Difference Method Research Articles

Model description of combined numerical and stochastic groundwater flow in Bandung-Soreang Groundwater Basin, West Java, Indonesia.

Model description of combined numerical and stochastic groundwater flow in Bandung-Soreang Groundwater Basin, West Java, Indonesia.

Application of the generalized equations of the finite difference method in the analysis of constant stiffness isotropic thin Shells

Application of the generalized equations of the finite difference method in the analysis of constant stiffness isotropic thin Shells

Nonuniqueness of lattice Boltzmann schemes derived from finite difference methods

Nonuniqueness of lattice Boltzmann schemes derived from finite difference methods

A fast calculation approach for thermal distribution of aircraft piston pumps based on finite difference method

A fast calculation approach for thermal distribution of aircraft piston pumps based on finite difference method

A study on brain tumor dynamics in two-dimensional irregular domain with variable-order time-fractional derivative.

A study on brain tumor dynamics in two-dimensional irregular domain with variable-order time-fractional derivative.

High-order fractional central difference method for multi-dimensional integral fractional Laplacian and its applications

High-order fractional central difference method for multi-dimensional integral fractional Laplacian and its applications

On Godunov-type finite volume methods for seismic wave propagation

The computational complexity of simulating seismic waves demands continual exploration of more efficient numerical methods. While Finite Volume methods are widely acclaimed for tackling general nonlinear hyperbolic (wave) problems, their application in realistic seismic wave simulation remains uncommon, with rare investigations in the literature. Furthermore, seismic wavefields are influenced by sharp subsurface interfaces frequently encountered in realistic models, which could, in principle, be adequately solved with Finite Volume methods. In this study, we delved into two Finite Volume (FV) methods to assess their efficacy and competitiveness in seismic wave simulations, compared to traditional Finite Difference schemes. We investigated Gudunov-type FV methods: an upwind method called wave propagation algorithm (WPA), and a Central-Upwind type method (CUp). Our numerical analysis uncovered that these finite volume methods could provide less dispersion (albeit increased dissipation) compared to finite differences for seismic problems characterized by velocity profiles with abrupt transitions in the velocity. However, when applied to more realistic seismic models, finite volume methods yielded unfavorable outcomes compared to finite difference methods, the latter offering lower computational costs and higher accuracy. This highlights that despite the potential advantages of finite volume methods, such as their conservative nature and aptitude for accurately capturing shock waves in specific contexts, our results indicate that they are only advantageous for seismic simulations when unrealistic abrupt transitions are present in the velocity models.

A COUPLED FDM-MPM METHOD FOR MODELING MULTIPLE PHENOMENA OF FLUID AND AND SOIL

This study proposes an advanced analytical framework for investigating soil behavior under hydraulic loading by establishing a coupled soil–water analysis that discretizes fluid flow using the Finite Difference Method (FDM) and soil deformation using the Material Point Method (MPM). Numerical methods for coupled soil–water analysis can generally be classified into grid-based approaches (e.g., the Finite Element Method, FEM) and particle-based approaches (e.g., the Distinct Element Method, DEM). Grid-based approaches often encounter difficulties in dealing with large deformations and collapse phenomena, whereas particle-based approaches can be computationally expensive and may struggle to accurately capture seepage. To overcome these limitations, a coupled fluid–soil analysis model has been developed using MPM, an intermediate method that combines strengths from both grid- and particle-based techniques. The model has been applied to a two-dimensional consolidation and seepage problem. The results indicate that the proposed method can accurately reproduce large deformations and seepage—phenomena that pose challenges for traditional grid-based methods—while also aligning well with earlier findings from particle-based analyses.

A Fast Method for the Acceleration Response Analysis of Two-Dimensional Sites Under Seismic Excitations

The mode superposition method has been widely used for the seismic response analysis of two-dimensional (2D) sites to enhance computational efficiency. However, this method lacks a guideline of modal truncation to control errors of acceleration responses. In this paper, the mode contribution coefficient of acceleration is proposed to be used as a criterion for modal truncation in the seismic acceleration response analysis of soil layers. Comparative analysis with the modal participation mass and modal contribution factor demonstrates the effectiveness of the proposed factor for the modal truncation of acceleration responses. The computational accuracy of the method for calculating acceleration from displacement using the central difference scheme is verified, which would further improve the computational efficiency in calculating site acceleration responses. A homogeneous soil site and a scarp topography site show that the proposed factor for modal truncation effectively controls the computational error of soil acceleration responses. Additionally, computing acceleration time histories from displacement time histories via the central difference method yields errors comparable to those from directly computing generalized coordinate accelerations. However, modal truncation based on modal participation mass or modal contribution factor results in fewer modes retained and larger computational errors.

