Finite Difference Method Research Articles

An Efficient Replacement for Schrödinger's Partial Differential Equation

The classical finite difference method for solving time-dependent partial differential equations has become quite tedious and requires the use of off-the-shelf algorithms such as those in MATLAB. The treatment by finite difference method then solution of the n resulting first order algebraic equations is quite difficult since the underlying matrix is singular. We propose an alternative revolutionary statistical technique using Bmatrix chains, which are a product of the Cairo techniques. This technique is valid for both classical macroscopic physics such as the heat diffusion equation and modern microscopic quantum mechanics such as Schrödinger's PDE. He completely neglects partial differential equations as if they never existed. The numerical results of the proposed numerical statistical method show stability, accuracy and superiority over conventional PDE methods.

A quantum computing concept for 1-D elastic wave simulation with exponential speedup

Summary Quantum computing has attracted considerable attention in recent years because it promises speedups that conventional supercomputers cannot offer, at least for some applications. Though existing quantum computers are, in most cases, still too small to solve significant problems, their future impact on domain sciences is already being explored now. Within this context, we present a quantum computing concept for 1-D elastic wave propagation in heterogeneous media with two components: a theoretical formulation and an implementation on a real quantum computer. The method rests on a finite-difference approximation, followed by a sparsity-preserving transformation of the discrete elastic wave equation to a Schrödinger equation, which can be simulated directly on a gate-based quantum computer. An implementation on an error-free quantum simulator verifies our approach and forms the basis of numerical experiments with small problems on the real quantum computer IBM Brisbane. The latter produce simulation results that qualitatively agree with the error-free version but are contaminated by quantum decoherence and noise effects. Complementing the discrete transformation to the Schrödinger equation by a continuous version allows the replacement of finite differences by other spatial discretization schemes, such as the spectral-element method. Anticipating the emergence of error-corrected quantum chips, we analyze the computational complexity of the best quantum simulation algorithms for future QCs. This analysis suggests that our quantum computing approach may lead to wave field simulations that run exponentially faster than simulations on classical computers.

Mixed, quantum-classical description of electron density transfer in the collision process

In this work, we investigate an ion-atom model describing the time-dependent evolution of electron density during the collision. For a S 3 + − H system, numerical simulations are based on classical trajectory calculations, and the electron density behaviour is described with the time-dependent Schrödinger equation. We apply the finite difference method to obtain quantitative insights into the charge transfer dynamics, providing detailed information about the spatial and temporal evolution of the collision process. The results are given for representative examples of the collision, from eV to keV range of energies, in head-on collision as well as for different values of impact parameter. A validity and precision of the proposed model and interpretation of the particle collision in terms of eigenstates are also discussed.

Non-central m-point formula in method of lines for solving the Korteweg-de Vries (KdV) equation

AbstractThe present study is committed to devising efficient spatial discretization with two non-central difference formulae incorporated in the method of lines (MOL). The method is then implemented numerically on the renowned dispersive evolution equation, the Korteweg-de Vries (KdV) model while infusing Euler and fourth-order Rung-Kutta (RK4) methods, respectively. The resulting schemes are proven to be numerically stable using Fourier’s stability approach, with the MOL matrix admitting negative real eigenvalues. Moreover, this proposal has been assessed on certain initial-boundary value problems of the KdV model by examining so many factors, like the percentage errors, absolute error differences, the error norms, and the invariants $$I_{k},$$ I k , for $$k=1,2,3,$$ k = 1 , 2 , 3 , among others. In this work, we solve KdV by applying a new approach of MOL to improve the results presented in some published articles. Lastly, the assessment revealed that our proposal was found to be better than those of the exponential finite-difference method (Exponential) and the Heat Balance Integral (HBI) methods, respectively, which serve as competing approaches for validation, in terms of both the accuracy and efficiency. The numerical examples are obtained by using MATLAB.

