Difference Method Research Articles

Uniform Error Bounds of a Conservative Compact Finite Difference Method for the Quantum Zakharov System in the Subsonic Limit Regime

Uniform Error Bounds of a Conservative Compact Finite Difference Method for the Quantum Zakharov System in the Subsonic Limit Regime

A high-order upwind compact difference scheme for solving the streamfunction-velocity formulation of the unsteady incompressible Navier–Stokes equations

In this paper, we propose an upwind compact difference method with fourth-order spatial accuracy and second-order temporal accuracy for solving the streamfunction-velocity formulation of the two-dimensional unsteady incompressible Navier–Stokes equations. The streamfunction and its first-order partial derivatives (velocities) are treated as unknown variables. Three types of compact difference schemes are employed to discretize the first-order partial derivatives of the streamfunction. Specifically, these schemes include the fourth-order symmetric scheme, the fifth-order upwind scheme, and the sixth-order symmetric scheme derived by combining the two parts of the fifth-order upwind scheme. As a result, the fourth-order spatial discretization schemes are established for the Laplacian term, the biharmonic term, and the nonlinear convective term, along with the Crank–Nicolson scheme for the temporal discretization. The unconditional stability characteristic of the scheme for the linear model is proved by discrete von Neumann analysis. Moreover, six numerical experiments involving three test problems with the analytic solutions, and three flow problems including doubly periodic double shear layer, lid-driven cavity flow, and dipole-wall interaction are carried out to demonstrate the accuracy, robustness, and efficiency of the present method. The results indicate that the present method not only has good numerical performance but also exhibits quite efficiency.

Deep Learning With Physics-Embedded Neural Network for Full Waveform Ultrasonic Brain Imaging.

The convenience, safety, and affordability of ultrasound imaging make it a vital non-invasive diagnostic technique for examining soft tissues. However, significant differences in acoustic impedance between the skull and soft tissues hinder the successful application of traditional ultrasound for brain imaging. In this study, we propose a physics-embedded neural network with deep learning based full waveform inversion (PEN-FWI), which can achieve reliable quantitative imaging of brain tissues. The network consists of two fundamental components: forward convolutional neural network (FCNN) and inversion sub-neural network (ISNN). The FCNN explores the nonlinear mapping relationship between the brain model and the wavefield, replacing the tedious wavefield calculation process based on the finite difference method. The ISNN implements the mapping from the wavefield to the model. PEN-FWI includes three iterative steps, each embedding the F CNN into the ISNN, ultimately achieving tomography from wavefield to brain models. Simulation and laboratory tests indicate that PEN-FWI can produce high-quality imaging of the skull and soft tissues, even starting from a homogeneous water model. PEN-FWI can achieve excellent imaging of clot models with constant uniform distribution of velocity, randomly Gaussian distribution of velocity, and irregularly shaped randomly distributed velocity. Robust differentiation can also be achieved for brain slices of various tissues and skulls, resulting in high-quality imaging. The imaging time for a horizontal cross-sectional imag e of the brain is only 1.13 seconds. This algorithm can effectively promote ultrasound-based brain tomography and provide feasible solutions in other fields.

A finite difference method on crack resistance of reinforced glass beam with non-linear adhesive

Reinforced glass beam (RG beam) is a type of structural glass member that has been developed in recent years. This RG beam consists of a glass beam to which reinforcement material, such as steel, is adhesively attached in the tensile side to improve crack resistance. The adhesive plays a critical role in the structural performance of RG beam. However, the effect of non-linear shear-slip behavior of the adhesive layer remains unclear and has been neglected in most previous studies, which can result in inaccurate estimations. To address this issue, this paper presents a finite difference model (FDM) that utilizes an explicit step-by-step method and trial-and-error iterative method for interfacial stress analysis. The model predicts the static structural response of a simply-supported RG beam that simulates the adhesive with non-linear stress-slip behavior. Furthermore, the model describes the adherend’s shear deformation in elastic. The FDM results are then compared with the finite element method (FEM), adopting discrete nonlinear connectors (springs) to simulate the adhesion, and available analytical methods. Detailed parametric studies are further conducted to investigate the influences of glass strength, load pattern, reinforcement ratio and height-to-span ratio for proposing design recommendations.

