Finite Difference Method Research Articles

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.

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.

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

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.

Deep Learning for Solving Partial Differential Equations: A Review of Literature

Partial Differential Equations (PDEs) are fundamental in modeling various phenomena in physics, engineering, and finance. Traditional numerical methods for solving PDEs, such as finite element and finite difference methods, often face limitations when applied to high-dimensional and complex systems. In recent years, deep learning has emerged as a promising alternative for approximating solutions to PDEs, offering potential improvements in both efficiency and scalability. This paper provides a comprehensive review of the literature on deep learning-based methods for solving PDEs, focusing on key approaches such as Physics-Informed Neural Networks (PINNs), deep Galerkin methods, and neural operators. These methods leverage the expressiveness of neural networks to capture underlying physics while avoiding the curse of dimensionality associated with classical techniques. We explore the theoretical foundations, advantages, and limitations of these deep learning models, along with their applications in diverse fields like fluid dynamics, quantum mechanics, and financial modeling. Additionally, this review examines recent advancements in hybrid models that combine traditional numerical methods with deep learning approaches to enhance accuracy and stability. Through this review, we highlight key trends and open challenges in the field, paving the way for future research at the intersection of deep learning and computational mathematics.

Comparing Performance in the Codes AREPO, GIZMO and GAMER-2

Modern astrophysical simulation is the main aspect of human getting insight from the cosmos. Numerical simulations became irreplaceable tools for generating various phases of the universe, the formation of galaxies, and other astrophysical phenomena. As the advances of computational power, validity and diversity of astrophysical modeling have been growing. In addition to the development of the processor performance, the diversity of the software community is also rising therefore, research between various simulation codes is worthwhile. It allowed researchers to know the nature of different simulation techniques linked to their basic algorithms. This study summarizes the basics of common simulation methods used by researchers including N-body Simulations, Smoothed Particle Hydrodynamics, Monte Carlo Methods, Finite Difference and Finite Element Methods. Furthermore, this research analyzes the basics of AREPO, GIZMO, and GAMER-2 codes, and compared the performance of these codes. Last, according to the analysis, this study discusses and demonstrates the advantages and limitations of these codes. These results shed light on guiding further exploration of cosmology simulations.

An efficient fourth-order convergent scheme based on half-step spline function for two-point mixed boundary value problems

The present work aims to introduce a novel numerical method for solving second-order two-point mixed boundary value problems. Mixed boundary value problems occur in various scientific fields, including quantum mechanics, fluid dynamics, and chemical reactor theory. The proposed method provides a highly accurate, fourth-order convergent numerical solution, achieved by implementing a compact finite difference method based on non-polynomial spline in tension approximations. Notably, the discretisation involves the use of half-step grid points, eliminating the need to modify the method at singularities while dealing with singular boundary value problems. The article also includes a comprehensive convergence analysis that demonstrates the theoretical order of convergence of the method. Additionally, the results obtained from solving six diverse mixed boundary value problems, including Burger's equation and Bratu-type equation, are compared to showcase the superiority and effectiveness of the proposed method.

Discretization of Fractional Fully Nonlinear Equations by Powers of Discrete Laplacians

AbstractWe study discretizations of fractional fully nonlinear equations by powers of discrete Laplacians. Our problems are parabolic and of order $$\sigma \in (0,2)$$ σ ∈ ( 0 , 2 ) since they involve fractional Laplace operators $$(-\Delta )^{\sigma /2}$$ ( - Δ ) σ / 2 . They arise e.g. in control and game theory as dynamic programming equations – HJB and Isaacs equation – and solutions are non-smooth in general and should be interpreted as viscosity solutions. Our approximations are realized as finite-difference quadrature approximations and are 2nd order accurate for all values of $$\sigma $$ σ . The accuracy of previous approximations of fractional fully nonlinear equations depend on $$\sigma $$ σ and are worse when $$\sigma $$ σ is close to 2. We show that the schemes are monotone, consistent, $$L^\infty $$ L ∞ -stable, and convergent using a priori estimates, viscosity solutions theory, and the method of half-relaxed limits. We also prove a second order error bound for smooth solutions and present many numerical examples.

High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries

Abstract In this article, a fourth-order finite-difference ghost-point method for the Poisson equation on regular Cartesian mesh is presented. The method can be considered the high-order extension of the second-order ghost method introduced earlier by the authors. Three different discretizations are considered, which differ in the stencil that discretizes the Laplacian and the source term. It is shown that only two of them provide a stable method. The accuracy of such stable methods is numerically verified on several test problems.

