of relevance for the optimal design of new drugs with pharmacological and therapeutic applications. Suicide substrate inactivation usually occurs via a two-step mechanism, although there are enzymes such as peroxidase and catalase [...] Read more.
Understanding the kinetic mechanism of enzyme inactivation by suicide substrate is of relevance for the optimal design of new drugs with pharmacological and therapeutic applications. Suicide substrate inactivation usually occurs via a two-step mechanism, although there are enzymes such as peroxidase and catalase in which the suicide inactivation by H2O2 happens in a single step. The approximate solution of the ordinary differential equation (ODE) system of the one step suicide substrate inactivation kinetics for a uni–uni reaction following the irreversible Michaelis–Menten model was previously analytically solved when accumulation of the substrate–enzyme complex was negligible, however not for more complex models, such as a ping-pong reaction, in which the enzyme is present in two active states during the catalytic turnover. To solve this issue, a theoretical approach was followed, in which the standard quasi-steady state and reactant stationary approximations were invoked. These approximations allowed for solving the ODE system of a ping-pong reaction with one substrate undergoing disproportionation when suicide inactivation was also present. Although the approximate analytical solutions were rather unwieldy, they were still valuable in qualitative analyses to explore the time course of the reaction products and identify the enzyme active state that irreversibly reacted with the suicide substrate during the reaction. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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32 pages, 21519 KiB  
Article
Theoretical and Numerical Study of Self-Organizing Processes in a Closed System Classical Oscillator and Random Environment
by Ashot S. Gevorkyan, Aleksander V. Bogdanov, Vladimir V. Mareev and Koryun A. Movsesyan
Mathematics 2022, 10(20), 3868; https://0-doi-org.brum.beds.ac.uk/10.3390/math10203868 - 18 Oct 2022
Cited by 2 | Viewed by 1081
Abstract
A self-organizing joint system classical oscillator–random environment is considered within the framework of a complex probabilistic process that satisfies a Langevin-type stochastic differential equation. Various types of randomness generated by the environment are considered. In the limit of statistical equilibrium (SEq), second-order partial [...] Read more.
A self-organizing joint system classical oscillator–random environment is considered within the framework of a complex probabilistic process that satisfies a Langevin-type stochastic differential equation. Various types of randomness generated by the environment are considered. In the limit of statistical equilibrium (SEq), second-order partial differential equations (PDE) are derived that describe the distribution of classical environmental fields. The mathematical expectation of the oscillator trajectory is constructed in the form of a functional-integral representation, which, in the SEq limit, is compactified into a two-dimensional integral representation with an integrand: the solution of the second-order complex PDE. It is proved that the complex PDE in the general case is reduced to two independent PDEs of the second order with spatially deviating arguments. The geometric and topological features of the two-dimensional subspace on which these equations arise are studied in detail. An algorithm for parallel modeling of the problem has been developed. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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20 pages, 1544 KiB  
Article
Bifurcation Analysis and Numerical Study of Wave Solution for Initial-Boundary Value Problem of the KdV-BBM Equation
by Teeranush Suebcharoen, Kanyuta Poochinapan and Ben Wongsaijai
Mathematics 2022, 10(20), 3825; https://0-doi-org.brum.beds.ac.uk/10.3390/math10203825 - 16 Oct 2022
Cited by 1 | Viewed by 1389
Abstract
In this work, we study the bifurcation and the numerical analysis of the nonlinear Benjamin-Bona-Mahony-KdV equation. According to the bifurcation theory of a dynamic system, the various kinds of traveling wave profiles are obtained including the behavior of solitary and periodic waves. Additionally, [...] Read more.
In this work, we study the bifurcation and the numerical analysis of the nonlinear Benjamin-Bona-Mahony-KdV equation. According to the bifurcation theory of a dynamic system, the various kinds of traveling wave profiles are obtained including the behavior of solitary and periodic waves. Additionally, a two-level linear implicit finite difference algorithm is implemented for investigating the Benjamin-Bona-Mahony-KdV model. The application of a priori estimation for the approximate solution also provides the convergence and stability analysis. It was demonstrated that the current approach is singularly solvable and that both time and space convergence are of second-order precision. To confirm the computational effectiveness, two numerical simulations are prepared. The findings show that the current technique performs admirably in terms of delivering second-order accuracy in both time and space with the maximum norm while outperforming prior schemes. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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16 pages, 439 KiB  
Article
An Approximation Method to Compute Highly Oscillatory Singular Fredholm Integro-Differential Equations
by SAIRA and Wen-Xiu Ma
Mathematics 2022, 10(19), 3628; https://0-doi-org.brum.beds.ac.uk/10.3390/math10193628 - 04 Oct 2022
Viewed by 1050
Abstract
This paper appertains the presentation of a Clenshaw–Curtis rule to evaluate highly oscillatory Fredholm integro-differential equations (FIDEs) with Cauchy and weak singularities. To calculate the singular integral, the unknown function approximated by an interpolation polynomial is rewritten as a Taylor series expansion. A [...] Read more.
