Research

39 pages, 490 KiB

Open AccessArticle

Speed of Convergence of Time Euler Schemes for a Stochastic 2D Boussinesq Model

by Hakima Bessaih and Annie Millet

Mathematics 2022, 10(22), 4246; https://doi.org/10.3390/math10224246 - 13 Nov 2022

Viewed by 835

We prove that an implicit time Euler scheme for the 2D Boussinesq model on the torus D converges. The various moments of the

W^{1, 2}

-norms of the velocity and temperature, as well as their discretizations, were computed. We obtained the [...] Read more.

We prove that an implicit time Euler scheme for the 2D Boussinesq model on the torus D converges. The various moments of the

W^{1, 2}

-norms of the velocity and temperature, as well as their discretizations, were computed. We obtained the optimal speed of convergence in probability, and a logarithmic speed of convergence in

L^{2} (Ω)

. These results were deduced from a time regularity of the solution both in

L^{2} (D)

and

W^{1, 2} (D)

, and from an

L^{2} (Ω)

convergence restricted to a subset where the

W^{1, 2}

-norms of the solutions are bounded. Full article

(This article belongs to the Special Issue Computational Methods in Nonlinear Analysis and Their Applications)

14 pages, 4486 KiB

Open AccessArticle

The Effects of Variable Thermal Conductivity in Thermoelastic Interactions in an Infinite Material with and without Kirchhoff’s Transformation

by Aatef Hobiny and Ibrahim Abbas

Mathematics 2022, 10(22), 4176; https://doi.org/10.3390/math10224176 - 08 Nov 2022

Cited by 1 | Viewed by 1059

Abstract

In this paper, the problem of an unbonded material under variable thermal conductivity with and without Kirchhoff’s transformations is investigated. The context of the problem is the generalized thermoelasticity model. The boundary plane of the medium is exposed to a thermal shock that [...] Read more.

In this paper, the problem of an unbonded material under variable thermal conductivity with and without Kirchhoff’s transformations is investigated. The context of the problem is the generalized thermoelasticity model. The boundary plane of the medium is exposed to a thermal shock that is time-dependent and considered to be traction-free. Because nonlinear formulations are difficult, the finite element method is applied to solve the problem without Kirchhoff’s transformations. In a linear case, when using Kirchhoff’s transformations, the problem’s solution is derived using the Laplace transforms and the eigenvalue approach. The effect of variable thermal conductivity is discussed and compared with and without Kirchhoff’s transformations. The graphical representations of numerical results are shown for the distributions of temperature, displacement and stress. Full article

(This article belongs to the Special Issue Computational Methods in Nonlinear Analysis and Their Applications)

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22 pages, 2952 KiB

Open AccessArticle

A Boundary Shape Function Method for Computing Eigenvalues and Eigenfunctions of Sturm–Liouville Problems

by Chein-Shan Liu, Jiang-Ren Chang, Jian-Hung Shen and Yung-Wei Chen

Mathematics 2022, 10(19), 3689; https://doi.org/10.3390/math10193689 - 09 Oct 2022

Cited by 3 | Viewed by 1312

Abstract

In the paper, we transform the general Sturm–Liouville problem (SLP) into two canonical forms: one with the homogeneous Dirichlet boundary conditions and another with the homogeneous Neumann boundary conditions. A boundary shape function method (BSFM) was constructed to solve the SLPs of these [...] Read more.

In the paper, we transform the general Sturm–Liouville problem (SLP) into two canonical forms: one with the homogeneous Dirichlet boundary conditions and another with the homogeneous Neumann boundary conditions. A boundary shape function method (BSFM) was constructed to solve the SLPs of these two canonical forms. Owing to the property of the boundary shape function, we could transform the SLPs into an initial value problem for the new variable with initial values that were given definitely. Meanwhile, the terminal value at the right boundary could be entirely determined by using a given normalization condition for the uniqueness of the eigenfunction. In such a manner, we could directly determine the eigenvalues as the intersection points of an eigenvalue curve to the zero line, which was a horizontal line in the plane consisting of the zero values of the target function with respect to the eigen-parameter. We employed a more delicate tuning technique or the fictitious time integration method to solve an implicit algebraic equation for the eigenvalue curve. We could integrate the Sturm–Liouville equation using the given initial values to obtain the associated eigenfunction when the eigenvalue was obtained. Eight numerical examples revealed a great advantage of the BSFM, which easily obtained eigenvalues and eigenfunctions with the desired accuracy. Full article

