Finite Difference Method Research Articles

Effects of the Shaft Speed on Stiffness and Damping Coefficients of Hydrodynamic Bearing-Shaft System under Variable Viscosity

The rotating shaft-hydrodynamic bearings systems operated with high speed and/or heavy load conditions expose serious rotor-dynamics instability problems due to characteristics of the supporting bearings. The stability and the dynamics of these systems, directly relate to the lubricant properties that are directly affect by the heat generation. In this study, the dynamic characteristics of a shaft-hydrodynamic journal bearing and its stability were investigated under variable viscosity. The equations of lubricant flow were derived by Dowson’s equation under variable viscosity, and the perturbation equations were obtained for 2 degrees-of-freedom system. The heat transfers between oil and the journal surface was modelled in a 3-dimensional energy equation, and the heat transfer on the journal structure was also modelled with heat conduction equation. An algorithm based on finite difference scheme with successive over relaxation method was developed to solve the theoretical models, simultaneously, and a serial simulation was performed to investigate the variations of the dynamic coefficients of the bearing-shaft system concerning the rotating speed for different radial clearance values. It was determined that the high speed increases the lubricant temperature, and so the static and dynamic performance characteristics decrease, moreover, this effect is more dominant for the smaller radial clearance.

A structure-preserving explicit numerical scheme for the Allen–Cahn equation with a logarithmic potential

This paper presents a stability analysis of a structure-preserving explicit finite difference method (FDM) for the Allen–Cahn (AC) equation with a logarithmic potential that has two arguments. Firstly, we compute the temporal step constraint that guarantees that if the initial condition is bounded by the two arguments of the minimum, then the numerical solutions are always bounded by them, i.e., the explicit numerical scheme satisfies the maximum principle. Secondly, we compute the temporal step constraint that guarantees that the discrete total energy of the system is non-increasing over time. To validate the preservation of the maximum principle and the decrease in discrete total energy, we perform numerical experiments.

Comparison analysis of entropy and entransy under various conditions of a hollow fiber membrane humidifier (HFMH)

The hollow fiber membrane heat and moisture transfer model is converted equivalently into a series of parallel-plate membranes stacked along the z-axis to simplify the model. A detailed modeling process is described based on the concept of finite difference method. The comparison of experimental values with calculated values ensures the validity of the model. Subsequently, an entransy dissipation model for the heat and moisture transfer process in a hollow fiber membrane humidifier (HFMH) is developed by analogizing heat and moisture conduction to electrical conduction. The variation of heat/moisture transfer capability within the HFMH is investigated from the perspective of “heat/moisture potential energy”. In addition, an entropy generation model for the HFMH is established to study the irreversible losses in the heat and moisture transfer process caused by irreversibility. The performance, entropy generation rate, and entransy dissipation rate of the HFMH under different conditions are comparatively analyzed. It can be found that the heat/moisture transfer performance of the humidifier can be improved by adjusting the operating conditions to increase the heat/moisture transfer driving force. However, the irreversible losses increase due to a higher heat/moisture transfer energy grade. The sensible and latent effectiveness can be enhanced by 20.9%–85.2 %, while the equivalent resistance can be reduced by 16.2%–42.0 % via optimizing the geometry structures and membrane properties. In the hot-dry region, the heat and moisture transfer increase by 25.9%–89.8 %, while the equivalent resistances decrease by 20.8%–66.0 %. In addition, the minimum equivalent resistances correspond to the optimal HFMH performance. The entransy analysis avoids the “paradox” of entropy analysis.

Application of spatial filtering high order FDTD method in the multi-physics field coupled equations

The traditional finite-difference time-domain method (FDTD(2, 2)) is widely employed to solve the coupled equations of quantum-electromagnetic multi-physics. However, the time step of the traditional explicit FDTD(2, 2) method is limited by the Courant-Friedrichs-Lewy (CFL) condition. Besides, the FDTD(2, 2) method only achieves the second-order accuracy, which can result in significant error accumulation. To overcome this problem, we propose the SF-HO-FDTD(2, q) method with scalable time stability conditions for solving the multi-physics field coupled equations (q denotes q-th order spatial difference scheme). This approach integrates spatial filtering (SF) and higher-order FDTD (HO-FDTD) methods. Moreover, the proposed method requires no further derivation of the iterative equations of the traditional HO-FDTD(2, q) method. It only needs to introduce the SF operation in each numerical iteration process to filter out the spatial high-frequency components arising from the use of time steps that do not satisfy the CFL stability condition. This condition is applied to ensure the stability of the numerical method. Therefore, the proposed method has a high compatibility with the traditional HO-FDTD(2, q) method. Then, the numerical dispersion analysis of the SF-HO-FDTD(2, q) method is theoretically studied. Finally, numerical examples are carried out to confirm the accuracy and efficiency of the strategy suggested in this study.