This paper appertains the presentation of a Clenshaw–Curtis rule to evaluate highly oscillatory Fredholm integro-differential equations (FIDEs) with Cauchy and weak singularities. To calculate the singular integral, the unknown function approximated by an interpolation polynomial is rewritten as a Taylor series expansion. A system of linear equations of FIDEs obtained by using equally spaced points as collocation points is solved to obtain the unknown function. The proposed method attains higher accuracy rates, which are proven by error analysis and some numerical examples as well. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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18 pages, 759 KiB  
Article
A Direct Prediction of the Shape Parameter in the Collocation Method of Solving Poisson Equation
by Lin-Tian Luh
Mathematics 2022, 10(19), 3583; https://0-doi-org.brum.beds.ac.uk/10.3390/math10193583 - 01 Oct 2022
Cited by 1 | Viewed by 895
Abstract
In this paper, we totally discard the traditional trial-and-error algorithms of choosing the acceptable shape parameter c in the multiquadrics −c2+∥x∥2 when dealing with differential equations, for example, the Poisson equation, with the RBF collocation method. [...] Read more.
In this paper, we totally discard the traditional trial-and-error algorithms of choosing the acceptable shape parameter c in the multiquadrics −c2+∥x∥2 when dealing with differential equations, for example, the Poisson equation, with the RBF collocation method. Instead, we choose c directly by the MN-curve theory and hence avoid the time-consuming steps of solving a linear system required by each trial of the c value in the traditional methods. The quality of the c value thus obtained is supported by the newly born choice theory of the shape parameter. Experiments demonstrate that the approximation error of the approximate solution to the differential equation is very close to the best approximation error among all possible choices of c. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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13 pages, 357 KiB  
Article
A Collocation Method for Mixed Volterra–Fredholm Integral Equations of the Hammerstein Type
by Sanda Micula
Mathematics 2022, 10(17), 3044; https://0-doi-org.brum.beds.ac.uk/10.3390/math10173044 - 23 Aug 2022
Viewed by 1084
Abstract
This paper presents a collocation method for the approximate solution of two-dimensional mixed Volterra–Fredholm integral equations of the Hammerstein type. For a reformulation of the equation, we consider the domain of integration as a planar triangle and use a special type of linear [...] Read more.
This paper presents a collocation method for the approximate solution of two-dimensional mixed Volterra–Fredholm integral equations of the Hammerstein type. For a reformulation of the equation, we consider the domain of integration as a planar triangle and use a special type of linear interpolation on triangles. The resulting quadrature formula has a higher degree of precision than expected, leading to a collocation method that is superconvergent at the collocation nodes. The convergence of the method is established, as well as the rate of convergence. Numerical examples are considered, showing the applicability of the proposed scheme and the agreement with the theoretical results. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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44 pages, 38349 KiB  
Article
Nonlinear Large-Scale Perturbations of Steady Thermal Convective Dynamo Regimes in a Plane Layer of Electrically Conducting Fluid Rotating about the Vertical Axis
by Simon Ranjith Jeyabalan, Roman Chertovskih, Sílvio Gama and Vladislav Zheligovsky
Mathematics 2022, 10(16), 2957; https://0-doi-org.brum.beds.ac.uk/10.3390/math10162957 - 16 Aug 2022
Cited by 3 | Viewed by 1426
Abstract
We present results of numerical investigation of regimes of steady thermal convective dynamo in a plane layer of electrically conducting fluid rotating about the vertical axis and subjected to large-scale perturbations. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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9 pages, 767 KiB  
Article
An Approximate Proximal Numerical Procedure Concerning the Generalized Method of Lines
by Fabio Silva Botelho
Mathematics 2022, 10(16), 2950; https://0-doi-org.brum.beds.ac.uk/10.3390/math10162950 - 16 Aug 2022
Cited by 2 | Viewed by 901
Abstract
This article develops an approximate proximal approach for the generalized method of lines. We recall that for the generalized method of lines, the domain of the partial differential equation in question is discretized in lines (or in curves) and the concerning solution is [...] Read more.
