Finite Difference Method Research Articles

The influence of spatial dispersion on the steady-state characteristics and thermodynamic instability fluctuations of one-dimensional iron particle flames

This study utilizes a simplified one-dimensional discrete model to analyze the characteristic parameters involved in the flame propagation of iron particles. It focuses on the influence of dispersive "micro flames" within these flames on propagation dynamics, investigating stable and unstable scenarios. The model adopts the form of particle suspension delineating alternant reaction intervals and inert intervals. The spatial dispersion rate (Γ) which describes the spatial extent of the "micro flames" is introduced, with Γ=1 for the continuum model and Γ>1 for the discrete model. Theoretical equations, combining kinetic and diffusion equations, are solved with the finite difference method. The solution is evaluated preliminarily to distinguish numerical instability and thermodynamic instability. Additionally, in the preset time and space range, conditions for different equivalence ratios, particle radius and spatial dispersion rates are analyzed emphatically, with a comparison of typical simulation results and experimental data. As shown in the numerical simulation, the flame maintains stable propagation when ϕ≥0.7. The flame front, where the particle temperature rises above the gas temperature, extends backward with the increase of particle radius. The increase of Γ tends to extend the flame front of the fuel-lean flame and constringe that of the fuel-rich flame. Thermodynamic instability occurs in fuel-lean suspension with its manifestation preliminarily classified to distinct fluctuation, faint fluctuation and the final cessation. The increase of Γ also extends the flame propagation time under the dominance of thermodynamic instability, indicating different temperature structure evolution from the continuum model.

An Eulerian formulation for the computational modeling of phase-contrast MRI.

Computational simulation of phase-contrast MRI (PC-MRI) is an attractive way to physically interpret properties and errors in MRI-reconstructed flow velocity fields. Recent studies have developed PC-MRI simulators that solve the Bloch equation, with the magnetization transport being modeled using a Lagrangian approach. Because this method expresses the magnetization as spatial distribution of particles, influences of particle densities and their spatial uniformities on numerical accuracy are well known. This study developed an alternative method for PC-MRI modeling using an Eulerian approach in which the magnetization is expressed as a spatially smooth continuous function. The magnetization motion was described using the Bloch equation with an advection term and computed on a fixed grid using a finite difference method, and k-space sampling was implemented using the spoiled gradient echo sequence. PC-MRI scans of a fully developed flow in straight and stenosed cylinders were acquired to provide numerical examples. Reconstructed flow in a straight cylinder showed excellent agreement with input velocity profiles and mean errors were less than 0.5% of the maximum velocity. Numerical cases of flow in a stenosed cylinder successfully demonstrated the velocity profiles, with displacement artifacts being dependent on scan parameters and intravoxel dephasing due to flow disturbances. These results were in good agreement with those obtained using the Lagrangian approach with a sufficient particle density. The feasibility of the Eulerian approach to PC-MRI modeling was successfully demonstrated.

How to build and solve continuous-time heterogeneous agents models in asset pricing? The martingale approach and the finite difference method

How to build and solve continuous-time heterogeneous agents models in asset pricing? The martingale approach and the finite difference method

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.

A numerical algorithm based on extended cubic B-spline functions for solving time-fractional convection-diffusion-reaction equation with variable coefficients

In this research, we develop an innovative and efficient numerical method for solving a nonlinear one-dimensional time-fractional convection-diffusion-reaction equation (TFCDRE) with a variable coefficient. This method integrates the Crank-Nicolson finite difference scheme with extended cubic B-spline (ExCBS) basis functions. The Caputo fractional derivative is applied for temporal discretization, while the ExCBS functions are utilized for spatial discretization. The stability of the method was discussed by the von Neumann method, which shows unconditional stability; and the convergence analysis secure an order of $O(h^2+\triangle t^{2-\alpha})$. We demonstrated the efficiency and simplicity of our method through three numerical experiments and verified its accuracy using the absolute error $(L_2)$ and maximum error $(L_\infty)$ norms in temporal and spatial dimensions. The results are also graphically represented and show substantial agreement with the approximated and exact solution. We confirmed the effectiveness and accuracy of our approach over the local discontinuous Galerkin finite element and the compact finite difference methods, respectively, from the literature.

Analytical and Numerical Methods for the Solution to the Rigid Punch Contact Integral Equations

In this article, the problem of a rigid punch pressed onto the surface of an elastic half-plane is studied. In a first instance, it is reminded that the standard frictionless hard contact situation, where the contact pressure is unbounded at contact ends, exhibits an analytical solution to the governing singular integral equation with Cauchy kernel. Thereafter, it is shown that the situation of contact regularization results in a singular integro-differential equation with Cauchy kernel. This latter case leads to bounded contact pressures at both contact ends, even in the frame of linear elasticity, which is of great interest in the presence of “peaking” phenomenon. However, this contact regularization requires a numerical treatment as opposed to the former. To that end, a novel simple but efficient numerical procedure, based on numerical integration in conjunction with a centered finite differences scheme, is presented and numerically illustrated through two examples at the end of the paper.