Research on the Stability of Lining Structures Under Different Fault Moments Based on FDM-DEM

Currently, research on employing finite difference method and discrete element method (FDM-DEM) coupling to assess the stability of tunnel lining structures is limited. This study utilized the FDM-DEM coupling approach, with the F2 fault of the East Tianshan Tunnel as a case study, to develop a numerical model in conjunction with PFC3D 6.0 and FLAC3D 6.0 software. We conducted a comprehensive analysis of the displacement deformation and crack progression of the tunnel lining structure under varying dislocation momentum conditions, unveiling the underlying mechanisms. The findings indicated that as the dislocation increased, the extent of damage to the vault intensified, and the particle contact force within the tunnel lining shifted from compression to tension, significantly contributing to the crack formation. Fault dislocation influenced the gradual expansion of cracks from the vault to the spandrel and arch waist, with the crack width increasing alongside the rising dislocation momentum. In particular, under substantial dislocation momentum, the overall stability of the tunnel lining was markedly diminished. The safety factor at the tunnel section declined progressively as the dislocation momentum escalated, with values of 2.53, 2.49, 2.43, 2.39, and 2.32 corresponding to dislocation momenta of 0.01 m, 0.05 m, 0.1 m, 0.15 m, and 0.2 m, respectively. This research offers valuable insights and a reference framework for investigating the stability of tunnel lining structures in proximity to fault dislocations, pinpointing potential failure points, and bolstering the structural integrity of tunnels.

A conformable fractional finite difference method for modified mathematical modeling of SAR-CoV-2 (COVID-19) disease.

In this research, the ongoing COVID-19 disease by considering the vaccination strategies into mathematical models is discussed. A modified and comprehensive mathematical model that captures the complex relationships between various population compartments, including susceptible (Sα), exposed (Eα), infected (Uα), quarantined (Qα), vaccinated (Vα), and recovered (Rα) individuals. Using conformable derivatives, a system of equations that precisely captures the complex interconnections inside the COVID-19 transmission. The basic reproduction number (R0), which is an essential indicator of disease transmission, is the subject of investigation calculating using the next-generation matrix approach. We also compute the R0 sensitivity indices, which offer important information about the relative influence of various factors on the overall dynamics. Local stability and global stability of R0 have been proved at a disease-free equilibrium point. By designing the finite difference approach of the conformable fractional derivative using the Taylor series. The present methodology provides us highly accurate convergence of the obtained solution. Present research fills research addresses the understanding gap between conceptual frameworks and real-world implementations, demonstrating the vaccination therapy's significant possibilities in the struggle against the COVID-19 pandemic.

Effect of Suction/Injection on Unsteady MHD Natural Convection Flow of Heat Mass Transfer in Porous Channel in the Presence of Soret Term

This research paper explores the effect of suction/injection on unsteady MHD natural convection flow of heat mass transfer in porous channel in the presence of Soret term. The governing partial differential equations are converted to non- dimensional forms and solved numerically by using unconditionally stable and convergent implicit finite difference method. A parametric study illustrating the influence of various physical parameters is performed. It is reported that the velocity profile increases as Soret term, Grashof number, Solutal Grashof number and Porous parameters values increase, while Magnetic field parameter decreases the velocity profile. The temperature profile rises by the influence in increasing values of Variable Thermal Conductivity and decreases by increasing values of Prandtl number and Radiation parameters. While concentration profile increase by the increasing values of Soret term and Chemical reaction. The dependence of the skin friction coefficient, rate of heat transfer and mass transfer on these parameter has been discussed.

Simulation of Dendrite Remelting via the Phase-Field Method

The solidification of alloys is a key physical phenomenon in advanced material-processing techniques including, but not limited to, casting and welding. Mastering and controlling the solidification process and the way in which microstructure evolution occurs constitute the key to obtaining excellent material properties. The microstructure of a solidified liquid metal is dominated by dendrites. The growth process of these dendrites is extremely sensitive to temperature changes, and even a small change in temperature can significantly affect the growth rate of the dendrite tip. Dendrite remelting is inevitable when the temperature exceeds the critical threshold. In this study, a temperature-induced-dendrite remelting model was established, which was implemented through the coupling of the phase field method (PFM) and finite difference method (FDM). The transient evolution law of dendrite remelting was revealed by simulating dendritic growth and remelting processes. The phase field model showed that the lateral dendrites melt first, the main dendrites melt later, and the main dendrites only shrink but do not melt when the lateral dendrites have not completely melted or the root is not broken. The long lateral branches break into fragments, while the short lateral branches shrink back into the main dendrites. The main dendrites fracture and melt in multiple stages due to inhomogeneity.