This article develops an approximate proximal approach for the generalized method of lines. We recall that for the generalized method of lines, the domain of the partial differential equation in question is discretized in lines (or in curves) and the concerning solution is developed on these lines, as functions of the boundary conditions and the domain boundary shape. Considering such a context, in the text we develop an approximate numerical procedure of proximal nature applicable to a large class of models in physics and engineering. Finally, in the last sections, we present numerical examples and results related to a Ginzburg–Landau-type equation. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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10 pages, 418 KiB  
Article
The Shape Parameter in the Shifted Surface Spline—An Easily Accessible Approach
by Lin-Tian Luh
Mathematics 2022, 10(16), 2844; https://0-doi-org.brum.beds.ac.uk/10.3390/math10162844 - 10 Aug 2022
Cited by 1 | Viewed by 943
Abstract
In this paper, we present an easily accessible approach to finding a suitable shape parameter in the shifted surface spline for function interpolation. We aim at helping more readers, including mathematicians and non-mathematicians, to use our method to solve practical problems. Hence, some [...] Read more.
In this paper, we present an easily accessible approach to finding a suitable shape parameter in the shifted surface spline for function interpolation. We aim at helping more readers, including mathematicians and non-mathematicians, to use our method to solve practical problems. Hence, some highly complicated mathematical theorems and definitions are avoided. The major requirement, as in our previous approach, that the data points should be evenly spaced in the domain is also relaxed. This means that the data points are purely scattered without restrictions. The drawback is that the shape parameter thus obtained is not exactly the same as the theoretically predicted optimal value, which can always be achieved by using our previous rigorous approach. However, experiments show that the gap is quite small and the final interpolation errors thus obtained are satisfactory. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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20 pages, 652 KiB  
Article
Optimization of Turbulence Model Parameters Using the Global Search Method Combined with Machine Learning
by Konstantin Barkalov, Ilya Lebedev, Marina Usova, Daria Romanova, Daniil Ryazanov and Sergei Strijhak
Mathematics 2022, 10(15), 2708; https://0-doi-org.brum.beds.ac.uk/10.3390/math10152708 - 31 Jul 2022
Cited by 2 | Viewed by 1754
Abstract
The paper considers the slope flow simulation and the problem of finding the optimal parameter values of this mathematical model. The slope flow is modeled using the finite volume method applied to the Reynolds-averaged Navier–Stokes equations with closure in the form of the [...] Read more.
The paper considers the slope flow simulation and the problem of finding the optimal parameter values of this mathematical model. The slope flow is modeled using the finite volume method applied to the Reynolds-averaged Navier–Stokes equations with closure in the form of the k−ωSST turbulence model. The optimal values of the turbulence model coefficients for free surface gravity multiphase flows were found using the global search algorithm. Calibration was performed to increase the similarity of the experimental and calculated velocity profiles. The Root Mean Square Error (RMSE) of derivation between the calculated flow velocity profile and the experimental one is considered as the objective function in the optimization problem. The calibration of the turbulence model coefficients for calculating the free surface flows on test slopes using the multiphase model for interphase tracking has not been performed previously. To solve the multi-extremal optimization problem arising from the search for the minimum of the loss function for the flow velocity profile, we apply a new optimization approach using a Peano curve to reduce the dimensionality of the problem. To speed up the optimization procedure, the objective function was approximated using an artificial neural network. Thus, an interdisciplinary approach was applied which allowed the optimal values of six turbulence model parameters to be found using OpenFOAM and Globalizer software. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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48 pages, 3798 KiB  
Article
Efficient Long-Term Simulation of the Heat Equation with Application in Geothermal Energy Storage
by Martin Bähr and Michael Breuß
Mathematics 2022, 10(13), 2309; https://0-doi-org.brum.beds.ac.uk/10.3390/math10132309 - 01 Jul 2022
Cited by 1 | Viewed by 1659
Abstract
Long-term evolutions of parabolic partial differential equations, such as the heat equation, are the subject of interest in many applications. There are several numerical solvers marking the state-of-the-art in diverse scientific fields that may be used with benefit for the numerical simulation of [...] Read more.