Stochastic Diffusive Modeling of CO₂ Emissions with Population and Energy Dynamics

Climate change, primarily driven by CO2 emissions from energy and non-energy sectors, necessitates effective mitigation strategies. This study develops a stochastic diffusive model to capture the complex dynamics of CO2 concentration, human population growth, and energy production. The objectives are to enhance the predictive accuracy of existing models by incorporating diffusion effects and stochastic variability, offering insights for sustainable environmental policies. A novel numerical scheme, an extension of the Euler-Maruyama algorithm, is proposed to solve stochastic time-dependent partial differential equations governing the model. The scheme's consistency and stability are rigorously analyzed in the mean square sense. Findings reveal that increasing emission rate coefficients in energy and non-energy sectors exacerbates CO2 levels, emphasizing the need for stringent controls. The proposed scheme demonstrates superior accuracy to the non-standard finite difference method, establishing its efficacy in modeling complex environmental processes. This research contributes a robust computational tool to improve existing predictive models, aiding decision-making for long-term ecological sustainability. By addressing uncertainties in the environmental process, the work advances the understanding of interactions between population growth, energy production, and CO2 emissions, offering a significant improvement over the traditional modeling approach. The novelty lies in integrating stochastic dynamics with diffusion to better inform CO2reduction strategies. Doi: 10.28991/ESJ-2025-09-01-012 Full Text: PDF

On IFDM simulation of Oldroyd 8-constant fluid flowing due to motile microorganisms

This work studies the motion of bacteria on the Oldroyd 8-constant slime layer to determine the fundamental mechanism of bacterium gliding over immune system cells based on creeping flow and long wavelength approximation applicable to Stokes equations. Galilean transformation is also utilized to convert the problem from the lab to a wave frame. The bacterial gliding speed and the fluid flow rate are computed using an implicit finite difference method (IFDM), and a root-finding algorithm. These computed pairs are utilized to get power dissipation. Our results show that, due to the effect of slime rheology, characterized by relaxation time and retardation time, the gliding speed, flow rate of slime, and energy dissipation of bacteria are greatly modulated. Dynamic analysis shows that bacteria glide more quickly with reduced power dissipation when relaxation time is less than retardation time. On the other hand, relaxation time greater than the retardation time leads to slower bacterial swimming with increased energy consumption. The work presented here emphasizes that changes in the rheology of slime, specifically added time (relaxation and retardation time), can promote or inhibit bacterial motility. Additionally, the outcomes are confirmed using a separate technique based on bvp4c solver. A new comprehensive understanding of how this gliding happens (by controlling slime rheology and gliding gait) opens up possibilities for bio-inspired designs in the form of micro-robots or anti-fouling surfaces based on this work.

AFiD-MHD: A finite difference method for magnetohydrodynamic flows

AFiD-MHD: A finite difference method for magnetohydrodynamic flows

Advances in Artificial Intelligence and Computational Methods: Enhancing Modeling, Prediction, and Optimization

bstract: This area of artificial intelligence and computational methods has witnessed tremendous development, especially with the solution of highly complex problems across scientific disciplines. This review brings together the latest advancements in the integration of AI with traditional computational techniques, focusing on their applications in numerical simulations, system optimization, and predictive modeling. Key contributions include AI-augmented finite difference methods for solving partial differential equations, advancements in material and energy systems optimization, and innovative approaches to highperformance computing. Future directions emphasize scalability, interdisciplinary applications, and the integration of emerging technologies such as quantum computing and reconfigurable hardware.

Radiation impact on heat and mass transfer of an unsteady MHD nanofluid flow past an exponentially accelerating vertical plate with heat generation

This paper comprehensively analyzes the consequences of thermal radiation on natural convection, electrical and thermally conducted water-based flowing nanofluid past an exponentially accelerating surface. The paper investigates the thermal and momentum boundary layers with the effects of radiation and heat generation on the nanofluid flow. The magnetic field acts perpendicular to the flow direction. The non-dimensional governing equations were numerically solved using the Finite Difference Method (FDM) and illustrated through the use of MATLAB software. Thermal radiation enhances the nanofluid temperature and velocity, an essential phenomenon for designing solar collectors. The heat generation factor reduces the skin friction of the flow. It improves heat transfer by convection rather than conduction mode, enhancing the velocity and dropping the temperature of the nanofluid. The influence of some dimensionless parameters on the nanofluid velocity and temperature led to this paper's conclusion. Mass and heat transmission are fundamental due to their daily applications, including human body temperature regulations, electronic devices, and heating/cooling systems.

