Abstract: This paper achieves effective and precise meshless modeling of three-dimensional (3D) single-phase coupled heat and mass transfer problems based on the generalized finite difference method (GFDM). It utilizes the Taylor formula and the weighted least squares method in the node influence domains to derive a generalized finite difference scheme for spatial derivatives of pressure and temperature. Consequently, a sequential coupled discrete scheme for the pressure diffusion equation and heat convection–conduction equation is formulated, resulting in the determination of pressure and temperature. An example conducts sensitivity analysis with different schemes of node collocation and different radius of influence domains. The calculation results demonstrate that this method exhibits good convergence. Two 3D model examples with regular and irregular boundaries illustrate the advantages of the GFDM in handling complex geometric problems within the computational domain, showcasing its superior flexibility and simplicity. This paper demonstrates the significant potential of GFDM in addressing complex geometric multi-physics field coupling challenges, offering innovative ideas for geothermal resource development, groundwater management, and thermal recovery in oil and gas reservoirs.

Abstract: This study presents an (ε, μ)−uniform numerical method for a two-parameter singularly perturbed time-delayed parabolic problems. The proposed approach is based on a fitted operator finite difference method. The Crank–Nicolson method is used on a uniform mesh to discretize the time variables initially. Subsequently, the resulting semi-discrete scheme is further discretized in space using Simpson's 1/3rd rule. Finally, the finite difference approximation of the first derivatives is applied. The method is unique in that it is not dependent on delay terms, asymptotic expansions, or fitted meshes. The fitting factor's value, which is used to account for abrupt changes in the solution, is calculated using the theory of singular perturbations. The developed scheme is demonstrated to be second-order accurate and uniformly convergent. The proposed method's applicability is validated by three model examples, which yielded more accurate results than some other methods found in the literature.

Abstract: In force-based beam finite elements, cross-section transverse displacements are often needed for post-processing purposes and for geometrically nonlinear structural analysis. This involves the complex integration of the cross-section strains along the beam axis, typically done by the Curvature and Shear Based Displacement Interpolation (CSBDI) technique. Although, the CSBDI is sufficiently accurate for standard applications, this may cause numerical issues when many quadrature cross-sections are placed along the element length. This work presents a novel technique for computing the transverse displacements of a 3D Timoshenko beam, based on a finite difference approximation of the bending and shear compatibility conditions, which avoids the issues of the CSBDI. The proposed technique is introduced in a force-based finite element formulation with moderately large deformations, endowed with a corotational approach, suitable for analyzing geometrically nonlinear framed structures. Detailed investigation of the accuracy and efficiency of the proposed technique is conducted comparing its performance with that of the CSBDI approach.

Abstract: With the rapid development of anti-mine technologies such as mine sweeping and noise reduction, the cost-effectiveness ratio of conventional mines has significantly decreased. To enhance the target detection capability of mines, it is necessary to research other technologies. Therefore, it is necessary to study the propagation characteristics of shallow sea seismic waves on the surface of buried bodies. In this paper, the staggered grid finite difference method is used to simulate the propagation of shallow sea seismic waves on the surface of buried bodies. The main analysis focuses on the influence of burial depth, burial body size, and physical parameters of burial bodies on the propagation characteristics of shallow sea seismic waves on the surface of buried bodies. The results show that burial depth has a relatively small impact on the shallow sea seismic waves on the surface of buried bodies, The size and physical parameters of the burial body have a significant impact on the shallow sea seismic waves on the surface of the burial body.

Abstract: Physics-Informed Neural Networks (PINN) has surged in popularity within deep learning. However, automatic differentiation (AD) based PINN may encounter inaccuracies in the explicit functional expression of neural networks regarding inputs, leading to imprecise computations. Moreover, AD is tied to neural network complexity, incurring substantial training costs for solving complex problems. This study adopts finite difference based PINN (FD-PINN) to solve electromagnetic forward problems, reducing computational costs by disconnecting from the influence of neural network scale and prediction accuracy on derivative computation precision. Demonstrated through examples, the feasibility of this approach is solely linked to grid size, offering insights for rapid electromagnetic inverse design in subsequent studies.