Long-term evolutions of parabolic partial differential equations, such as the heat equation, are the subject of interest in many applications. There are several numerical solvers marking the state-of-the-art in diverse scientific fields that may be used with benefit for the numerical simulation of such long-term scenarios. We show how to adapt some of the currently most efficient numerical approaches for solving the fundamental problem of long-term linear heat evolution with internal and external boundary conditions as well as source terms. Such long-term simulations are required for the optimal dimensioning of geothermal energy storages and their profitability assessment, for which we provide a comprehensive analytical and numerical model. Implicit methods are usually considered the best choice for resolving long-term simulations of linear parabolic problems; however, in practice the efficiency of such schemes in terms of the combination of computational load and obtained accuracy may be a delicate issue, as it depends very much on the properties of the underlying model. For example, one of the challenges in long-term simulation may arise by the presence of time-dependent boundary conditions, as in our application. In order to provide both a computationally efficient and accurate enough simulation, we give a thorough discussion of the various numerical solvers along with many technical details and own adaptations. By our investigation, we focus on two largely competitive approaches for our application, namely the fast explicit diffusion method originating in image processing and an adaptation of the Krylov subspace model order reduction method. We validate our numerical findings via several experiments using synthetic and real-world data. We show that we can obtain fast and accurate long-term simulations of typical geothermal energy storage facilities. We conjecture that our techniques can be highly useful for tackling long-term heat evolution in many applications. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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19 pages, 6884 KiB  
Article
Momentum Distribution Functions and Pair Correlation Functions of Unpolarized Uniform Electron Gas in Warm Dense Matter Regime
by Alexander Larkin, Vladimir Filinov and Pavel Levashov
Mathematics 2022, 10(13), 2270; https://0-doi-org.brum.beds.ac.uk/10.3390/math10132270 - 29 Jun 2022
Cited by 1 | Viewed by 1086
Abstract
In this paper we continued our research of the uniform electron gas in a warm dense matter regime, focusing on the momentum distribution functions and pair correlation functions. We use the single–momentum path integral Monte Carlo method, based on the Wigner formulation of [...] Read more.
In this paper we continued our research of the uniform electron gas in a warm dense matter regime, focusing on the momentum distribution functions and pair correlation functions. We use the single–momentum path integral Monte Carlo method, based on the Wigner formulation of quantum statistics to calculate both momentum- and coordinate-depending distributions and average values of quantum operators for many-fermion Coulomb systems. We discovered that the single-particle momentum distribution function deviates from the ideal Fermi distribution and forms the so-called “quantum tails” at high momenta, if non-ideality is strong enough in both degenerate and non-degenerate cases. This effect is always followed by the appearance of the short-range order on pair correlation functions and can be explained by the tunneling through the effective potential wells surrounding the electrons. Furthermore, we calculated the average kinetic and potential energies in the wide range of states, expanding our previous results significantly. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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32 pages, 7251 KiB  
Article
Optimal Tuning of the Speed Control for Brushless DC Motor Based on Chaotic Online Differential Evolution
by Alejandro Rodríguez-Molina, Miguel Gabriel Villarreal-Cervantes, Omar Serrano-Pérez, José Solís-Romero and Ramón Silva-Ortigoza
Mathematics 2022, 10(12), 1977; https://0-doi-org.brum.beds.ac.uk/10.3390/math10121977 - 08 Jun 2022
Cited by 9 | Viewed by 2471
Abstract
The efficiency in the controller performance of a BLDC motor in an uncertain environment highly depends on the adaptability of the controller gains. In this paper, the chaotic adaptive tuning strategy for controller gains (CATSCG) is proposed for the speed regulation of BLDC [...] Read more.
The efficiency in the controller performance of a BLDC motor in an uncertain environment highly depends on the adaptability of the controller gains. In this paper, the chaotic adaptive tuning strategy for controller gains (CATSCG) is proposed for the speed regulation of BLDC motors. The CATSCG includes two sequential dynamic optimization stages based on identification and predictive processes, and also the use of a novel chaotic online differential evolution (CODE) for providing controller gains at each predefined time interval. Statistical comparative results with other tuning approaches evidence that the use of the chaotic initialization based on the Lozi map included in CODE for the CATSCG can efficiently handle the disturbances in the closed-loop system of the dynamic environment. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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19 pages, 1308 KiB  
Article
Accurate Goertzel Algorithm: Error Analysis, Validations and Applications
by Chuanying Li, Peibing Du, Kuan Li, Yu Liu, Hao Jiang and Zhe Quan
Mathematics 2022, 10(11), 1788; https://0-doi-org.brum.beds.ac.uk/10.3390/math10111788 - 24 May 2022
Cited by 1 | Viewed by 1814
Abstract
The Horner and Goertzel algorithms are frequently used in polynomial evaluation. Each of them can be less expensive than the other in special cases. In this paper, we present a new compensated algorithm to improve the accuracy of the Goertzel algorithm by using [...] Read more.
