Finite Difference Method Research Articles

Effect of skin thickness on electromagnetic dosimetry analysis of a human body model up to 100 GHz

Abstract The accuracy of electromagnetic (EM) exposure assessments depends mainly on the resolution of a voxel human body model. The resolution of the conventional human body model is limited to a few millimeters. In the millimeter wave (mmWave) frequency range, EM waves are absorbed by the superficial tissues in the human body. Therefore, resolution and skin thickness of the human body model are important for accuracy of the EM wave exposure metrics recommended by international human safety guidelines. Realistic thickness modeling of the skin tissue on the human body may provide greater accuracy in the EM exposure assessment, especially at mmWave frequency range. In this paper, effects of the skin thickness on the EM exposure metrics in one-dimensional multi-layered models obtained from different regions of the body model are investigated using the dispersive algorithm based on the finite-difference time-domain method over the frequency range from 1 to 100 GHz. Furthermore, effects of eyelid tissue in a human eye on the EM exposure metrics are studied over the frequency range. The EM exposure metrics such as absorbed power density, heating factor, and power transmission coefficient are calculated up to 100 GHz to evaluate the limits of EM wave exposure.

Comparative numerical study of the energy performance of two different configurations of a hybrid photovoltaic/thermal air collector

Solar energy is a renewable alternative to fossil fuels. When solar energy is converted into electricity by photovoltaic (PV) solar modules, the heat generated increases the temperature of the PV solar cells and, as a result, reduces their electrical efficiency. Several techniques are used to cool PV solar cells in order to reduce their operating temperature. One technique is the use of hybrid photovoltaic/thermal (PVT) solar collectors, which simultaneously produce electricity and useful heat. In the airflow channel, the heat and mass transfer equations and those derived from the energy balance on the solid layers of the collector were discretized using the finite difference method with an implicit alternating directions (ADI) scheme. All the algebraic equations obtained after discretization were solved using the Thomas algorithm. The aim is to make a numerical comparison of the energy performance of the Verre-PV-Tedlar and Verre-PV-Verre flat-plate air PVT hybrid solar collectors in the climatic conditions of Lomé, Togo. Under the same operating conditions, the average electrical efficiencies of the Verre-PV-Verre and Verre-PV-Tedlar flat-plate air PVT hybrid collectors are 15.34% and 13.85% respectively. There is a 1.49% improvement in electrical efficiency for a Verre-PV-Verre air-source PVT hybrid collector compared with a Verre-PV-Tedlar PVT hybrid collector. The Verre-PV-Verre flat plate PVT air hybrid collector has an average daily electrical energy gain of 0.564 kWh and a thermal energy gain of 0.839 kWh, while the Verre-PV-Tedlar flat plate PVT air hybrid collector has an average daily electrical energy gain of 0.548 kWh and a thermal energy gain of 1.403 kWh.

An Integrated Taylor Expansion and Least Squares Approach to Enhanced Acoustic Wave Staggered Grid Finite Difference Modeling

The staggered grid finite difference method has emerged as one of the most commonly used approaches in finite difference methodologies due to its high computational accuracy and stability. Inevitably, discretizing over time and space domains in finite difference methods leads to numerical artifacts. This paper introduces a novel approach that combines the widely used Taylor series expansion with the least squares method to effectively suppress numerical dispersion. We have derived the coefficients for the staggered grid finite difference method by integrating Taylor series expansions with the least squares method. To validate the effectiveness of our approach, we conducted analyses on accuracy, dispersion, and stability, alongside simple and complex numerical examples. The results indicate that our method not only inherits the capabilities of the original Taylor series and least squares methods in suppressing numerical dispersion across small and medium wavenumber ranges but also surpasses the original methods. Moreover, it demonstrates robust dispersion suppression capabilities at high wavenumber ranges.

Heat transfer and entropy generation analysis of ternary nanofluid

Abstract In light of the rising demand for improved heat transfer in thermal systems, this work offers a unique technique for boosting heat transfer capacity and reducing entropy generation using a ternary nanofluid. Moreover, constant pressure gradient, magnetic field, Joule heating, and thermal radiation also contribute to the flow dynamics. The 2D mathematical model is solved numerically using a finite difference method and the simulations are done in MATLAB. The obtained results show that ternary nanoparticles not only increase the thermal rates but are also helpful in maintaining the irreversibility of a system. It is also observed that the heat transfer of the base fluid increased by 7.5%, 8.3%, and 8.5% on adding TiO2, CuO-TiO2, and MgO-CuO-TiO2 nanoparticles, respectively.