Numerical modeling of mixed convective nanofluid flow with fractal stochastic heat and mass transfer using finite differences

This study presents the first comprehensive numerical simulation of heat and mass transfer in fractal-like mixed convective nanofluid flows. The flow of non-Newtonian nanofluids over flat and oscillating sheets is modelled mathematically, and a finite difference scheme is used to solve this model. The two-stage scheme can tackle fractal and fractal stochastic mathematical models of partial differential equations. The consistency in the mean square is proved, and Fourier series stability analysis is adopted to find stability conditions for fractal stochastic partial differential equation. The scheme is applied to solve the unsteady Casson nanofluid flow over the flat and oscillatory sheet, which affects thermal radiation, heat source, and chemical reaction. The existence of the solution is also provided for the Navier-Stokes equation of the considered flow model using fractal time derivative. The graph illustrates that the proposed fractal technique achieves faster convergence than the Crank-Nicolson approach. Applications in energy systems, materials science, and environmental engineering are just a few of the domains that could benefit from a better understanding of mixed convective nanofluid flows with fractal features, and that is what this research study hopes to accomplish. Scientists and engineers may better develop efficient and environmentally friendly systems by simulating and analyzing these complicated processes with the suggested finite difference technique.

A linearly polarized light emission with a composite nanowire grating in whole white band

Abstract To obtain a highly linearly polarized light, a composite model consisting of white light emission, anti-reflection film, and metal-dielectric-metal nanowire grating was designed, analyzed, optimized, and fabricated. Based on the finite-difference time-domain method, the impacts of material, period, height, and incidence angle on the polarization performance of the composite model were discussed. The metal-dielectric-metal nanowire grating was fabricated on blue chip and fluorescent ceramics using nanoimprint technology. The employed materials of metal-dielectric-metal nanowire grating were aluminum and PMMA, with the period of 200 nm, wire width of 100 nm, and the height of metal and dielectric were 100 nm and 120 nm. Additionally, the anti-reflection film consisting of PMMA with the thickness of 45 nm was incorporated on fluorescent ceramics to enhance energy efficiency. Finally, through a series of test experiments, the composite model can be realized by the extinction ratio of 40 dB, while the transmittance of TM mode exceeds 50% at 450-750 nm. The theoretical analysis of this study is verified by experiments, and it has significant potential in the pursuit of high brightness, ultra-thin micro displays.

A new efficient two-step iterative method for solving absolute value equations

PurposeThis study presents a two-step numerical iteration method specifically designed to solve absolute value equations. The proposed method is valuable and efficient for solving absolute value equations. Several numerical examples were taken to demonstrate the accuracy and efficiency of the proposed method.Design/methodology/approachWe present a two-step numerical iteration method for solving absolute value equations. Our two-step method consists of a predictor-corrector technique. The new method uses the generalized Newton method as the predictor step. The four-point open Newton-Cotes formula is considered the corrector step. The convergence of the proposed method is discussed in detail. This new method is highly effective for solving large systems due to its simplicity and effectiveness. We consider the beam equation, using the finite difference method to transform it into a system of absolute value equations, and then solve it using the proposed method.FindingsThe paper provides empirical insights into how to solve a system of absolute value equations.Originality/valueThis paper fulfills an identified need to study absolute value equations.

Analysis of a Narrow-Stencil Finite Difference Method for Approximating Viscosity Solutions of Fully Nonlinear Second Order Parabolic PDEs

Analysis of a Narrow-Stencil Finite Difference Method for Approximating Viscosity Solutions of Fully Nonlinear Second Order Parabolic PDEs

Development of a Novel Fractional Magneto-Viscoelastic Dynamic Model for an Adaptive Beam Featuring Functional Composite Magnetoactive Elastomers: Simulations and Experimental Studies