The Horner and Goertzel algorithms are frequently used in polynomial evaluation. Each of them can be less expensive than the other in special cases. In this paper, we present a new compensated algorithm to improve the accuracy of the Goertzel algorithm by using error-free transformations. We derive the forward round-off error bound for our algorithm, which implies that our algorithm yields a full precision accuracy for polynomials that are not too ill-conditioned. A dynamic error estimate in our algorithm is also presented by running round-off error analysis. Moreover, we show the cases in which our algorithms are less expensive than the compensated Horner algorithm for evaluating polynomials. Numerical experiments indicate that our algorithms run faster than the compensated Horner algorithm in those cases while producing the same accurate results, and our algorithm is absolutely stable when the condition number is smaller than 1016. An application is given to illustrate that our algorithm is more accurate than MATLAB’s fft function. The results show that the relative error of our algorithm is from 1015 to 1017, and that of the fft was from 1012 to 1015. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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22 pages, 1091 KiB  
Article
Operator Calculus Approach to Comparison of Elasticity Models for Modelling of Masonry Structures
by Klaus Gürlebeck, Dmitrii Legatiuk and Kemmar Webber
Mathematics 2022, 10(10), 1670; https://0-doi-org.brum.beds.ac.uk/10.3390/math10101670 - 13 May 2022
Viewed by 1478
Abstract
The solution of any engineering problem starts with a modelling process aimed at formulating a mathematical model, which must describe the problem under consideration with sufficient precision. Because of heterogeneity of modern engineering applications, mathematical modelling scatters nowadays from incredibly precise micro- and [...] Read more.
The solution of any engineering problem starts with a modelling process aimed at formulating a mathematical model, which must describe the problem under consideration with sufficient precision. Because of heterogeneity of modern engineering applications, mathematical modelling scatters nowadays from incredibly precise micro- and even nano-modelling of materials to macro-modelling, which is more appropriate for practical engineering computations. In the field of masonry structures, a macro-model of the material can be constructed based on various elasticity theories, such as classical elasticity, micropolar elasticity and Cosserat elasticity. Evidently, a different macro-behaviour is expected depending on the specific theory used in the background. Although there have been several theoretical studies of different elasticity theories in recent years, there is still a lack of understanding of how modelling assumptions of different elasticity theories influence the modelling results of masonry structures. Therefore, a rigorous approach to comparison of different three-dimensional elasticity models based on quaternionic operator calculus is proposed in this paper. In this way, three elasticity models are described and spatial boundary value problems for these models are discussed. In particular, explicit representation formulae for their solutions are constructed. After that, by using these representation formulae, explicit estimates for the solutions obtained by different elasticity theories are obtained. Finally, several numerical examples are presented, which indicate a practical difference in the solutions. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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19 pages, 8649 KiB  
Article
Blood Flow in Multi-Sinusoidal Curved Passages with Biomimetic Rheology: An Application of Blood Pumping
by Gamal Hassan Sewify, Khurram Javid, Muhammad Adeel, Aamar Abbasi, Sami Ullah Khan, Mohamed Omri and Lioua Kolsi
Mathematics 2022, 10(9), 1579; https://0-doi-org.brum.beds.ac.uk/10.3390/math10091579 - 07 May 2022
Cited by 7 | Viewed by 1504
Abstract
The unsteady flow of biological liquid through non-uniform pumps under porosity impacts is considered. The Jeffrey fluid is used as blood in the current study, which is also characterized as viscoelastic fluid because of its dual characteristics: on the one hand, its viscosity [...] Read more.