Change point detection of health stage and remaining useful lifetime prediction for electro-mechanical brake unit on trains

Electro-mechanical brake units (EMBUs) have complex degradation process. There are challenges such as component fault coupling, unclear degradation characteristics, random disturbance and so on. A remaining useful life (RUL) framework for EMBUs is developed in this paper. To eliminate the issue of time lag, a data pre-processing method based on zero-phase filtering is proposed for health indicator (HI) construction. Then, the 3 health stages (HSs) characteristics of degradation process is proposed, and is modeled using the Wiener process. To overcome perturbation of large diffusion coefficient and random fluctuation, the change point (CP) detection methods based on the variance ratio hypothesis test and maximal overlap discrete wavelet transform — multi-resolution analysis (MODWT-MRA) are developed. Besides, a consecutive anomaly window detection strategy using sliding windows is employed. Finally, the MODWT-MRA-based RUL prediction methods is proposed. The proposed methods are verified by the durability test data. The proposed data pre-processing method can avoid 10% time error compared to the common difference method. The proposed MODWT-MRA-based CP detection and RUL prediction method can overcome the interference of large random fluctuations in the non-healthy state of EMBU.

An Efficient Finite Difference Method for Stochastic Linear Second-Order Boundary-Value Problems Driven by Additive White Noises

An Efficient Finite Difference Method for Stochastic Linear Second-Order Boundary-Value Problems Driven by Additive White Noises

Numerical Analysis for Predicting the Rate of Graphitization on Wood Biomass During the Plasma-Assisted Pyrolysis

Abstract The plasma-assisted pyrolysis process is a powerful treatment for converting wood biomass to graphitic carbon. In order to make this process more precise and effective in time, the rate of graphitization data is needed, especially for predicting the effective time of treatment. In this study, numerical analysis is conducted to predict the rate of graphitization. An explicit finite difference method is applied for the numerical analysis. Some initial parameters used are the wood thickness (5 mm), coefficient of diffusivity (0.082 mm2/s), input plasma temperature (4,000 C), initial wood temperature (35 C), and room temperature (27 C). The analysis was conducted with a variation of time differences of 1 s, 2 s, 3 s, 4 s, and 5 s. The assumption used in this analysis is that the minimum temperature for graphitic carbon conversion is 2,000 C. The result of this study is the values of graphitization rates and exposure times summarized in TABLE 1. Therefore, this numerical analysis can successfully be used to predict the rate of graphitization and plasma exposure time for different wood biomass thicknesses.

Stereologically corrected particle size distributions for polymer-mounted additive manufacturing powders

Well-defined particle size distributions are required for good flowability and powder packing properties of additive manufacturing powders. Mounting powders within a polymer and using standard metallurgical preparation techniques to cross-section and prepare powder particles for optical analysis allows for simple characterisation processes. However, measured diameters of cross-sectioned particles are typically underestimates of actual particle diameters and hence require stereological correction. The effectiveness of three stereological corrections are investigated in this work, namely the Scheil-Schwartz-Saltykov method, the Goldsmith-Cruz-Orive method and a Finite Difference Method. These methods are investigated against plasma-atomised, gas-atomised and ultrasonically processed Ti-6Al-4 V powders. The corrected outputs are compared to laser size diffraction, benchmark data for each powder. Although all three stereological corrections produce improved estimations of the particle size distributions, the Finite Difference Method is recommended producing cumulative mean absolute error values of 2.4%, 3.1% and 7.5% for the plasma-atomised, gas-atomised and ultrasonically processed powders respectively.

Two Decoupled and Linearized Block-Centered Finite Difference Methods for the Nonlinear Symmetric Regularized Long Wave Equation

Two Decoupled and Linearized Block-Centered Finite Difference Methods for the Nonlinear Symmetric Regularized Long Wave Equation

A Fast Compact Block-Centered Finite Difference Method on Graded Meshes for Time-Fractional Reaction-Diffusion Equations and Its Robust Analysis

A Fast Compact Block-Centered Finite Difference Method on Graded Meshes for Time-Fractional Reaction-Diffusion Equations and Its Robust Analysis

Efficiently linear and unconditionally energy-stable time-marching schemes with energy relaxation for the phase-field surfactant model