Comparison of Contact Angle Models in Two‐Phase Flow Simulations Using a Conservative Phase Field Equation

ABSTRACTIn phase field methods based on a second‐order Allen‐Cahn (AC) equation, contact angles are prescribed mostly via a geometric formulation. However, it is of great interest to utilize the surface‐energy formulation, which is often employed in the Cahn‐Hilliard (CH) phase field method, in the AC phase field method. This article thus put forward a surface‐energy formulation of contact angles. The model was compared with the geometric one in a number of impact problems, including both normal and oblique impacts. The governing equations were discretized using a finite difference method on a half‐staggered grid. The Navier–Stokes equation was tackled using an explicit projection method. The major findings are as follows. First, the geometric model can maintain a fixed contact angle throughout contact line motion, while the surface‐energy one predicts a changeable contact angle, with a fluctuation of about 5°. In the oblique drop impact, contact angle hysteresis was captured even if a static contact angle was applied in the surface‐energy formulation.

Numerical Investigation of Turbulent Convection Flow in a Rectangular Closed Cavity

Natural turbulent convection in closed cavities has many practical applications in the field of engineeringsuch as the design of electronic computer chips, atomic installation and industrial cooling among others. Inparticular, it enables in achieving a desired micro-climate and efficient ventilation in a building. Recent studiesshow that turbulent flow is affected by variations in Rayleigh numbers, aspect ratio, and heater positionamong others. Temperature is kept constant in all these studies hence inadequate literature on the effectsof temperature on a turbulent flow. In this study, aspect ratio and Rayleigh numbers are kept constant at2 and 1012 respectively and natural turbulent convection flow in a closed rectangular cavity is investigatednumerically as the operating temperature is varied from 285.5K to 293K. The rectangular cavity’s lower wallwas heated and cooling done at the top face wall while the rest of the vertical walls were kept in adiabaticcondition. Material properties such as density of the fluid kept on changing at any given temperature. Thethermal profile data generated influenced the nature of the turbulent flow. The non-linear averaged continuity,momentum, and energy equation terms were modeled by the SST k − ω model to generate streamlines,isotherms, and velocity magnitude for a different operating temperature and presented graphically. The finitedifference method and FLUENT were used to solve two SST k − ω model equations, vortices, and energy withboundary conditions. It was discovered that, as the operating temperature increased turbulence decreaseddue to a decrease in the velocity of the elements and vortices became more parallel and smaller.

Developing PDE-constrained optimal control of multicomponent contamination flows in porous media

This paper develops a robust and efficient PDE-constrained optimal control model for multicomponent pollutions in porous media, which takes into account nonlinear multi-component contamination flows of groundwater. The objective of the pollution optimal control is to identify the optimal injection rates from the top part boundary of domain, which can minimize the least squares error between the concentrations that are simulated and the allowable observed concentrations at observation sites, combining with the affection of costs associated with reducing emissions at injection locations. To discrete the constrained governing system of nonlinear multi-component flows, the splitting improved upwind finite difference scheme is developed for multicomponent PDEs system involving nonlinear chemical reactions of multicomponent pollutants and the pollutant injected rates on the upper part boundary. We employ the differential evolution (DE) optimization algorithm to solve the optimization. We numerically demonstrate the effectiveness of our model by analyzing the flow simulation on a simple geometric aquifer and identifying the optimal injection rates by minimizing the concentration derivation and the abatement costs. We also investigate the simulation of the contamination flow in a more realistic-shaped aquifer, which further validates our model's robustness and efficacy. The developed PDE-constrained control model and algorithm can be applied to applications of groundwater pollution control.

Solving Multi‐Term Time‐Fractional Diffusion Equations Using a High‐Order Fractional Finite Difference Scheme Accompanied by Neural Network Techniques

ABSTRACTThis article is devoted to solving multi‐term time‐fractional diffusion equations with a high‐order finite difference scheme. We approximate the temporal fractional derivatives by the L1‐2‐3 formula, an expansion of the L1 formula with higher accuracy. We employ neural network strategies and automatic differentiation techniques for approximating integer‐order derivatives. In the present work, we propose very important lemmas and theorems to prove the unconditional stability and convergence of the proposed scheme. Extensive numerical experiments for one‐ and two‐dimensional problems, especially for curved boundaries, confirm the theoretical convergence orders. Comparisons with fractional physics‐informed neural networks and finite difference methods in solving the multi‐term time‐fractional diffusion equations demonstrate the superiority of the proposed method. It is also illustrated that the proposed scheme can efficiently and accurately extract the pattern of the solutions even when noise corruption up to ten percent is imposed on the forcing term.