The unsteady flow of biological liquid through non-uniform pumps under porosity impacts is considered. The Jeffrey fluid is used as blood in the current study, which is also characterized as viscoelastic fluid because of its dual characteristics: on the one hand, its viscosity in nature; on the other hand, its elastic effect. Rheological equations are framed in a curvilinear coordinates system, and porosity influences are simulated with the body force term in momentum equations. The flow system has been transformed from fixed to wave frame using a linear–mathematical transformation between these two frames. In the next mathematical steps, these transformed equations are given in non-dimensional form using physical variables. The system of PDE is reduced to an ODE under lubrication theory and long wavelength approximation. Solutions to reduced ordinary differential equations are obtained numerically in MATLAB software via a BVP4C scheme. The physical impacts of the involved parameters on flow features, such as curvature, porosity (Darcy’s number), non-uniformity, and viscoelastic parameters, have been visualized graphically. Multi-sinusoidal waves are used in the boundary wall of the curved pump for peristaltic pumping. The magnitude of velocity profile for a saw-tooth wave (trapezoidal wave) is larger (smaller) than all other natures of peristaltic waves. The larger intensity of Darcy’s number has a dynamic role in the reduction of peristaltic pumping, whereas the opposite behavior is noticed when increasing the non-uniform nature of a channel. A comparison between all multi-sinusoidal waves is also addressed. The results of the present research shall be very productive for the manufacture of peristaltic pumps for drug delivery and bio-medical systems. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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19 pages, 2133 KiB  
Article
Computer-Aided Methods for Molecular Classification
by Alina Bărbulescu, Lucica Barbeș and Cristian Ștefan Dumitriu
Mathematics 2022, 10(9), 1543; https://0-doi-org.brum.beds.ac.uk/10.3390/math10091543 - 04 May 2022
Cited by 1 | Viewed by 1381
Abstract
The study aims to analyze the degree of similarity of some molecules belonging to two subgroups of Aminoalkylindoles. After extracting the molecules’ characteristics using Cheminformatics methods, and the computation of the Tanimoto coefficients, dendrograms and heatmaps were built to reveal the degree of [...] Read more.
The study aims to analyze the degree of similarity of some molecules belonging to two subgroups of Aminoalkylindoles. After extracting the molecules’ characteristics using Cheminformatics methods, and the computation of the Tanimoto coefficients, dendrograms and heatmaps were built to reveal the degree of similarity of the analyzed drugs. Some atom-pair similarities between the molecules in the same group were detected. The clusters determined by the k-means method divided the Benzoylindoles into two subgroups but kept all the Phenylacetylindoles together in the same set. The activity spectrum of the elements in each group was also analyzed, and similarities have been emphasized. The clustering has been validated using the Kruskal–Wallis test on the series of computed probabilities of the main effects. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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19 pages, 359 KiB  
Article
On the Efficiency of Staggered C-Grid Discretization for the Inviscid Shallow Water Equations from the Perspective of Nonstandard Calculus
by Marcel Zijlema
Mathematics 2022, 10(9), 1387; https://0-doi-org.brum.beds.ac.uk/10.3390/math10091387 - 21 Apr 2022
Viewed by 1394
Abstract
This paper provides a rationale for the commonly observed numerical efficiency of staggered C-grid discretizations for solving the inviscid shallow water equations. In particular, using the key concepts of nonstandard calculus, we aim to show that the grid staggering of the primitive variables [...] Read more.
This paper provides a rationale for the commonly observed numerical efficiency of staggered C-grid discretizations for solving the inviscid shallow water equations. In particular, using the key concepts of nonstandard calculus, we aim to show that the grid staggering of the primitive variables (surface elevation and normal velocity components) is capable of dealing with flow discontinuities. After a brief introduction of hyperreals through the notion of infinitesimal increments, a nonstandard rendition of the governing equations is derived that essentially turns into a finite procedure and permits a convenient way of modeling the hydraulic jumps in open channel flow. A central result of this paper is that the discrete formulations thus obtained are distinguished by the topological structures of the solution fields and subsequently provide a natural framework for the staggered discretization of the governing equations. Another key of the present study is to demonstrate that the discretization naturally regularizes the solution of the inviscid flow passing through the hydraulic jump without the need of non-physical dissipation. The underlying justification is provided by analytically studying the distributions of the flow variables across an infinitesimal thin hydraulic jump along with the use of hyperreal Heaviside step functions. This main finding is shown to be useful to comprehend the importance of the application of staggered finite difference schemes to accurately predict rapidly varying free-surface flows. A numerical experiment is provided to confirm this result. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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Graphical abstract

12 pages, 795 KiB  
Article
Zeroing Neural Network for Pseudoinversion of an Arbitrary Time-Varying Matrix Based on Singular Value Decomposition
by Mariya Kornilova, Vladislav Kovalnogov, Ruslan Fedorov, Mansur Zamaleev, Vasilios N. Katsikis, Spyridon D. Mourtas and Theodore E. Simos
Mathematics 2022, 10(8), 1208; https://0-doi-org.brum.beds.ac.uk/10.3390/math10081208 - 07 Apr 2022
Cited by 19 | Viewed by 1731
Abstract
Many researchers have investigated the time-varying (TV) matrix pseudoinverse problem in recent years, for its importance in addressing TV problems in science and engineering. In this paper, the problem of calculating the inverse or pseudoinverse of an arbitrary TV real matrix is considered [...] Read more.