A phase velocity preserving fourth-order finite difference scheme for the Helmholtz equation with variable wavenumber

Numerically solving the Helmholtz equation with large wavenumbers can be challenging due to the highly oscillatory nature of the solution. This paper proposes a 3-point finite difference scheme for numerically solving the 1D Helmholtz equation with variable wavenumber. The proposed scheme preserves the phase velocity, making it well-suited for problems with large wavenumbers. Convergence analysis shows that the proposed scheme is guaranteed to have fourth-order accuracy. Numerical results demonstrate that the proposed scheme maintains high accuracy even for problems with large wavenumbers.

Modeling acoustic wavefields from moving sources in the presence of a time-varying free surface

Marine vibrators are increasingly being recognized as a viable alternative to seismic airguns for ocean-bottom acquisition due to their ability to generate more low-frequency content and their less adverse impact on marine wildlife. However, their use introduces processing challenges, such as the Doppler effect and time-dependent source-receiver offsets, which are negligible in conventional airgun acquisition. In addition, the time-varying nature of the sea surface during the multisecond acquisition time introduces further challenges for processing and inversion. To accurately account for source motion and time-varying sea-surface effects in seismic data processing, we develop a reliable and robust numerical modeling tool. We use a mimetic finite-difference method in a generalized coordinate system to model the full acoustic wavefield triggered by a moving source in the presence of a time-varying sea surface. Our approach uses a time- and space-dependent coordinate transformation, which tracks the source movement and conforms to the irregular time-varying sea surface, to map an irregular physical domain in Cartesian coordinates to a regular computational domain in generalized coordinates. We formulate this coordinate transformation such that the coordinate systems conformally match below the ocean-bottom level. Numerical examples demonstrate that this approach is accurate and stable, even for an unrealistically exaggerated sea state. This computational tool is not limited to modeling but also could be used to develop advanced processing techniques for marine vibrator data, such as imaging and inversion.

Analyzing the dynamic patterns of COVID-19 through nonstandard finite difference scheme

This paper presents a novel approach to analyzing the dynamics of COVID-19 using nonstandard finite difference (NSFD) schemes. Our model incorporates both asymptomatic and symptomatic infected individuals, allowing for a more comprehensive understanding of the epidemic's spread. We introduce an unconditionally stable NSFD system that eliminates the need for traditional Runge–Kutta methods, ensuring dynamical consistency and numerical accuracy. Through rigorous numerical analysis, we evaluate the performance of different NSFD strategies and validate our analytical findings. Our work demonstrates the benefits of using NSFD schemes for modeling infectious diseases, offering advantages in terms of stability and efficiency. We further illustrate the dynamic behavior of COVID-19 under various conditions using numerical simulations. The results from these simulations demonstrate the effectiveness of the proposed approach in capturing the epidemic's complex dynamics.

Numerical paradigm to explore the chemically reacting Williamson nanofluid flow with the influence of bioconvection effects using neural networks

This research focuses on using a type of advanced computer program called artificial neural networks (ANNs) using the Levenberg–Marquardt backpropagation technique (LMBPT-NNs) to stimulate the behavior of bioconvective effects on heat and mass transport in non-Newtonian chemically Williamson nanofluid through stretched surface (BEHMT-NNCRWNF-SS). Williamson nanofluid flow refers to specially engineered fluids with nanoparticles dispersed within them and moving when subjected to external forces. Understanding their flow behavior is crucial to effectively use them in various applications, especially in the field of heat and mass transfer and fluid dynamics. Before being solved numerically, a couple of governing partial differential equations are transformed into ordinary differential equations through suitable similarity functions. The finite difference method (FDM) (Lobatto IIIA) is applied for the solution of a nonlinear nanofluidic system by selecting different collocation points. These collocation points play a crucial role in approximating the solution and ensuring accurate results. To address various fluidic problems, it's important to use the appropriate results of FDM to create a reference dataset for the LMBPT-NNs. Statistical analysis should then be performed on the training, testing, and validation of the reference datasets to obtain the optimal scheme for different fluidic flow issues.. The precision of the LMBPT-NNs is checked through ANN tools like mean square error (MSE), regression analysis, curve fitness, and histogram error.