Spatio-temporal higher-order finite-difference solution for the scalar acoustic wave equation based on the global difference coefficient

The partial derivatives obtained through the difference approximation in the finite-difference method for solving the scalar acoustic wave equation may give rise to computational errors, which have the potential to induce numerical dispersion. Typically, the temporal or spatial higher-order difference format is employed, whereby the difference order between the computational region and the perfectly matched layer (PML) boundaries can result in boundary reflections. In this study, we derive the acoustic wave equation and its PML boundary conditions in the finite difference format of the temporal fourth-order and the spatial 2Nth-order, based on the Lax-Wendroff method. Subsequently, the stability conditions of the two finite difference formats are presented and analyzed under different parameters. This effectively addresses the issue of temporal dispersion. Furthermore, the high-order PML temporal boundary conditions effectively suppress the boundary reflection phenomenon generated by the computational regions and the different difference orders of the PML boundaries. Moreover, the time-space dispersion relation of the acoustic wave equation is employed to globally optimize the difference coefficients via the least-squares method, thereby suppressing the spatial dispersion. The numerical solution experiments of the acoustic wave equation for the horizontal laminar model and the Marmousi model demonstrate the efficacy of the presented algorithm.

Mathematical modeling of electroosmotic flow and thermal transport in stenotic tapered arteries using finite difference method

This study investigates the electroosmotic flow and thermal transport of nanofluids, specifically aluminum oxide and titanium dioxide, within tapered arteries with stenosis. Using the finite difference method (FDM), we modeled the flow dynamics and heat transfer mechanisms under the assumption of low Reynolds numbers and mild stenosis. The study focuses on the effects of nanoparticle volume fraction, heat absorption parameters, and electroosmotic forces on blood flow, velocity profiles, and temperature distribution. Results show that increasing nanoparticle concentration and heat absorption significantly reduce flow velocity, particularly in divergent artery geometries. Additionally, the inclusion of electroosmotic effects enhances heat transfer, while the Grashof number influences central velocity. These findings offer valuable insights into optimizing nanoparticle-based therapeutic interventions for cardiovascular treatments, providing a framework for enhancing drug delivery systems and improving the efficacy of heat-based therapies. The study’s outcomes could lead to improved diagnostic tools and therapeutic strategies for managing cardiovascular diseases.

3-D semivectorial bidirectional marching solver with fourth-order accuracy for optical waveguides

In this paper, we develop and analyze a new, to the best of our knowledge, semivectorial bidirectional operator marching method with fourth-order accuracy (Bi-OMM4) for three-dimensional optical waveguides. The fourth-order semivector exponential scheme reproduces the exact reformulations for equations with pairs of bidirectional reflection operators based on the Dirichlet-to-Neumann (DtN) mapping. Implementations for large range step sizes in both directions are presented, and exact bidirectional range marching formulas are derived for each range-independent segment. The study compares the results obtained from the Bi-OMM4 with the fourth-order bidirectional beam propagation method based on the finite difference scheme (FD-Bi-BPM4) and the bidirectional operator marching method with second-order accuracy (Bi-OMM2) to validate the accuracy and effectiveness of Bi-OMM4 by analyzing several examples of uniform and longitudinally varying waveguides. The results show that the Bi-OMM4 is numerically faster than the FD-Bi-BPM4 by almost seven times for different transverse grid sampling points, and it offers higher accuracy than Bi-OMM2 without a significant increase in computation resources.

Dynamics of water pollution concentration with uniform and exponential increment of pollutants

The study focuses on the one dimensional Advection-Dispersion Equation (ADE) to study the dynamics of concentration of water pollution when the pollutants at the source is increasing uniformly and exponentially. Analytical solutions are obtained by using Laplace transform and numerical solutions are by Finite Difference Method (FDM). The steady state case is studied. The dynamics of pollution concentration along the length of river channel are studied through two dimensional plots by varying the rate of added pollutants, cross sectional area of river and water flow velocity. The pollution concentration decreases along the length of river (downstream) for each analysis. The analytical and numerical solutions are shown in three dimensional plots. The analytical and numerical solutions are compared with the help of relative error. The relative errors calculated for uniform and exponential increment of pollutants at the source are compared while studying the dynamics of the concentration. The environmental and chemical engineering may substantially benefit from such studies.

A Vieta–Lucas collocation and non-standard finite difference technique for solving space-time fractional-order Fisher equation

The purpose of the article is to analyze an accurate numerical technique to solve a space-time fractional-order Fisher equation in the Caputo sense. For this purpose, the spectral collocation technique is used, which is based on the Vieta–Lucas approximation. By using the properties of Vieta–Lucas polynomials, this technique reduces the nonlinear equations into a system of ordinary differential equations (ODEs). The non-standard finite difference (NSFD) method converts this system of ODEs into algebraic equations which have been solved numerically. Moreover, the error estimate is investigated for the proposed method. To show the accuracy and efficiency of the technique, the obtained numerical results are compared with the analytical results and existing results of the particular forms of the considered fractional order models through error analysis. The important feature of this article is the exhibition of variations of the field variable for various values of spatial and temporal fractional order parameters for different particular cases.