(This article belongs to the Special Issue Computational Methods in Nonlinear Analysis and Their Applications)

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43 pages, 3032 KiB

Open AccessArticle

Splitting Methods for Semi-Classical Hamiltonian Dynamics of Charge Transfer in Nonlinear Lattices

by Jānis Bajārs and Juan F. R. Archilla

Mathematics 2022, 10(19), 3460; https://doi.org/10.3390/math10193460 - 22 Sep 2022

Cited by 1 | Viewed by 1377

Abstract

We propose two classes of symplecticity-preserving symmetric splitting methods for semi-classical Hamiltonian dynamics of charge transfer by intrinsic localized modes in nonlinear crystal lattice models. We consider, without loss of generality, one-dimensional crystal lattice models described by classical Hamiltonian dynamics, whereas the charge [...] Read more.

We propose two classes of symplecticity-preserving symmetric splitting methods for semi-classical Hamiltonian dynamics of charge transfer by intrinsic localized modes in nonlinear crystal lattice models. We consider, without loss of generality, one-dimensional crystal lattice models described by classical Hamiltonian dynamics, whereas the charge (electron or hole) is modeled as a quantum particle within the tight-binding approximation. Canonical Hamiltonian equations for coupled lattice-charge dynamics are derived, and a linear analysis of linearized equations with the derivation of the dispersion relations is performed. Structure-preserving splitting methods are constructed by splitting the total Hamiltonian into the sum of Hamiltonians, for which the individual dynamics can be solved exactly. Symmetric methods are obtained with the Strang splitting of exact, symplectic flow maps leading to explicit second-order numerical integrators. Splitting methods that are symplectic and conserve exactly the charge probability are also proposed. Conveniently, they require only one solution of a linear system of equations per time step. The developed methods are computationally efficient and preserve the structure; therefore, they provide new means for qualitative numerical analysis and long-time simulations for charge transfer by nonlinear lattice excitations. The properties of the developed methods are explored and demonstrated numerically considering charge transport by mobile discrete breathers in an example model previously proposed for a layered crystal. Full article

(This article belongs to the Special Issue Computational Methods in Nonlinear Analysis and Their Applications)

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12 pages, 293 KiB

Open AccessArticle

Blow-Up of the Solution for a Semilinear Parabolic Problem with a Mixed Source

by Wai Yuen Chan

Mathematics 2022, 10(17), 3156; https://doi.org/10.3390/math10173156 - 02 Sep 2022

Cited by 2 | Viewed by 827

Abstract

A semilinear parabolic equation with the Dirichlet boundary condition is examined. The reaction source is a mixed nonlinear function. This paper investigates the existence and uniqueness of a solution. A sufficient condition for the solution being blow-up in finite time is found. Full article

(This article belongs to the Special Issue Computational Methods in Nonlinear Analysis and Their Applications)

24 pages, 9946 KiB

Open AccessArticle

Sawi Transform Based Homotopy Perturbation Method for Solving Shallow Water Wave Equations in Fuzzy Environment

by Mrutyunjaya Sahoo and Snehashish Chakraverty

Mathematics 2022, 10(16), 2900; https://doi.org/10.3390/math10162900 - 12 Aug 2022

Cited by 6 | Viewed by 1316

Abstract

In this manuscript, a new hybrid technique viz Sawi transform-based homotopy perturbation method is implemented to solve one-dimensional shallow water wave equations. In general, the quantities involved with such equations are commonly assumed to be crisp, but the parameters involved in the actual [...] Read more.