Numerical approximation of a two-dimensional fractional neutron diffusion model describing dynamics of neutron flux in a nuclear reactor

The goal of this work is to develop a high-order numerical technique to solve a two-dimensional fractional neutron diffusion (FND) model describing dynamical behavior of a lead-cooled fast reactor. This method is based on L1 technique for temporal discretization and a high-order compact Alternating Direction Implicit (ADI) finite difference scheme for the spatial discretization. Numerical simulations are conducted to demonstrate the applicability of the method. We calculate the neutronic flux in the core and examine how the order of the temporal-fractional derivative influences the neutronic flux. The results obtained with FND model for different values of anomalous diffusion coefficient are compared against those obtained with P1 model and classical neutron diffusion equation. In addition, the computational time (CPU time) is provided in order to justify the computational efficiency of the proposed method.

Efficient Numerical Solutions for Fuzzy Time Fractional Diffusion Equations Using Two Explicit Compact Finite Difference Methods

This article introduces an extension of classical fuzzy partial differential equations, known as fuzzy fractional partial differential equations. These equations provide a better explanation for certain phenomena. We focus on solving the fuzzy time diffusion equation with a fractional order of 0 < α ≤ 1, using two explicit compact finite difference schemes that are the compact forward time center space (CFTCS) and compact Saulyev’s scheme. The time fractional derivative uses the Caputo definition. The double-parametric form approach is used to transfer the governing equation from an uncertain to a crisp form. To ensure stability, we apply the von Neumann method to show that CFTCS is conditionally stable, while compact Saulyev’s is unconditionally stable. A numerical example is provided to demonstrate the practicality of our proposed schemes.

Reliability analysis and risk assessment of pile-reinforced slopes considering spatial soil variability and site investigation

ABSTRACT Site investigation data is of great significance for practical slope stability analyses and designs. Previous studies on the probabilistic analysis of pile-reinforced slopes, however, rarely considered the influence of site-specific information, which may lead to a biased judgment of the stability of the soil-pile system, thus resulting in unreasonable designs. This paper proposes a new integrated probabilistic framework to analyse the probabilistic characteristics and quantitative risk of the pile-reinforced slopes considering site investigation data. Particularly, with the aid of the framework, the responses of the stabilising piles under different schemes of investigation boreholes are systematically studied. The Random Finite Difference Method within the framework of Monte Carlo Simulation (MCS) and Subset simulation is employed to explore the uncertainty within the pile-reinforced slope and the associated pile responses. A soil slope is taken as an illustrative example to validate the effectiveness of the proposed framework through various parametric studies on the investigation scheme of boreholes (e.g., numbers, locations, and sampling intervals). It is found that the stability results obtained from the deterministic analysis with mean soil properties and the conventional unconditional random field MCS may be biased compared to the results by the proposed framework considering real-site situations.

Comparison of Single-Phase Mathematical Models for Solid-State Packed Beds for Thermal Energy Storage

This article presents an analytical solution for the evaluation of the thermal performance of packed bed sensible heat storage. The numerical model developed was tested for four different solid storage mediums. The thermal energy equation is solved numerically by deploying the finite difference method. The presented analytical solution is based on a novel mathematical approach. The numerical model was validated using the computer simulation package Comsol Multiphysics v5.3. Our numerical model results are in good agreement with the published experimental data, with an overall difference of ~10%. Hence, the numerical model is an efficient way of evaluating the thermal performance of packed bed thermal energy storage systems compared to other numerical strategies or computer simulation techniques. This proves that the novel analytical model has shown to be a reliable and broadly accurate approach to acquire the thermal performance of sensible heat storage.