Investigation of nonlinear dynamic behaviors of vertical rotor system supported by aerostatic bearings

Abstract A common issue associated with gas bearing-rotor systems is the tendency to generate self-excited vibrations, leading to instability. To address this problem, the fluid-structure coupling model of an aerostatic bearing-rotor system is established in this paper. Then a hybrid method combining the finite difference method and direct integration method, is employed to solve the bearing lubrication equation and rotor motion equation simultaneously. Furthermore, based on the orbit of rotor center,frequency spectrum diagram, Poincaré map, waterfall diagram and bifurcation diagram, the effects of rotational speed, rotor mass, orifice diameter and nominal clearance on the nonlinear dynamic behaviors of the bearing-rotor system are investigated. The results indicate that the system exhibits rich nonlinear behaviors with increasing rotational speed and rotor mass, including the occurrence of typical half-speed whirl. However, the nonlinear vibrations of the system can be restricted by selecting appropriate bearing structural parameters, providing theoretical guidance for the design of aerostatic bearing-rotor systems.

A Novel Fourth-Order Finite Difference Scheme for European Option Pricing in the Time-Fractional Black–Scholes Model

This paper addresses the valuation of European options, which involves the complex and unpredictable dynamics of fractal market fluctuations. These are modeled using the α-order time-fractional Black–Scholes equation, where the Caputo fractional derivative is applied with the parameter α ranging from 0 to 1. We introduce a novel, high-order numerical scheme specifically crafted to efficiently tackle the time-fractional Black–Scholes equation. The spatial discretization is handled by a tailored finite point scheme that leverages exponential basis functions, complemented by an L1-discretization technique for temporal progression. We have conducted a thorough investigation into the stability and convergence of our approach, confirming its unconditional stability and fourth-order spatial accuracy, along with (2−α)-order temporal accuracy. To substantiate our theoretical results and showcase the precision of our method, we present numerical examples that include solutions with known exact values. We then apply our methodology to price three types of European options within the framework of the time-fractional Black–Scholes model: (i) a European double barrier knock-out call option; (ii) a standard European call option; and (iii) a European put option. These case studies not only enhance our comprehension of the fractional derivative’s order on option pricing but also stimulate discussion on how different model parameters affect option values within the fractional framework.

Terahertz Non-Destructive Testing of Porosity in Multi-Layer Thermal Barrier Coatings Based on Small-Sample Data

Accurately characterizing the internal porosity rate of thermal barrier coatings (TBCs) was essential for prolonging their service life. This work concentrated on atmospheric plasma spray (APS)-prepared TBCs and proposed the utilization of terahertz non-destructive detection technology to evaluate their internal porosity rate. The internal porosity rates were ascertained through a metallographic analysis and scanning electron microscopy (SEM), followed by the reconstruction of the TBC model using a four-parameter method. Terahertz time-domain simulation data corresponding to various porosity rates were generated employing the time-domain finite difference method. In simulating actual test signals, white noise with a signal-to-noise ratio of 10 dB was introduced, and various wavelet transforms were utilized for denoising purposes. The effectiveness of different signal processing techniques in mitigating noise was compared to extract key features associated with porosity. To address dimensionality challenges and further enhance model performance, kernel principal component analysis (kPCA) was employed for data processing. To tackle issues related to limited sample sizes, this work proposed to use the Siamese neural network (SNN) and generative adversarial network (GAN) algorithms to solve this challenge in order to improve the generalization ability and detection accuracy of the model. The efficacy of the constructed model was assessed using multiple evaluation metrics; the results indicate that the novel hybrid WT-kPCA-GAN model achieves a prediction accuracy exceeding 0.9 while demonstrating lower error rates and superior predictive performance overall. Ultimately, this work presented an innovative, convenient, non-destructive online approach that was safe and highly precise for measuring the porosity rate of TBCs, particularly in scenarios involving small sample sizes facilitating assessments regarding their service life.