Petrov–Galerkin finite element method for solving the time-fractional Rosenau–Hyman equation

Abstract This study explores the fractional-order Rosenau–Hyman equation, which models nonlinear wave dynamics in dispersive and memory-dependent media. A Petrov–Galerkin finite element method (PG-FEM) is employed, incorporating the Grünwald–Letnikov fractional derivative to efficiently capture long-range memory effects and non-local interactions. The proposed numerical scheme ensures high accuracy and computational efficiency, overcoming the limitations of traditional finite difference and spectral methods in handling compacton solutions and nonlinearities. Numerical simulations demonstrate that decreasing the fractional order $$(\alpha < 1)$$ ( α < 1 ) leads to broader wave dispersion and slower attenuation, mimicking physical behaviors observed in viscoelastic materials, fluid flow through porous media, and biological diffusion processes. The results further validate that fractional-order models outperform integer-order counterparts in accurately describing power-law wave decay and anomalous transport phenomena. These findings reinforce the necessity of fractional calculus in modeling wave propagation, biomechanics, and material science applications, paving the way for future research in adaptive computational techniques and experimental validation.

From Climate to Cloud: Advancing Fog Detection Through Satellite Imagery

The broad spatiotemporal coverage provided by satellite remote sensing is fundamental for monitoring fog events, a phenomenon that impacts transportation, agriculture, and ecosystem functioning. Despite advances in remote sensing technology, significant knowledge gaps remain regarding the application of these techniques to fog detection, especially over terrestrial ecosystems. This scoping review synthesizes the trends in methods used for fog detection by analyzing 38 papers retrieved from Scopus and Web of Science. Only studies that utilized satellite imagery to analyze the spatiotemporal dynamics of fog were included. Articles that employed non-satellite methodologies or focused on processes other than the detection, formation, or identification of fog events were excluded. In addition to a term co-occurrence analysis of abstracts using VOSviewer, this study examines key parameters of the detection methods—including sensor type, spectral bands, temporal resolution, and algorithmic approaches (e.g., threshold methods and deep learning techniques)—to evaluate their evolution and current limitations. Our results reveal that while approximately 53% of studies rely on geostationary satellite data (95% CI: 36.7–68.5%), favored for their high temporal resolution, the remaining 47% employ polar-orbiting sensors (95% CI: 31.5–63.2%) that offer superior spatial resolution. Notably, most research has concentrated on maritime fog detection, with few studies extending these techniques to complex terrestrial environments. The review highlights critical gaps in current approaches and proposes an integrated framework that combines traditional brightness temperature difference methods with emerging machine learning techniques, which could advance fog detection in diverse settings.

An efficient 3D curvilinear structured grid generation method for large-scale seismic wave simulation in regions with irregular topography

SUMMARY The finite difference method (FDM) based on curvilinear structured grids is an efficient numerical algorithm that retains the advantages of conventional FDM, such as computational efficiency and ease of implementation, while effectively simulating seismic wave propagation in regions with irregular topography. Grid quality plays a crucial role in the finite difference simulations, as it directly affects the reliability, stability, and efficiency of the simulation. Moreover, large-scale 3D simulations often require billion-point grids, making the efficient generation of high-quality grids a significant challenge. The curvilinear grid used in seismic wave simulations has a special feature compared to other computational problems: only the shape of the free surface is fixed, while the other boundaries remain flexible. Based on this feature, we developed a straightforward parabolic grid generation method that starts from the free surface and expands the grid layer by layer along the depth direction toward the predefined bottom boundary. Additionally, this method is highly efficient, easy to parallelize, and capable of rapidly generating large-scale grids. We tested the grid generation method and its application in seismic wave simulations using five irregular topography models. Grid quality assessments demonstrate that the proposed method efficiently generates high-quality grids. Furthermore, seismic wave simulations reveal that, compared to vertically stretched grids, the generated grids reduce computation time while maintaining sufficient simulation accuracy.