Analysis of Spatial Curved Bi-Fixed Beam with Varying Curvature and Varying Cross-Sectional Area Using Finite Displacement Transfer Method

This article is focused on determination of fixed-end reactions, displacements, internal forces and internal moments for the spatial curved bi-fixed beam with varying curvature and varying cross-sectional area. The objective is to analyze spatially curved beam without involving analytical differentiation and integration of the governing equations. A finite displacement transfer method to analyze spatially curved beam is used to obtain numerically approximate analysis results and to eliminate analytical differentiation and integration, completely. The results of the analysis procedure are compared with the results of other methods found in the literature. For the specific case of the circular helical beam, the fixed end reactions' values have absolute maximum and minimum percentage differences of 1.23% and 0.06%, respectively. For the specific case of the elliptic-helical beam, the fixed end reactions' values have absolute maximum and minimum percentage differences of 1.13% and 0.0%, respectively. The finite difference method, along with the recursive scheme, gives reasonably accurate results without involving complex calculations

Numerical simulation of coupled Klein–Gordon–Schrödinger equations: RBF partition of unity

The coupled Klein–Gordon–Schrödinger equations have significant implications in quantum field theory, particle physics, cosmology, and nonlinear dynamics. In this study, we propose an efficient method for numerically simulating this system. The proposed approach involves employing the radial basis function partition of unity for spatial discretization. This method utilizes scaled Lagrange basis functions with polyharmonic spline kernels, taking advantage of the scalability property of the polyharmonic kernel to ensure stability in the approximation process. The resulting spatially discretized system yields a nonlinear time-dependent set of differential equations. To solve this system, we combine an implicit central finite difference scheme with a predictor–corrector procedure to overcome the nonlinearity. In the numerical results section, we assess the efficiency, accuracy, and versatility of the proposed method by conducting several simulations in large domains over extended periods. We present and compare these results with those obtained using alternative methods, demonstrating the effectiveness of the proposed approach in accurately solving the coupled Klein–Gordon–Schrödinger equations.

Two linear energy-preserving compact finite difference schemes for coupled nonlinear wave equations

In this paper, we propose and analyze two highly efficient compact finite difference schemes for coupled nonlinear wave equations containing coupled sine-Gordon equations and coupled Klein-Gordon equations. To construct energy-preserving, high-order accurate and linear numerical methods, we first utilize the scalar auxiliary variable (SAV) approach and introduce three auxiliary functions to rewrite the original problem as a new equivalent system. Then we make use of the compact finite difference method and the Crank-Nicolson method to propose an efficient fully-discrete scheme (SAV-CFD-CN). The modified energy conservation and the convergence of the SAV-CFD-CN scheme are proved in detail, which has fourth-order convergence in space and second-order convergence in time. In order to preserve the discrete energy of original system, we further combine Lagrange multiplier approach, compact finite difference method and the Crank-Nicolson method to propose the second fully-discrete scheme (LM-CFD-CN). The proposed two schemes are high-order accurate, linear and highly efficient, only four symmetric positive definite systems with constant coefficients are required to be solved at each time level. Numerical experiments for the coupled nonlinear wave equations are given to confirm theoretical findings.

A time-domain SGFD-FK hybrid method for 2D teleseismic elastic wave modeling and inversion

AbstractFull waveform inversion (FWI) has proved to be a reliable tool for high-resolution imaging of lithospheric structures at various depths down to the upper mantle. However, when the size of the model is large, the computational burden is significative and applications are restricted to low frequencies. To tackle this issue, we developed a new 2D time-domain hybrid method to simulate high-frequency teleseismic body waves propagating through a local heterogeneous elastic Earth model: The frequency-wavenumber (FK) integration method is coupled with the staggered grid finite difference method (SGFD). The FK method is used to compute the wavefield due to obliquely incident plane P and SV waves in a 1D multilayered half space that excites a local heterogeneous region. Inside this region, the velocity-stress staggered grid FD method (SGFD) is used to accurately deal with wave propagation in heterogeneous media. Spurious waves that might be generated at the boundaries of the local region are avoided using convolutional perfectly matched layers (CPML). This new hybrid method inherits the low-memory requirements of the FK method and the accuracy, efficiency and easy implementation of the SGFD. The new hybrid method is benchmarked against the analytical FK method for some canonical models and shows good agreement with analytical solutions. Subsequently, our modeling tool is incorporated into a full waveform inversion algorithm adapted for teleseismic configurations to invert the incident P wave and its coda. The inversion is carried out using a gradient approach that is efficiently implemented via the adjoint-state method. The results suggest that our hybrid method and FWI algorithm represent a valuable tool for 2D forward and inverse regional applications using teleseismic data sets.