Optimization of coupling phase-change materials and thermal screens in façade-integrated hybrid photovoltaic collectors for optimal energy production and thermal comfort in buildings

The operation of building-integrated photovoltaic (BIPV) systems gives rise to a significant proportion of the solar radiation absorbed by the cells being unable to be converted into electricity. This phenomenon consequently increases the temperature. This temperature increase impacts the cells' electrical efficiency, leading to a reduction in their performance and accelerating their degradation. The combination of phase-change materials with insulating fluid blades, situated behind photovoltaic cells, represents a passive cooling solution that optimizes the performance of hybrid photovoltaic systems when incorporated into facades. The present study assesses a system that incorporates paraffin as a PCM and an argon layer in a PV-PCM-argon layer physical model (PVT/ArPCM), in comparison with a PV-PCM system (PVT/PCM), to enhance the thermoelectric performance of photovoltaic systems mounted on façades while ensuring optimal thermal comfort within buildings. The discrete heat transfer equations were solved using the Thomas algorithm and the iterative Gauss-Seidel method in conjunction with the implicit finite difference method. The findings illustrate that the electrical efficiency experienced only a slight increase, estimated at 0.01%, while there was a notable enhancement in the indoor thermal comfort experienced by occupants, with a 65% improvement observed due to the incorporation of an argon-filled thermal screen. The incorporation of an argon layer led to a minor reduction in temperature of 0.01°C in the photovoltaic cells, resulting in a minimal improvement of 0.014% in electrical power production. The phase-changing material incorporated into PVT/ArPCM demonstrated superior thermal management capabilities in comparison to the same material employed in PVT/PCM.

Effects of Anisotropy, Convection, and Relaxation on Nonlinear Reaction-Diffusion Systems

The effects of relaxation, convection, and anisotropy on a two-dimensional, two-equation system of nonlinearly coupled, second-order hyperbolic, advection–reaction–diffusion equations are studied numerically by means of a three-time-level linearized finite difference method. The formulation utilizes a frame-indifferent constitutive equation for the heat and mass diffusion fluxes, taking into account the tensorial character of the thermal diffusivity of heat and mass diffusion. This approach results in a large system of linear algebraic equations at each time level. It is shown that the effects of relaxation are small although they may be noticeable initially if the relaxation times are smaller than the characteristic residence, diffusion, and reaction times. It is also shown that the anisotropy associated with one of the dependent variables does not have an important role in the reaction wave dynamics, whereas the anisotropy of the other dependent variable results in transitions from spiral waves to either large or small curvature reaction fronts. Convection is found to play an important role in the reaction front dynamics depending on the vortex circulation and radius and the anisotropy of the two dependent variables. For clockwise-rotating vortices of large diameter, patterns similar to those observed in planar mixing layers have been found for anisotropic diffusion tensors.

Susceptibility of hydraulic structures to internal erosion: modeling of suffusion process

Internal erosion is a primary cause of hydraulic structure failure, manifesting as two key phenomena: piping erosion and suffusion. These processes differ in their geometrical and hydraulic boundary conditions, resulting in varying levels of failure risk. Piping erosion is the more hazardous and rapid process, often leading to immediate failure if not promptly addressed. In contrast, suffusion gradually alters the permeability of the medium, potentially culminating in failure after a prolonged period of development. In this study, we propose a model for suffusion based on Darcy’s law, Papamichos' erosion law, and Einstein's viscosity evolution relation. The model describes the temporal evolution of the average porosity in the porous medium, the concentration of eroded particles in the fluid, and the mass of particles eroded, while highlighting the impact of hydraulic and mechanical parameters, such as the erosion coefficient and maximum porosity, on this evolution. The numerical solution is obtained using the finite difference method, with the sample discretized into elementary layers. For discretization, an explicit off-centered scheme was adopted. The simulation results reveal that suffusion is strongly influenced by soil hydraulic and mechanical parameters, particularly the erosion coefficient (λ) and final porosity (Ømax).

Heat transfer of mid-deep ground source heat pump for crude oil gathering and transportation

Wax precipitation in crude oil gathering and transportation (COG&T) causes substantial decrease in oil flowability and thus blockages of pipelines, presenting high safety risks and dramatic economic loss. Conventional oil/gas blending and low-temperature transport are inefficient and energy-intensive. Geothermal energy released from abandoned oil wells may be used as renewable heat source, but remains unexplored. This study focused on feasibility of utilizing geothermal energy from abandoned oil wells by mid-deep ground source heat pump (MD-GSHP), so as to alleviate wax precipitation for short-distance COG&T. Numerical models of ground heat exchanger (GHE) were established by using finite difference method (FDM) based on validation of engineered examples. The results demonstrated that, to enhance geothermal extraction, it was essential to increase fluid flow rate, lower inlet temperature, use materials with high thermal resistance, and optimize the pipe dimensions. Additionally, the impacts of operational parameters of GHE and COG&T subsystems on MD-GSHP system performances were also analyzed. Increasing inlet temperature by 8 °C resulted in 39.5 % reduction in system load and 89.4 % increase in coefficient of performance (COP). This study provides a proof-in-concept demonstration for extracting geothermal energy from abandoned wells by MD-GSHP to mitigate wax precipitation in COG&T project.