Recently, the rapid emergence of functional composite magnetoactive elastomers with integrated hard magnetic particles (known as hard magnetoactive elastomers) has attracted significant interest in fields such as soft robotics and material science. It is of paramount importance to develop an efficient model that can accurately predict the nonlinear dynamic behavior of adaptive structures for their practical application and the synthesis of effective control strategies. In this study, a novel nonlinear fractional magneto-viscoelastic model of an adaptive cantilevered beam featuring hard magnetoactive elastomers has been developed. This model effectively characterizes the beam’s nonlinear and rate-dependent response behavior under magnetic stimuli of varying frequencies and magnitudes. Considering the fractional Kelvin-Voigt energy dissipation model and large deformation nonlinearity, the governing equations for the cantilevered hard-magnetic soft beam were derived using the Hamilton principle. The finite difference method, combined with the Galerkin modal decomposition scheme, was subsequently utilized to discretize the time and space domains, respectively. A hardware-in-the-loop experimental framework was finally designed to experimentally investigate the beam’s nonlinear response behavior and validate the simulation results. The simulated nonlinear quasi-static responses of the beam, subjected to magnetic loading up to 30 mT, exhibited excellent agreement with the experimental data collected. Moreover, the simulated dynamic responses of the beam, subjected to various amplitudes of the applied magnetic field and magnetic frequencies up to 2 Hz, were thoroughly investigated. These included time-histories, phase-plane, and hysteretic responses, all demonstrating strong alignment with the experimental observations.

A GFDM based computational model for the analysis of tunnels under gravitational loadings

Based on an impedance relation, a new computational model is developed for the analysis of tunnels under gravitational loading. The shape of tunnel is arbitrary. The impedance matrix, representing the soil resistance, is evaluated through integration of the equations of elasticity by generalized finite difference method (GFDM). The rigidity matrix of tunnel ring is constructed by using two types of elements: plate and shell elements. The equations of these elements are evaluated through integration of the equations of higher order plate and shell theories again by GFDM. These element equations accommodate not only axial and bending deformations, but, also shear deformations in the ring. In the study, the rotational joint deformation is simulated through the use of rotational spring model and the slippage is considered by modifying the impedance matrix so that the tangential soil resistance force between the ring and soil medium vanishes. Some numerical results are presented, in nondimensional forms, for circular and square tunnels where the following three cases are considered for circular tunnels: a) the soil medium is infinite, the change in σv (vertical (in situ) compressional stress) in the vicinity of tunnel is disregarded b) the soil medium is infinite, the change in σv is taken into account c) the soil medium is halfspace (HS). In connection with the case of (a), a comparison is presented for the ring flexibility curves of radial displacement and section forces of circular ring with those reported in literature. In view of the interpretations of the results and comparisons, we think, the proposed model may be used reliably in the analysis of tunnels.

Accurate Description of Coulombic Interactions in Organic Field‐Effect Transistors Enabled by Efficient 3D Poisson's Equation Solver with Mixed Boundary Conditions

Accurate Description of Coulombic Interactions in Organic Field‐Effect Transistors Enabled by Efficient 3D Poisson's Equation Solver with Mixed Boundary Conditions

Electro-fluid-dynamics (EFD) of soft-bodied organisms swimming through mucus having dilatant, viscous, and pseudo-plastic properties

The sperm propelling mechanism has been proposed as a possible resource for soft micro-robots in confined spaces, with potential applications in biomedical engineering. Human sperm cells essentially swim through the non-Newtonian liquid (cervical mucus) to reach their target. Thus, sperm cells swimming through non-Newtonian fluids is not vital only for physiology, but also for the fabrication of swimming micro-robots. Inspired by these remarkable applications, we examine the basic mechanics of spermatozoa motility using an undulating sheet model. This undulating sheet is bounded between two rigid walls which is self-propeling in the negative axial direction. The liquid around the spermatozoa is taken as Carreau fluid with electro-osmotic properties. The application of the lubrication approximation results in the reduction of momentum equations. The resulting ODE is solved numerically via the finite difference method and MATLAB’s built-in routine bvp5c. The unknowns that are present in the boundary conditions are refined by the root-finding algorithm. Power losses, cell speed, flow rate, velocity of the fluid, and streamline pattern are visualized by graphs. The findings of this study have important implications for the designing and optimization of electrically controlled microswimmers.

Analysis of the Sapphire Fiber Fabry-Perot Interferometer Fringe Visibility under Different Excitation Conditions

In this paper, a theoretical analysis of how the excitation conditions affect the sapphire fiber Fabry-Perot interferometer (SFPI) visibility was performed. The conditions were considered, in which an SFPI was excited by a single-mode fiber (SMF), a multimode fiber (MMF), and a fiber collimator. The finite difference method (FDM) was used to realize the numerical solution of the modal electric fields, and then, the modal excited distributions in the sapphire fiber and the SFPI visibility were calculated. The results showed that different numbers of modes were excited in sapphire fibers under different excitation conditions and finally affected the fringe visibility of the SFPI. The fiber collimator excited the fewest modes and the visibility remained at the highest level. Finally, an experiment was performed, and the experimental results agreed well with the theoretical results.