Many researchers have investigated the time-varying (TV) matrix pseudoinverse problem in recent years, for its importance in addressing TV problems in science and engineering. In this paper, the problem of calculating the inverse or pseudoinverse of an arbitrary TV real matrix is considered and addressed using the singular value decomposition (SVD) and the zeroing neural network (ZNN) approaches. Since SVD is frequently used to compute the inverse or pseudoinverse of a matrix, this research proposes a new ZNN model based on the SVD method as well as the technique of Tikhonov regularization, for solving the problem in continuous time. Numerical experiments, involving the pseudoinversion of square, rectangular, singular, and nonsingular input matrices, indicate that the proposed models are effective for solving the problem of the inversion or pseudoinversion of time varying matrices. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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9 pages, 329 KiB  
Article
A Tensor Splitting AOR Iterative Method for Solving a Tensor Absolute Value Equation
by Yuhan Chen and Chenliang Li
Mathematics 2022, 10(7), 1023; https://0-doi-org.brum.beds.ac.uk/10.3390/math10071023 - 23 Mar 2022
Viewed by 1492
Abstract
In this paper, a tensor splitting AOR iterative method for solving the H+-tensor absolute value equation is presented. Some sufficient conditions for the existence of the solution to the tensor absolute value equation are given. Under suitable conditions, the new method [...] Read more.
In this paper, a tensor splitting AOR iterative method for solving the H+-tensor absolute value equation is presented. Some sufficient conditions for the existence of the solution to the tensor absolute value equation are given. Under suitable conditions, the new method is proved to be convergent. Finally, some numerical examples demonstrate that our new method is effective. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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15 pages, 465 KiB  
Article
On a New Family of Runge–Kutta–Nyström Pairs of Orders 6(4)
by Vladislav N. Kovalnogov, Ruslan V. Fedorov, Dmitry A. Generalov, Ekaterina V. Tsvetova, Theodore E. Simos and Charalampos Tsitouras
Mathematics 2022, 10(6), 875; https://0-doi-org.brum.beds.ac.uk/10.3390/math10060875 - 09 Mar 2022
Cited by 8 | Viewed by 1466
Abstract
In this study, Runge–Kutta–Nyström pairs of orders 6(4) using six stages per step are considered. The main contribution of the present work is that we introduce a new family of pairs (i.e., new methodology of solution for order conditions) that possesses seven free [...] Read more.
In this study, Runge–Kutta–Nyström pairs of orders 6(4) using six stages per step are considered. The main contribution of the present work is that we introduce a new family of pairs (i.e., new methodology of solution for order conditions) that possesses seven free parameters instead of four, as used by similar pairs until now. Using these extra coefficients efficiently we may construct methods with better properties. Here, we exploit the free parameters in order to derive a pair with extended imaginary stability interval. This type of method may furnish better results on problems with periodic solutions. Extended numerical tests justify our effort. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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8 pages, 255 KiB  
Article
A Simple Affine-Invariant Spline Interpolation over Triangular Meshes
by László L. Stachó
Mathematics 2022, 10(5), 776; https://0-doi-org.brum.beds.ac.uk/10.3390/math10050776 - 28 Feb 2022
Viewed by 1306
Abstract
Given a triangular mesh, we obtain an orthogonality-free analogue of the classical local Zlámal–Ženišek spline procedure with simple explicit affine-invariant formulas in terms of the normalized barycentric coordinates of the mesh triangles. Our input involves first-order data at mesh points, and instead of [...] Read more.
Given a triangular mesh, we obtain an orthogonality-free analogue of the classical local Zlámal–Ženišek spline procedure with simple explicit affine-invariant formulas in terms of the normalized barycentric coordinates of the mesh triangles. Our input involves first-order data at mesh points, and instead of adjusting normal derivatives at the side middle points, we constructed the elementary splines by adjusting the Fréchet derivatives at three given directions along the edges with the result of bivariate polynomials of degree five. By replacing the real line R with a generic field K, our results admit a natural interpretation with possible independent interest, and the proofs are short enough for graduate courses. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
12 pages, 300 KiB  
Article
Runge-Kutta-Nyström Pairs of Orders 8(6) with Coefficients Trained to Perform Best on Classical Orbits
by Houssem Jerbi, Mohamed Omri, Mourad Kchaou, Theodore E. Simos and Charalampos Tsitouras
Mathematics 2022, 10(4), 654; https://0-doi-org.brum.beds.ac.uk/10.3390/math10040654 - 19 Feb 2022
Cited by 2 | Viewed by 1665
Abstract
In this study, we consider eight stages per step family of explicit Runge-Kutta-Nyström pairs of orders eight and six. The pairs from this family effectively use eight stages for each step. The coefficients provided by such a method are much less than the [...] Read more.