Numerical Study of Multi-Term Time-Fractional Sub-Diffusion Equation Using Hybrid L1 Scheme with Quintic Hermite Splines

Anomalous diffusion of particles has been described by the time-fractional reaction–diffusion equation. A hybrid formulation of numerical technique is proposed to solve the time-fractional-order reaction–diffusion (FRD) equation numerically. The technique comprises the semi-discretization of the time variable using an L1 finite-difference scheme and space discretization using the quintic Hermite spline collocation method. The hybrid technique reduces the problem to an iterative scheme of an algebraic system of equations. The stability analysis of the proposed numerical scheme and the optimal error bounds for the approximate solution are also studied. A comparative study of the obtained results and an error analysis of approximation show the efficiency, accuracy, and effectiveness of the technique.

The influence of parasitic modes on stable lattice Boltzmann schemes and weakly unstable multi-step Finite Difference schemes

Numerical analysis for linear constant-coefficient multi-step Finite Difference schemes is a longstanding topic, developed approximately fifty years ago. It relies on the stability of the scheme, and thus—within the L2 setting—on the absence of multiple roots of the amplification polynomial on the unit circle. This allows for the decoupling, while discussing the convergence of the method, of the study of the consistency of the scheme from the precise knowledge of its parasitic/spurious modes, so that the methods can be essentially studied as if they had only one step. Furthermore, stability alleviates the need to delve into the complexities of floating-point arithmetic on computers, which can be challenging topics to address. In this paper, we demonstrate that in the case of “weakly” unstable Finite Difference schemes with multiple roots on the unit circle, although the schemes may remain stable, considering parasitic modes is essential in studying their consistency and, consequently, their convergence. This research was prompted by unexpected numerical results on stable lattice Boltzmann schemes, which can be rewritten in terms of multi-step Finite Difference schemes. Unlike Finite Difference schemes, rigorous numerical analysis for lattice Boltzmann schemes is a contemporary topic with much left for future discoveries. Initial expectations suggested that third-order initialization schemes would suffice to maintain the accuracy of fourth-order schemes. However, this assumption proved incorrect for weakly unstable Finite Difference schemes and for stable lattice Boltzmann methods. This borderline scenario underscores that particular care must be adopted for lattice Boltzmann schemes, and the significance of genuine stability in facilitating the construction of Lax-Richtmyer-like theorems and in mastering the impact of round-off errors concerning Finite Difference schemes. Despite the simplicity and apparent lack of practical usage of the linear transport equation at constant velocity considered throughout the paper, we demonstrate that high-order lattice Boltzmann schemes for this equation can be used to tackle nonlinear systems of conservation laws relying on a Jin-Xin approximation and high-order splitting formulæ.

Numerical Methods and Computation in Financial Mathematics: Solving PDEs, Monte Carlo Simulations, and Machine Learning Applications

Abstract. This paper delves into three fundamental numerical methods and computational techniques in financial mathematics: Finite Difference Methods (FDM), Monte Carlo Simulations (MCS), and Machine Learning (ML) applications. Finite Difference Methods are widely utilized for solving partial differential equations (PDEs) in option pricing, with various schemes offering different stability and convergence properties. Monte Carlo Simulations provide a powerful approach for pricing complex derivatives and risk management, addressing the challenges of high-dimensionality and computational complexity. Machine Learning has revolutionized predictive modeling in finance, enabling sophisticated analysis of large datasets to uncover hidden patterns and enhance trading strategies. Through a detailed examination of these methods, including specific examples and data, this paper highlights their theoretical foundations, practical implementations, and the advancements they bring to computational finance. By bridging theoretical approaches with practical applications, we aim to offer insights into the future directions and challenges in financial mathematics.

Dynamics of a delayed discrete-time predator prey model proposed from a nonstandard finite difference scheme

Existing literature established stability in a delayed logistic Lotka–Volterra predator- prey model in terms of equilibrium analysis. However, several researchers did not construct the time-series analysis in such models. It has also been observed that both the RK4 methods and the inbuilt ‘dde23’ MATLAB solver were unable to generate stable solutions. This motivated us to develop a nonstandard scheme to capture the numerical solutions which are well consistent with the analytical equilibrium analysis of continuous delayed predator–prey model. In this paper, we will propose a nonstandard finite difference (NSFD) scheme for a delayed predator–prey model. We shall prove that the developed scheme preserves the qualitative behavior of the system, including the local stability of the equilibrium, and stability switching for any step size h=1m,m∈Z+. It is observed that the discretized system shows the occurrence of a Neimark-Sacker bifurcation. Moreover, the convergence analysis of the numerical scheme establishes first-order convergence. The bifurcation diagram and comparison of delay τ−sequence generated by NSFD with the ones obtained by analytical means have been discussed graphically.