Temporal evolution, propagation and mutual interaction of cos-gaussian beams in photorefractive crystals exhibiting both the linear and quadratic electro-optic effect

Abstract We investigate for the first time, the temporal evolution, propagation and interaction of cos-gaussian beams in novel photorefractive crystals having both the linear and quadratic electro-optic effect. The dynamical evolution equation is set up as a function of time using the Helmholtz equation in the paraxial approximation using the photorefractive charge transport model. The analysis is performed theoretically using the finite difference method to solve the dyamical evolution equation. Diffraction effects are prominent initially since the photorefractive effect has not built up while at larger values of time, we see self trapping and formation of breather solitons and Y-type solitons. A detailed study is subsequently undertaken for the propagation of cos-gaussian beams of different cosine modulation parameters and at various different values of the external electric field once steady state has been reached. Self trapping and formation of optical spatial solitons including breathing solitons and Y-type breathing solitons are observed as the nonlinearity is strengthened by increasing the external electric field. The characteristics of the solitons are dependent upon the modulation parameter of the cos-gaussian beam. The PMN-0.33PT crystal is taken for illustration of the results. Further, the interplay between the linear and the quadratic electro-optic effect and the ensuing effect on the propagation of the cos-gaussian beams is investigated for various cosine modulation parameters. Finally, the interaction of two cos-gaussian beams is studied for in phase and out of phase beams considering various cosine modulation parameters and various separation distances.

Numerical approximation of space-fractional diffusion equation using Laguerre spectral collocation method

The space-fractional diffusion equation is extensively used to model various issues in engineering and mathematics. This paper addresses the numerical approximation of the space-fractional-order diffusion equation, using a fractional operator in the Caputo sense. The proposed equation is computed numerically through the Laguerre spectral collocation method combined with the finite difference scheme. The results were visualized using MATLAB R2016a, and the accuracy of the numerical scheme was validated by comparing the approximate results with those derived from exact values and the Legendre spectral collocation technique. The originality of this study lies in the application of the Laguerre spectral collocation method for fractional-order diffusion equations in the Caputo sense. Two numerical problems were included to demonstrate the validity and applicability of the proposed technique. The obtained results confirm that this method aligns well with the exact values and the Legendre spectral collocation method, as shown in tabular and graphical simulations.

Optimizing thermal performance in a non-Darcy porous ventilated cavity with heated diagonal baffles of varied heights

This study investigates the convection effects of adding non-Darcy porous media to open enclosures with diagonally placed hot baffles near the open ports. The cold wall is fixed at the right side of the cavity. The remaining sidewalls are adiabatic. The fluid enters the enclosure from the inlet in the left vertical wall and leaves from an outlet at the right vertical wall. The finite difference method is employed to discretize the governing equations. The problem is analyzed for various parameters, Rayleigh number ([Formula: see text]), length of heating baffles ([Formula: see text]), porosity of the porous medium ([Formula: see text]), Reynolds number ([Formula: see text]) and Darcy number ([Formula: see text]) at alternate configured vented cavities and the results are illustrated graphically. The observation shows maximum heat transfer attained at BT open enclosure regardless of various parameters. For [Formula: see text] considerable increase in the heat transfer occurs at the right sidewall when the porosity of porous medium increases. The impact of configurations on heat transfer along the right wall is negligible at low Reynolds numbers but becomes significant at high Reynolds numbers.

Bearing characteristics and continuous–discontinuous numerical analysis of a pressure grouting pile in a calcareous sand foundation

The complex mechanical characteristics of calcareous sand complicate comprehensive analysis of the bearing characteristics of pressure grouting piles. The bearing mechanism of pressure grouting piles is unclear. Field tests of the bearing characteristics of pressure grouting piles in calcareous sand foundations were carried out. On the basis of the field test data, the coupled finite difference method (FDM) and discrete element method (DEM) were applied to explore the settlement characteristics. The force transfer laws in calcareous sand foundations are analyzed using complex network theory. The results show that the ultimate bearing capacity of pressure grouting piles increases significantly with increasing pile diameter. An increase in the pile‒soil interface friction coefficient can increase the pile side friction resistance, which can then increase the ultimate bearing capacity of the pile foundation. The friction coefficient of calcareous sand foundations can also increase the ultimate bearing capacity of pile foundations. However, the settlement of the pile foundation also increases. The axial force of the grouting pile gradually increases as the burial depth increases. When the pile bottom is reached, the axial force of the pile body clearly increases. This study presents an effective tool for the comprehensive analysis of the bearing characteristics of pressure grouting in calcareous sand foundations.