In this manuscript, a new hybrid technique viz Sawi transform-based homotopy perturbation method is implemented to solve one-dimensional shallow water wave equations. In general, the quantities involved with such equations are commonly assumed to be crisp, but the parameters involved in the actual scenario may be imprecise/uncertain. Therefore, fuzzy uncertainty is introduced as an initial condition. The main focus of this study is to find the approximate solution of one-dimensional shallow water wave equations with crisp, as well as fuzzy, uncertain initial conditions. First, by taking the initial condition as crisp, the approximate series solutions are obtained. Then these solutions are compared graphically with existing solutions, showing the reliability of the present method. Further, by considering uncertain initial conditions in terms of Gaussian fuzzy number, the governing equation leads to fuzzy shallow water wave equations. Finally, the solutions obtained by the proposed method are presented in the form of Gaussian fuzzy number plots. Full article

(This article belongs to the Special Issue Computational Methods in Nonlinear Analysis and Their Applications)

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14 pages, 300 KiB

Open AccessArticle

Solving Nonlinear Second-Order Differential Equations through the Attached Flow Method

by Carmen Ionescu and Radu Constantinescu

Mathematics 2022, 10(15), 2811; https://doi.org/10.3390/math10152811 - 08 Aug 2022

Cited by 4 | Viewed by 3198

Abstract

The paper considers a simple and well-known method for reducing the differentiability order of an ordinary differential equation, defining the first derivative as a function that will become the new variable. Practically, we attach to the initial equation a supplementary one, very similar [...] Read more.

The paper considers a simple and well-known method for reducing the differentiability order of an ordinary differential equation, defining the first derivative as a function that will become the new variable. Practically, we attach to the initial equation a supplementary one, very similar to the flow equation from the dynamical systems. This is why we name it as the “attached flow equation”. Despite its apparent simplicity, the approach asks for a closer investigation because the reduced equation in the flow variable could be difficult to integrate. To overcome this difficulty, the paper considers a class of second-order differential equations, proposing a decomposition of the free term in two parts and formulating rules, based on a specific balancing procedure, on how to choose the flow. These are the main novelties of the approach that will be illustrated by solving important equations from the theory of solitons as those arising in the Chafee–Infante, Fisher, or Benjamin–Bona–Mahony models. Full article

(This article belongs to the Special Issue Computational Methods in Nonlinear Analysis and Their Applications)

17 pages, 327 KiB

Open AccessArticle

DCA for Sparse Quadratic Kernel-Free Least Squares Semi-Supervised Support Vector Machine

by Jun Sun and Wentao Qu

Mathematics 2022, 10(15), 2714; https://doi.org/10.3390/math10152714 - 01 Aug 2022

Cited by 1 | Viewed by 1006

Abstract

With the development of science and technology, more and more data have been produced. For many of these datasets, only some of the data have labels. In order to make full use of the information in these data, it is necessary to classify [...] Read more.

With the development of science and technology, more and more data have been produced. For many of these datasets, only some of the data have labels. In order to make full use of the information in these data, it is necessary to classify them. In this paper, we propose a strong sparse quadratic kernel-free least squares semi-supervised support vector machine (

S S Q L S S^{3} V M

), in which we add a

ℓ_{0}

norm regularization term to make it sparse. An NP-hard problem arises since the proposed model contains the

ℓ_{0}

norm and another nonconvex term. One important method for solving the nonconvex problem is the DC (difference of convex function) programming. Therefore, we first approximate the

ℓ_{0}

norm by a polyhedral DC function. Moreover, due to the existence of the nonsmooth terms, we use the sGS-ADMM to solve the subproblem. Finally, empirical numerical experiments show the efficiency of the proposed algorithm. Full article

(This article belongs to the Special Issue Computational Methods in Nonlinear Analysis and Their Applications)

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38 pages, 5905 KiB

Open AccessArticle

Staggered Semi-Implicit Hybrid Finite Volume/Finite Element Schemes for Turbulent and Non-Newtonian Flows

by Saray Busto, Michael Dumbser and Laura Río-Martín

Mathematics 2021, 9(22), 2972; https://doi.org/10.3390/math9222972 - 21 Nov 2021

Cited by 12 | Viewed by 2063

Abstract

This paper presents a new family of semi-implicit hybrid finite volume/finite element schemes on edge-based staggered meshes for the numerical solution of the incompressible Reynolds-Averaged Navier–Stokes (RANS) equations in combination with the

k - ε

turbulence model. The rheology for calculating the laminar [...] Read more.