Quantum efficiency enhancement of reflective GaAs photocathodes with exponential-doping structure generating a favorable built-in electric field

The rapid development of GaAs photocathodes has led to an increased focus on the attainment of high quantum efficiency. Three types of exponential-doping structures with a high to low doping concentration distribution from the interior to the surface are proposed for reflective GaAs emission layers. These three structures generate different built-in electric fields that facilitate photoelectron emission. The one-dimensional continuity equations for the increasing, constant, and decreasing types of built-in electric fields are derived, respectively. The electron concentration distribution and quantum efficiency varying with the wavelength are solved numerically by the finite difference method. The simulation results indicate that the quantum efficiency of the GaAs photocathode with the increasing type of built-in electric field is superior to that with the constant built-in electric field, while the GaAs photocathode with the decreasing type of built-in electric field shows the worst performance. Then, the designed GaAs photocathodes with the increasing and constant types of built-in electric fields are grown by metal-organic chemical vapor deposition and activated by cesium-oxygen alternating deposition. The measured spectral response curves show that the quantum efficiency of the GaAs photocathode with the increasing type of built-in electric field is higher in the whole band than that with the constant type of built-in electric field. In addition, the exponential-doping structure generating the increasing type of built-in electric field is beneficial for improving the surface potential barrier and increasing the surface electron escape probability.

A hybrid lattice Boltzmann and finite difference method for two-phase flows with soluble surfactants

A hybrid method is developed to simulate two-phase flows with soluble surfactants. In this method, the interface and bulk surfactant concentration equations of diffuse-interface form, which include source terms to consider surfactant adsorption and desorption dynamics, are solved in the entire fluid domain by the finite difference method, while two-phase flows are solved by a lattice Boltzmann color-gradient model, which can accurately simulate binary fluids with unequal densities. The flow and interface surfactant concentration fields are coupled by a modified Langmuir equation of state, which allows for surfactant concentration beyond critical micelle concentration. The capability and accuracy of the hybrid method are first validated by simulating three numerical examples, including the adsorption of bulk surfactants onto the interface of a stationary droplet, the droplet migration in a constant surfactant gradient, and the deformation of a surfactant-laden droplet in a simple shear flow, in which the numerical results are compared with theoretical solutions and available literature data. Then, the hybrid method is applied to simulate the buoyancy-driven bubble rise in a surfactant solution, in which the influence of surfactants is identified for varying wall confinement, density ratio, Eotvos number and Biot number. It is found that surfactants exhibit a retardation effect on the bubble rise due to the Marangoni stress that resists interface motion, and the retardation effect weakens as the Eotvos or Biot number increases. We further show that the weakened retardation effect at higher Biot numbers is attributed to a decreased non-uniform effect of surfactants at the interface. By comparing with the Cahn-Hilliard phase-field method, we also show that the present method conserves the mass for each fluid, improves numerical stability especially at high density ratio and Eotvos number, and does not need the selection of free parameters, thus breaking the limitations of the existing method.

Injected phase change seismic source construction and monitoring experiment

Scientific assessment of the transport channels and distribution of oil, gas, water, and other fluids is the key to enhancing oil and gas production capacity and ensuring the safe and efficient development of mineral resources. Traditional microseismic monitoring has been suffering from limitations such as ambiguous attribute characteristics of the seismic source and low signal-to-noise ratios of the data, leading to poor location accuracy and large uncertainties. We develop a phase change seismic source monitoring method that allows for a detailed characterization of the spatial distribution of subsurface fluid transport by accurately locating the equivalent seismic source within supportive fractures. A phase change seismic source is developed and constructed based on the phase change material, enabling control over phase change temperature, source energy, and source scale through the principle of phase change energy storage. The source introduces an active approach to seismic signal generation, diverging from the conventional passive method reliant on rock rupture. Then, 3D homogeneous models are constructed for ground monitoring and in-well monitoring using impulse source propagation theory. The numerical simulation of the wavefield uses the finite-difference method, and the equivalence of discrete sources is confirmed using the Pearson correlation function. The reliability of the monitoring method is verified through a seismic monitoring experiment and comprehensive feasibility analysis.