Abstract: This work examines a mathematical model of diabetes mellitus and its consequences in a population using fractional differential equations. It attempts to solve the problem using a nonstandard way because standard finite difference numerical methods can result in numerical instabilities. The nonstandard finite difference scheme (NSFDS), which satisfies dynamical consistency, is the recommended nonstandard method for discretising the model. To demonstrate the stability of the model at the equilibrium points, analyses of both discrete and continuous models are performed.Stability analysis is carried out at the discretised models equilibrium point using the Schur-Cohn criterion. Consequently, the models asymptotically stable state is demonstrated. Furthermore, by contrasting the stability for various step sizes with conventional techniques like Finite Difference Scheme (FDS), the benefits of the NSFDS are shown. The NSFDS has been shown to converge at bigger step sizes. Furthermore, a graphical comparison is shown between the numerical findings acquired by the NSFDS and the FDS. It is noted that the NSFDS is accurate.

Abstract: The Voxels-in-Cell (VIC) method was recently introduced for reducing the computational cost of the finite- difference time-domain (FDTD) method with objects composed with voxels. In this paper, we present some applications of VIC method in electromagnetic problems and show that the accuracy of the method is preserved while large reductions of the computational requirements can be achieved.

Abstract: Magnetotelluric (MT) forward modeling is essential in geophysical exploration, enabling the investigation of the Earth’s subsurface electrical conductivity. Traditional finite difference methods (FDMs) typically use uniform grids, which can be computationally inefficient and fail to accurately capture complex geological structures. This study addresses these challenges by introducing a non-uniform grid-based FDM for MT forward modeling. The proposed method optimizes computational resources by varying grid resolution, offering finer grids in areas with complex geology and coarser grids in more homogeneous regions. We apply this method to both typical synthetic models and a complex fault structure case study, demonstrating its capability to accurately resolve subsurface features while reducing computational costs. The results highlight the method’s effectiveness in capturing fine-scale details that are often missed by uniform grid approaches. The conclusions drawn from this study suggest that the non-uniform grid FDM not only improves the accuracy of MT modeling but also enhances its efficiency, making it a valuable tool for geophysical exploration in challenging environments.

Abstract: Some formulas have been derived / developed for mathematical representation of numerical data on a pair of variables by suitable mathematical equation / mathematical curve applying some commonly used operations in calculus of finite difference namely usual algebraic operation, forward difference operation, backward difference operation, divided difference operation, backward divided difference operation, difference & ratio operation and backward difference & ratio operation. This paper is a brief review on these recent developments of the formulas for representing numerical data on a pair of variables by mathematical curve.

Abstract: In this paper, we present an intensive investigation of the finite volume method (FVM) compared to the finite difference methods (FDMs). In order to show the main difference in the way of approaching the solution, we take the Burgers equation and the Buckley–Leverett equation as examples to simulate the previously mentioned methods. On the one hand, we simulate the results of the finite difference methods using the schemes of Lax–Friedrichs and Lax–Wendroff. On the other hand, we apply Godunov's scheme to simulate the results of the finite volume method. Moreover, we show how starting with a variational formulation of the problem, the finite element technique provides piecewise formulations of functions defined by a collection of grid data points, while the finite difference technique begins with a differential formulation of the problem and continues to discretize the derivatives. Finally, some graphical and numerical comparisons are provided to illustrate and corroborate the differences between these two main methods.

Abstract: <p><strong><span>Abstract</span></strong></p> <p><span>In this paper, we propose an efficient finite difference method to solve third order boundary value problems. Analysis proves that the proposed method have at least quadratic convergence. Numerical results confirm the accuracy and efficiency of the proposed method.</span></p> <p> </p>

Abstract: In this article, a numerical efficient method for fractional mobile/immobile equation is developed. The presented numerical technique is based on the compact finite difference method. The spatial and temporal derivatives are approximated based on two difference schemes of orders [Formula: see text] and [Formula: see text], respectively. The proposed method is unconditionally stable and the convergence is analyzed within Fourier analysis. Furthermore, the solvability of the compact finite difference approach is proved. The obtained results show the ability of the compact finite difference.