Analytical and numerical computations for bioconvective flow Maxwell nanofluid with variable thermal properties using homotopy analysis method and finite difference scheme

This continuation presents analytical and numerical simulations for unsteady laminar flow of Maxwell nanofluid subject to decomposition of microbes driven by a porous stretching surface. The fluctuation in mass and heat transmission is proceeded with variable assumptions of thermal conductivity and mass diffusivity. Moreover, the heat source and activation energy applications are taken into focused. The stretching surface also periodically accelerates with uniform magnitude. Based on stated assumptions, the problem is modeled in set of partial differential equations (PDE’s). Such PDE’s are solved with implementation of analytical and numerical schemes. The analytical computations are proceeded with help of homotopy analysis method (HAM) while numerical treatment is governed by finite difference method (FDM). A comparative inspection of both HAM and FDM simulations are performed. It is examined that upon increasing the iterations of HAM, an excellent accuracy is noted with numerical scheme. Moreover, physical aspects of flow parameters are visualized and studied exclusively. The heat transfer increases due to Deborah constant and porosity parameter. The consideration of variable thermal conductivity and mass diffusivity enhances the heat and mass transfer phenomenon.

Improved Splitting-Integrating Methods for Image Geometric Transformations: Error Analysis and Applications

Geometric image transformations are fundamental to image processing, computer vision and graphics, with critical applications to pattern recognition and facial identification. The splitting-integrating method (SIM) is well suited to the inverse transformation T−1 of digital images and patterns, but it encounters difficulties in nonlinear solutions for the forward transformation T. We propose improved techniques that entirely bypass nonlinear solutions for T, simplify numerical algorithms and reduce computational costs. Another significant advantage is the greater flexibility for general and complicated transformations T. In this paper, we apply the improved techniques to the harmonic, Poisson and blending models, which transform the original shapes of images and patterns into arbitrary target shapes. These models are, essentially, the Dirichlet boundary value problems of elliptic equations. In this paper, we choose the simple finite difference method (FDM) to seek their approximate transformations. We focus significantly on analyzing errors of image greyness. Under the improved techniques, we derive the greyness errors of images under T. We obtain the optimal convergence rates O(H2)+O(H/N2) for the piecewise bilinear interpolations (μ=1) and smooth images, where H(≪1) denotes the mesh resolution of an optical scanner, and N is the division number of a pixel split into N2 sub-pixels. Beyond smooth images, we address practical challenges posed by discontinuous images. We also derive the error bounds O(Hβ)+O(Hβ/N2), β∈(0,1) as μ=1. For piecewise continuous images with interior and exterior greyness jumps, we have O(H)+O(H/N2). Compared with the error analysis in our previous study, where the image greyness is often assumed to be smooth enough, this error analysis is significant for geometric image transformations. Hence, the improved algorithms supported by rigorous error analysis of image greyness may enhance their wide applications in pattern recognition, facial identification and artificial intelligence (AI).

An efficient structured grid generation scheme for seismic wave simulation in regions with extreme topography

Using curvilinear structured grids, the finite difference method (FDM) can effectively simulate seismic wave propagation in regions with irregular topography. However, generating high-quality curvilinear structured grids in regions with extreme topography, such as mountains or nearly vertical reservoir dams, is not an easy task and limits the application of FDM in seismic wave simulation. Grid quality is crucial for finite difference simulations, as it directly affects the accuracy, stability, and efficiency of the simulation. Therefore, achieving an efficient and automatic generation of high-quality structured grids is essential for applying FDM to seismic wave simulation in regions with topography. The grids used for seismic wave simulation have a special feature compared to other computational problems: only the shape of the free surface is fixed, while the shapes of the other boundaries are free to be set. Based on this feature, we propose an efficient and automatic grid generation scheme that integrates the hyperbolic grid generation method and the algebraic interpolation method. Initially, the grids are generated using the hyperbolic method, starting from the given free surface and expanding layer by layer along the depth direction. However, when dealing with extreme topographies, the resulting grids exhibit poor smoothness in the depth direction. To enhance the grid quality, we redistribute the grid points using an interpolation approach along the depth direction. We designed two models with extreme topography and used the Foothills model to test the grid generation scheme and its application in seismic wave simulation. The results of the grid quality assessment indicate that the scheme can efficiently generate high-quality grids. Seismic wave propagation simulations reveal that these high-quality grids significantly improve computational efficiency with sufficient simulation accuracy.