A novel temporal finite element method to solve static viscoelastic problems

A novel temporal finite element method is presented to solve static viscoelastic problems, which is more flexible and appropriate to describe temporal variation of displacement than conventional finite difference or numerical integral methods. By using virtual work principle, a spatial finite element based governing equation is derived in terms of displacement and its derivatives. Then two temporal finite element models are developed using Gurtin variational principle and weighted residual technique. A kind of hybrid shape functions with polynomial and trigonometric basis is stressed to give more flexible descriptions of time varying variables. A criterion of stability analysis is derived, which can numerically be conducted when the constitution of shape functions and step size are prescribed. A recursive algorithm toward end is developed by which the temporal FE analysis can be conducted via a matrix power product with the initial condition, instead of step marching. The proposed approach is available for the viscoelastic model described by linear differential equations with order h ≤ 2, and can conveniently be combined with well-developed numerical algorithms to tackle with boundary value problems, such as FEM, SBFEM etc. Various numerical examples, including those with static/harmonic loads, creep, stress singularity and heterogeneous structures, etc. are provided to illustrate the efficiency of the proposed approaches, and impacts of the temporal FE model, constitution of shape functions, and step size, etc. are taken into account. Satisfactory results are achieved in comparison with analytical or ABAQUS-based solutions.

Hybrid Modelling Using Lattice Boltzmann and Finite Difference Methods of Dikes Effect on Sediment Transport and Morphological Processes with a Quasi-Tridimensional Approach

Hybrid Modelling Using Lattice Boltzmann and Finite Difference Methods of Dikes Effect on Sediment Transport and Morphological Processes with a Quasi-Tridimensional Approach

Treatment and delay control strategy for a non-linear rift valley fever epidemic model

Rift Valley Fever (RVF) is a viral disease affecting animals and humans, causing symptoms such as fever, liver damage, and bleeding, particularly prevalent in Africa. This study focuses on numerical solutions for a non-linear delayed dynamic epidemiological model of RVF. It extends a control problem incorporating the susceptible, infected, treated, recovered vector to analyze the impact of measures such as mosquito repellent and treatment. The goal is to examine how time delays in implementing control measures affect the dynamics of an epidemic. The model considers delay factors such as mosquito replication, hospitalization, travel restrictions, and isolation due to the lack of proper vaccination. The study explores the model’s aspects, including the reproduction number, equilibrium points, and stability. Local and global implications are examined using techniques such as the Lyapunov function and the Brauer-F lemma. Numerical analysis employs the non-standard finite difference method, establishing the local stability of the equilibrium through the effective reproduction number Rrvf and sensitivity analysis. The research highlights the importance of treatment and delay strategies in reducing RVF transmission, emphasizing the critical need for immunization and preventive measures.

Solutions to elliptic and parabolic problems via finite difference based unsupervised small linear convolutional neural networks

In recent years, there has been a growing interest in leveraging deep learning and neural networks to address scientific problems, particularly in solving partial differential equations (PDEs). However, many neural network-based methods like PINNs rely on auto differentiation and sampling collocation points, leading to a lack of interpretability and lower accuracy than traditional numerical methods. As a result, we propose a fully unsupervised approach, requiring no training data, to estimate finite difference solutions for PDEs directly via small linear convolutional neural networks. Our proposed approach uses substantially fewer parameters than similar finite difference-based approaches while also demonstrating comparable accuracy to the true solution for several selected elliptic and parabolic problems compared to the finite difference method.

Global tsunami modelling on a spherical multiple-cell grid

A model of shallow water equations (SWEs) on a spherical multiple-cell (SMC) grid of 4-level (2.5-5-10-20 km) spatial resolutions is used to simulate tsunami propagation on global ocean surfaces. The unstructured SMC grid retains rectangular cells of the longitude-latitude grid so that efficient finite-difference schemes could be used. It also supports multi-resolutions like mesh refinement to resolve small islands and coastline details while keeping models compact enough to fit into available computers. Two earthquake-induced tsunami cases are simulated and compared with available observations. Results indicate that the modelled tsunami arrival times agree well with observations while tsunami wave heights are underestimated, particularly the observed runups on remote coastal lands. Possible reasons for this underestimation include the smoothing scheme used to suppress numerical oscillations and the missing of initial kinetic energy input from the earthquakes. Another reason is the limitation of the SWEs to describe coastal bores and breaking waves in coastal waters. A possible tsunami scenario induced by a landslide in the Canary Islands is also simulated to assess its potential impact on Atlantic coastal regions. Model results indicate that this kind of tsunami may cause severe damage to local areas but its effects on far fields, like the UK and American coastal regions are small. As the initial landslide disturbance is overly simplified, this study does not give a true representation of a real landslide tsunami but rather a qualitative assessment of its impact on the Atlantic Ocean. More realistic initial condition and improved model representation of coastal processes are needed for further studies of this possible landslide hazard.