Pulsatile flow of blood and Cattaneo Christov heat flux model in diseased arteries with variable properties

ABSTRACT The current effort is executed to interpret the heat transmission of combined stenosis and aneurysm in an artery, taking into account variable properties such as viscous dissipation and thermal radiation. We reduced the two directional energy and momentum equations to unidirectional flow by normalizing them under the mild stenotic assumption. After that, a mathematical framework for the physical problem is developed. An intricate nonlinear set of partial differential equations results from this formulation. A suitable numerical method of finite difference is employed on consequent equations to solve them. Afterward, graphical results predict the region of flow and the influence of involved parameters is shown in pictorial form. Furthermore, these solutions are employed to test how certain parameters, like streamlines, flow rate, and wall shear stress, affect blood flow. Using streamlines, we will also observe the general behavior of blood flow patterns.

A Time-Domain Wavenumber Integration Model for Underwater Acoustics Based on the High-Order Finite Difference Method

Simulating the acoustic field excited by pulse sound sources holds significant practical value in computational ocean acoustics. Two primary methods exist for modeling underwater acoustic propagation in the time domain: the Fourier synthesis technique based on frequency decomposition and the time-domain underwater acoustic propagation model (TD-UAPM). TD-UAPMs solve the wave equation in the time domain without requiring frequency decomposition, providing a more intuitive explanation of the physical process of sound energy propagation over time. However, time-stepping numerical methods can accumulate numerical errors, making it crucial to improve the algorithm’s accuracy for TD-UAPMs. Herein, the time-domain wavenumber integration model SPARC was improved by replacing the second-order finite element method (FEM) with the high-order accuracy finite difference method (FDM). Furthermore, the matched interface and boundary (MIB) method was used to process the seabed more accurately. The improved model was validated using three classic underwater acoustic benchmarks. By comparing the acoustic solutions obtained using the FDM and the FEM, it is evident that the improved model requires fewer grid points while maintaining the same level of accuracy, leading to lower computational costs and faster processing compared to the original model.

Entropy optimization of lid-driven micropolar hybrid nanofluid flow in a partially porous hexagonal-shaped cavity with relevance to energy efficient storage processes

Hydromagnetically associated heat convection can greatly enhance the performance of high-efficiency thermal appliances and renewable energy sources through an optimized design. This investigation examines the production of thermodynamic irreversibility and heat convection for a double lid-driven flow within a partially porous stratified hexagonal enclosure. The top and bottom-wall are moving in the opposite direction with an equal velocity U0. The top-wall and the bottom-wall are kept at temperature Tc and Th (Th >\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$>$$\\end{document} Tc) while the slanted walls are assumed to be thermally insulated. A constant magnetic field is employed in the horizontal x-direction. The hexagonal cavity was filled with a micropolar hybrid nanofluid Ag-MgO/water. The system of dimensionless equations was solved by the finite difference method (FDM) associated with successive over-relaxation (SOR), successive under-relaxation (SUR), and Gauss–Seidel iteration tactics and required results are computed with problem specific program in MATLAB code. The results indicate that the Ra and the thickness of the porous layer (Xp) significantly influences heat convection and thermal irreversibility processes. The Nuavg and STotal rises 6.299% and 3.373% as ‘ϕhnf\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\upphi }_{{\ ext{hnf}}}$$\\end{document}’ enhances from 0 to 4%, respectively. Furthermore, as the values of Ra, Ha, K0, and ϕhnf\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\upphi }_{{\ ext{hnf}}}$$\\end{document} increase, Beavg experiences a decline of 53.73%, 11.04%, 38.36%, and 0.09% respectively. Also, movement of wall has a significant impact on heat transfer rates and entropy production. The present study may be extended in numerous areas to mimic the problems like—(1) onset of thermo-mechanical process for solid–fluid interaction in a conduit. (2) Thermos-chemical process with extraction of ions in two-phase fluid for double layer plating on a continuously moving sheet, as region of porous stratum saturated with a class of fluid and region without porous medium occupied with other fluid.