Thermal Properties and Cooling Simulation of Red Dragon Fruit Using the Finite Difference Method

Red dragon fruit (Hylocereus polyrhizus) can be transformed into various processed products, such as chips, jam, and juice. However, thermal processing may significantly alter its physical and chemical properties. As the calorific properties vary among different fruits, this study aims to measure the thermal properties of red dragon fruit. By measuring its thermal properties, red dragon fruit can be processed with improved energy efficiency and optimized thermal management. This study aims to examine the thermal properties and perform a cooling simulation of red dragon fruit. The cooling simulation, based on one-dimensional radial thermal flow, was conducted using the spherical heat equation explicitly discretized by the Forward Time Central Space (FTCS) scheme within the Finite Difference Method (FDM). The specific heat, thermal diffusivity, and thermal conductivity of red dragon fruit were 3.827 kJ/kg oC, 9.18 x 10-4 cm2/s, and 0.34 W/m oC, respectively. The overall average coefficient of determination (R2) from the validation test of the simulation results was 0.9654. Based on these findings, the one-dimensional (radial) discretized heat transfer equation model can be used to predict the temperature distribution in spherical fruit over time, based on its thermal diffusivity. However, this approach has limitations. Future research would benefit from measuring the thermal diffusivity of whole red dragon fruit and applying a three-dimensional discretized heat transfer equation to obtain more accurate results and minimize potential errors.

Effect of Gas Oversaturation Degree on Flotation Separation Performance of Electrode Materials from Spent Lithium-Ion Batteries

The electrode materials from spent lithium-ion batteries consist of graphite and lithium cobalt oxides (LCO), which cannot be efficiently separated by the conventional flotation technique due to the fine size distributions of graphite and LCO. In this work, nanobubbles were introduced to the flotation system of electrode materials. Nanobubbles were produced with the method of temperature difference. Different degrees of gas oversaturation in the water/slurry were achieved by raising the temperature of cold water (stored at 4 °C for at least 72 h) to target values of 20 °C, 25 °C, and 30 °C. It was found that the height and lateral distance of nanobubbles increased with the degree of gas oversaturation of water. In addition, the larger graphite agglomerations were observed to form in the presence of nanobubbles. The D50 (chord length) of graphite agglomerations increased by 8 μm, 11 μm, and 21 μm, respectively, compared with the D50 of graphite in natural water. More graphite agglomerations adhered to a captive bubble with the aid of nanobubbles than in the case of no nanobubbles, which was indicated by increased wrapping angles of graphite (agglomerations) adhering to a captive bubble. Furthermore, the maximum adhesion force between a captive bubble and substrate increases to 220, 270, and 300 μN as cold water temperature increases to 20, 25, and 30 °C, respectively. The frost of nanobubbles on a graphite surface and the resulting graphite agglomerations through the bridging effect of nanobubbles are thought to be responsible for the improved flotation performance of electrode materials. The present results indicate that the flotation performance of fine minerals can be regulated by regulating the gas oversaturation degree of the slurry.

A Non-Standard Finite Difference Scheme for Time-Fractional Singularly Perturbed Convection–Diffusion Problems

This paper introduces a stable non-standard finite difference (NSFD) method to solve time-fractional singularly perturbed convection–diffusion problems. The fractional derivative in time is defined in the Caputo sense. The proposed method shows high efficiency when applied using a uniform mesh and can be easily extended to a Shishkin mesh in the spatial domain. We discuss error estimates to demonstrate the convergence of the numerical scheme. Additionally, various numerical examples are presented to illustrate the behavior of the solution for different values of the perturbation parameter ϵ and the order of the fractional derivative.

Numerical Simulation of KdV-Burgers' Equation in Dusty Plasmas

In this paper, the reductive perturbation method is used to derive the KdV Burgers’ equation in dusty plasmas in the presence of the Boltzmann distribution of electrons and ions and charged dust grains. A numerical solution to the KdV Burgers’ equation has been obtained by using the explicit finite difference method, and the solitary and shock structures have been studied at various values of the dispersion coefficient and the dissipation coefficient. A comparison of solitary and shock structures is also shown by plotting the analytical and numerical solutions. The accuracy and efficiency of the present method have been evaluated by comparing the absolute error of the numerical results obtained with the analytical solution. An analysis of Von Neumann stability is conducted and it shows that the scheme is unconditionally stable.

Numerical implementation of optimal water resource management problems in open channels using Python

2D flow modeling plays a crucial role in understanding and predicting hydrological processes such as floods and droughts. A mathematical model is formulated based on initial and boundary conditions using 2D Saint-Venant differential equations to simulate flood movements. The model is discretized using the explicit finite difference method and implemented using Python programming. For the experiment, a rectangular-shaped flow channel was selected. The water surface elevation z(m), flow velocity components u(m/s) and v(m/s) were computed over specific time intervals and visualized.