Slab waveguide communication study using Finite Difference Method (FDM) with fourth-order compact scheme

An accurate and efficient semi-vectorial mode solver, examining Electric (E) and Magnetic (H) fields, is utilized to explore the communication characteristics of various slab waveguides. This approach relies on the Finite Difference Method (FDM) using uniform grid points, employing a Higher (Fourth) Order Compact type technique coupled with Conjugate Gradient (CG) iteration. The present study is an extension of the previous work where various aspects of E- and H-field propagations through rib-strucured waveguide was analyzed successfully. By incorporating the refractive index profile of the slab structure, excellent functionality of the waveguide is demonstrated here in Electric field by utilizing the concepts of normalized index and effective or modal index, confinement factor, and the modal birefringence properties which proves to be critical for potential applications in photonic integrated circuits. The materials chosen for the purpose are AlGaAs-GaAs and GeSi which play a vital role in characterizing the waveguides’performances through simulation using MATLAB. The use of surface and contour plots aids in visualizing the distribution of corresponding field within the guided modes. This analysis significantly contributes in understanding the waveguide's behavior and confinement capability during the field propagation. Most importantly, the variations of the waveguides’performance indices with their structure parameters help to identify their optimized values for fabrication to enhance the transmission efficiency of the waveguide.

Developing Higher-Order Unconditionally Positive Finite Difference Methods for the Advection Diffusion Reaction Equations

This study introduces the higher-order unconditionally positive finite difference (HUPFD) methods to solve the linear, nonlinear, and system of advection–diffusion–reaction (ADR) equations. The stability and consistency of the developed methods are analyzed, which are necessary and sufficient for the numerical approach to converge to the exact solution. The problem under consideration is of the Cauchy type, and hence, Von Neumann stability analysis is used to analyze the stability of the proposed schemes. The HUPFD’s efficacy and efficiency are investigated by calculating the error, convergence rate, and computing time. For validation purposes, the higher-order unconditionally positive finite difference solutions are compared to analytical calculations. The numerical results demonstrate that the proposed methods produce accurate solutions to solve the advection diffusion reaction equations. The results also show that increasing the order of the unconditionally positive finite difference leads an implicit scheme that is conditionally stable and has a higher order of accuracy with respect to time and space.

Cosmic string influence on a 2D hydrogen atom and its relationship with the Rytova–Keldysh logarithmic approximation in semiconductors

A two-dimensional hydrogen atom offers a promising alternative for describing the quantum interaction between an electron and a proton in the presence of a cosmic string. Reducing the hydrogen atom to two dimensions enhances its suited to capture the cylindrical/conical symmetry associated with the cosmic string, providing a more appropriate description of the physical system. After solving Schrd̈inger’s equation, we calculate the eigenenergies, probability distribution function, and expected values for the hydrogen atom with logarithmic potential under the influence of the topological defect. The calculations for the 2D hydrogen atom are performed for the first time using the Finite Difference Method. The results are presented through graphics, tables, and diagrams to elucidate the system’s physical properties. We have verified that our calculations agree with a linear variational method result. Our model leads to an interesting analogy with excitons in a two-dimensional monolayer semiconductor located within a specific semiconductor region. To elucidate this analogy, we present and discuss some interaction potentials and their exciton eigenstates by comparing them with the results from the literature.

On a Stability of Non-Stationary Discrete Schemes with Respect to Interpolation Errors

The aim of this article is to analyze the efficiency and accuracy of finite-difference and finite-element Galerkin schemes for non-stationary hyperbolic and parabolic problems. The main problem solved in this article deals with the construction of accurate and efficient discrete schemes on nonuniform and dynamic grids in time and space. The presented stability and convergence analysis enables improving the existing accuracy estimates. The obtained stability results show explicitly the rate of accumulation of interpolation and projection errors that arise due to the movement of grid points. It is shown that the cases when the time grid steps are doubled or halved have different stability properties. As an additional technique to improve the accuracy of discretizations on non-stationary space grids, it is recommended to use projection operators instead of interpolation operators. This technique is used to solve a test parabolic problem. The results of specially selected computational experiments are also presented, and they confirm the accuracy of all theoretical error estimates obtained in this article.