MAINTAINING GAS RECOVERY IN LOW-PERMEABILITY FORMATIONS

The purpose of this work was to ensure efficient gas production in low-permeability and heterogeneous formations by numerically modeling the distribution of reservoir pressure drop around production and injection wells. To investigate this goal, we used a simulation based on the combined finite-element-difference method for nonstationary Leibenson piezoelectric conductivity problems. The method is based on the combination of elements of the already known finite difference and finite element methods for the nonstationary piezoelectric conductivity problem. The study found that there are many factors that affect the process of maintaining gas production in poorly permeable areas of a gas-bearing formation. Among them, first of all, after increasing the permeability of the corresponding section of the reservoir where production is created. Another important factor for maintaining gas recovery is the ability to naturally or artificially maintain gas phase infiltration within the affected reservoir area. This aspect is especially important for ensuring stable gas recovery over a long period of operation. During the intensive and long-term operation of the working area of the gas-bearing formation, additional injection wells were installed in the complex of production wells to avoid its depletion. It is important to note that provided a high level of gas phase penetration at the beginning of field operation, other factors of the production process, such as additional production wells and infiltration within the working area, have a lesser impact on the overall pressure distribution in this area. Thus, the study shows that with the passage of time and prolonged operation of the working section of the reservoir, the influence of its penetration decreases, but the importance of injected substances in the injection wells and gas phase infiltration within this section of the reservoir increases.

A general fourth-order mesoscopic multiple-relaxation-time lattice Boltzmann model and its macroscopic finite-difference scheme for two-dimensional diffusion equations

In this work, we first develop a general mesoscopic multiple-relaxation-time lattice Boltzmann (MRT-LB) model for the two-dimensional diffusion equation with the constant diffusion coefficient and source term, where the D2Q5 (five discrete velocities in two-dimensional space) lattice structure is considered. Then we exactly derive the corresponding macroscopic finite-difference scheme of the MRT-LB model. Additionally, we also propose a proper MRT-LB model for the diffusion equation with a linear source term, and obtain a macroscopic six-level finite-difference scheme. After that, we conduct the accuracy and stability analysis on the finite-difference scheme and the mesoscopic MRT-LB model, and find that at the diffusive scaling, both of them can achieve a fourth-order accuracy in space based on the Taylor expansion. The stability analysis also shows that they are both unconditionally stable. Finally, some numerical experiments are conducted, and the numerical results are also consistent with our theoretical analysis.

Numerical Solution of Volterra-Hammerstein Integral Equation of the First Kind by Finite Difference Method Decomposition with Nyström Method

Numerical Solution of Volterra-Hammerstein Integral Equation of the First Kind by Finite Difference Method Decomposition with Nyström Method

Lie group analysis on EMHD Jeffrey nanofluid flow with exponential heat source: Heat transfer optimization using RSM

The present research examines the behavior of a Jeffrey nanofluid flow across a stretching sheet under the effect of electric and magnetic fields. It comprises the Buongiorno model as well as an exponential heat source. The impact of chemical reaction has also been taken into consideration. While assuming no mass flux, the study considers boundary conditions for thermal convection and velocity slip. Lie group transformations are employed to transform the set of governing equations into a dimensionless system and later simulated using the finite difference scheme. It is found that the velocity profile rises as the Deborah number is enhanced whereas the ratio of relaxation to retardation time parameter has an inverse effect on the velocity profile. It is noted that per unit change in the Deborah number descends the drag coefficient by 31.29%. In this study, the response surface methodology and sensitivity analysis have been conducted by choosing heat transport as the dependent variable and the electric-field parameter   ( 0.01 ≤ E ≤ 0.03 ) , exponential heat source parameter ( 0.02 ≤ Qe ≤ 0.06 ) , and Biot number ( 0.15 ≤ Bi ≤ 0.25 ) as the independent variables. The Nusselt number escalates when the Bi number is increased and drops as the E values are raised. In the instance of the Biot number, the Nusselt number exhibits the maximum sensitivity.