This paper presents a new family of semi-implicit hybrid finite volume/finite element schemes on edge-based staggered meshes for the numerical solution of the incompressible Reynolds-Averaged Navier–Stokes (RANS) equations in combination with the

k - ε

turbulence model. The rheology for calculating the laminar viscosity coefficient under consideration in this work is the one of a non-Newtonian Herschel–Bulkley (power-law) fluid with yield stress, which includes the Bingham fluid and classical Newtonian fluids as special cases. For the spatial discretization, we use edge-based staggered unstructured simplex meshes, as well as staggered non-uniform Cartesian grids. In order to get a simple and computationally efficient algorithm, we apply an operator splitting technique, where the hyperbolic convective terms of the RANS equations are discretized explicitly at the aid of a Godunov-type finite volume scheme, while the viscous parabolic terms, the elliptic pressure terms and the stiff algebraic source terms of the

k - ε

model are discretized implicitly. For the discretization of the elliptic pressure Poisson equation, we use classical conforming

P^{1}

and

Q^{1}

finite elements on triangles and rectangles, respectively. The implicit discretization of the viscous terms is mandatory for non-Newtonian fluids, since the apparent viscosity can tend to infinity for fluids with yield stress and certain power-law fluids. It is carried out with

P^{1}

finite elements on triangular simplex meshes and with finite volumes on rectangles. For Cartesian grids and more general orthogonal unstructured meshes, we can prove that our new scheme can preserve the positivity of k and

ε

. This is achieved via a special implicit discretization of the stiff algebraic relaxation source terms, using a suitable combination of the discrete evolution equations for the logarithms of k and

ε

. The method is applied to some classical academic benchmark problems for non-Newtonian and turbulent flows in two space dimensions, comparing the obtained numerical results with available exact or numerical reference solutions. In all cases, an excellent agreement is observed. Full article

(This article belongs to the Special Issue Computational Methods in Nonlinear Analysis and Their Applications)

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17 pages, 12413 KiB

Open AccessArticle

Industrial Steel Heat Treating: Numerical Simulation of Induction Heating and Aquaquenching Cooling with Mechanical Effects

by José Manuel Díaz Moreno, Concepción García Vázquez, María Teresa González Montesinos, Francisco Ortegón Gallego and Giuseppe Viglialoro

Mathematics 2021, 9(11), 1203; https://doi.org/10.3390/math9111203 - 26 May 2021

Cited by 3 | Viewed by 2142

Abstract

This paper summarizes a mathematical model for the industrial heating and cooling processes of a steel workpiece corresponding to the steering rack of an automobile. The general purpose of the heat treatment process is to create the necessary hardness on critical parts of [...] Read more.

This paper summarizes a mathematical model for the industrial heating and cooling processes of a steel workpiece corresponding to the steering rack of an automobile. The general purpose of the heat treatment process is to create the necessary hardness on critical parts of the workpiece. Hardening consists of heating the workpiece up to a threshold temperature followed by a rapid cooling such as aquaquenching. The high hardness is due to the steel phase transformation accompanying the rapid cooling resulting in non-equilibrium phases, one of which is the hard microconstituent of steel, namely martensite. The mathematical model describes both processes, heating and cooling. During the first one, heat is produced by Joule’s effect from a very high alternating current passing through the rack. This situation is governed by a set of coupled PDEs/ODEs involving the electric potential, the magnetic vector potential, the temperature, the austenite transformation, the stresses and the displacement field. Once the workpiece has reached the desired temperature, the current is switched off an the cooling stage starts by aquaquenching. In this case, the governing equations involve the temperature, the austenite and martensite phase fractions, the stresses and the displacement field. This mathematical model has been solved by the FEM and 2D numerical simulations are discussed along the paper. Full article

(This article belongs to the Special Issue Computational Methods in Nonlinear Analysis and Their Applications)

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