In this study, we consider eight stages per step family of explicit Runge-Kutta-Nyström pairs of orders eight and six. The pairs from this family effectively use eight stages for each step. The coefficients provided by such a method are much less than the number of non linear order conditions required to be solved. Thus, we traditionally apply various simplified assumptions in order to address this drawback. The assumptions taken in the family we consider here deliver a subsystem where all the coefficients are evaluated successively and explicitly with respect to five free parameters. We train (adjust) these free parameters in order to derive a certain pair that outperforms other similar pairs of orders 8(6) in Keplerian type orbits, e.g., Kepler, perturbed Kepler, Arenstorf orbit, or Pleiades. Differential evolution technique is used for the training. The pair that we finally present offers about an additional digit of accuracy in a variety of orbits. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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19 pages, 955 KiB  
Article
Parallel Direct and Iterative Methods for Solving the Time-Fractional Diffusion Equation on Multicore Processors
by Murat A. Sultanov, Elena N. Akimova, Vladimir E. Misilov and Yerkebulan Nurlanuly
Mathematics 2022, 10(3), 323; https://0-doi-org.brum.beds.ac.uk/10.3390/math10030323 - 20 Jan 2022
Cited by 9 | Viewed by 2484
Abstract
The work is devoted to developing the parallel algorithms for solving the initial boundary problem for the time-fractional diffusion equation. After applying the finite-difference scheme to approximate the basis equation, the problem is reduced to solving a system of linear algebraic equations for [...] Read more.
The work is devoted to developing the parallel algorithms for solving the initial boundary problem for the time-fractional diffusion equation. After applying the finite-difference scheme to approximate the basis equation, the problem is reduced to solving a system of linear algebraic equations for each subsequent time level. The developed parallel algorithms are based on the Thomas algorithm, parallel sweep algorithm, and accelerated over-relaxation method for solving this system. Stability of the approximation scheme is established. The parallel implementations are developed for the multicore CPU using the OpenMP technology. The numerical experiments are performed to compare these methods and to study the performance of parallel implementations. The parallel sweep method shows the lowest computing time. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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12 pages, 558 KiB  
Article
Unified Convergence Analysis of Chebyshev–Halley Methods for Multiple Polynomial Zeros
by Stoil I. Ivanov
Mathematics 2022, 10(1), 135; https://0-doi-org.brum.beds.ac.uk/10.3390/math10010135 - 03 Jan 2022
Cited by 5 | Viewed by 1508
Abstract
In this paper, we establish two local convergence theorems that provide initial conditions and error estimates to guarantee the Q-convergence of an extended version of Chebyshev–Halley family of iterative methods for multiple polynomial zeros due to Osada (J. Comput. Appl. Math. [...] Read more.
In this paper, we establish two local convergence theorems that provide initial conditions and error estimates to guarantee the Q-convergence of an extended version of Chebyshev–Halley family of iterative methods for multiple polynomial zeros due to Osada (J. Comput. Appl. Math. 2008, 216, 585–599). Our results unify and complement earlier local convergence results about Halley, Chebyshev and Super–Halley methods for multiple polynomial zeros. To the best of our knowledge, the results about the Osada’s method for multiple polynomial zeros are the first of their kind in the literature. Moreover, our unified approach allows us to compare the convergence domains and error estimates of the mentioned famous methods and several new randomly generated methods. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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3 pages, 210 KiB  
Correction
Correction: Tolmachev et al. Algorithmic Aspects of Simulation of Magnetic Field Generation by Thermal Convection in a Plane Layer of Fluid. Mathematics 2023, 11, 808
by Daniil Tolmachev, Roman Chertovskih and Vladislav Zheligovsky
Mathematics 2023, 11(10), 2395; https://0-doi-org.brum.beds.ac.uk/10.3390/math11102395 - 22 May 2023
Viewed by 532
Abstract
There was an error in the original publication [...] Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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