In this article, we consider consistent, efficient, and accurate numerical approximations for a recently developed Cahn–Hilliard type phase-field surfactant model enjoying the bounded-from-below condition of total energy. The phase-field surfactant model naturally satisfies the energy dissipation property in time. Based on this feature, we construct linearly decoupled and unconditionally energy-stable time-marching schemes based on an efficient variant of scalar auxiliary variable approach. To enhance the consistency of discrete modified energy and discrete original energy, a practical energy relaxation technique is adopted. We analytically proved the unique solvability and unconditional energy stability of the proposed schemes. In each time step, only two linear elliptic type equations with constants coefficients need to be solved and several algebraic operations need to be performed. Therefore, the proposed schemes are highly efficient and easy to implement. The space is discretized by the finite difference method and the linear multigrid algorithm is used to accelerate the convergence. The numerical results indicate that the proposed method not only has desired accuracy in time but also satisfies the energy dissipation law even if larger time steps are used. Furthermore, the surfactant laden spinodal decompositions and surfactant absorptions can be well simulated.

Efficient dimension-by-dimension adaptive TENO-based limiter and troubled-cell indicator for nodal-based high-order spectral difference method

A new targeted essentially non-oscillatory (TENO) limiter with adaptive dissipation has been developed for the 3rd- and 4th-order spectral difference method. This limiter can potentially be useful for other nodal-based discontinuous high-order methods as well. Unlike traditional WENO limiters for these methods which limit low-order moments one by one, the new limiter is based on a direct convex combination of reconstruction polynomial candidates. This strategy ensures the good computational efficiency for high order reconstructions. The reconstruction stencils only involve nodal values from the target cell and its neighbors sharing faces with it. The reconstruction employs Hermite polynomials, making the new limiter compact. It can be implemented in a dimension-by-dimension manner for multi-dimensional problems, which is consistent with the spectral difference method. To further enhance efficiency, a TENO-based troubled-cell indicator is also developed to only activate limiters in troubled cells. Extensive one-dimensional and two-dimensional numerical experiments validate the performance of the new TENO-based limiter and indicator. In summary, the limiter is much less dissipative than WENO limiters and can resolve richer small-scale flow structures on coarse meshes. The indicator precisely marks out troubled cells, slightly improving over the KXRCF indicator.

Damage detection and model updating of offshore jacket platforms using a direct sensitivity relation based on structural response

One of the challenges of the finite element model updating using time domain data is the lack of a direct relationship between changes in structural parameters and structural responses. Computational techniques, such as the Finite Difference Method (FDM) and Direct Differentiation Method (DDM), are used to calculate the sensitivity of the structural response to variations of structural parameters in the time domain. The use of these methods is time-consuming for large and complex structure. In the present study, an exact sensitivity equation is presented that establishes a direct and linear relationship between changes of structural parameters and structural response variations. The proposed closed form equations, have high computational efficiency in calculating the sensitivity of time domain responses. The measured responses directly contribute to the calculation of the sensitivity matrix, so the finite element model updating procedure is improved in terms of accuracy and reliability. The proposed method updates the finite element model even using a single excitation frequency that distinguishes the presented sensitivity relationship from frequency response-based methods which use wide ranges of excitation frequencies for model updating. To evaluate the effectiveness of the proposed method, different damage scenarios with different intensities have been simulated on two offshore jacket platform models. The performance of the proposed method in identifying the location and severity of induced damages in the presence of measurement and modeling errors confirms the effectiveness of the proposed method.

Mixed Galerkin- Implicit Differences Methods for Solving Coupled Parabolic Boundary Value Problems with Variable Coefficients

Mixed Galerkin- Implicit Differences Methods for Solving Coupled Parabolic Boundary Value Problems with Variable Coefficients

Spatio-temporal change of winter wheat yield and its quantitative responses to compound frost-dry events – An example of the Huang-Huai-Hai Plain of China from 2001 to 2020

Extreme climate events such as frost and drought have great influence on wheat growth and yield. Understanding the effects of frost, drought and compound frost-dry events on wheat growth and yield is of great significance for ensuring national food security. In this study, wheat yield prediction model (SCYMvp) was developed by combining crop growth model (CGM), satellite images and meteorological variables. Wheat yield maps in the Huang-Huai-Hai Plain (HHHP) during 2001–2020 were generated using SCYMvp model. Meanwhile, accumulative frost days (AFD), accumulative dry days (ADD) and accumulative frost-dry days (AFDD) in different growth periods of wheat were calculated, and the effects of frost and drought on wheat yield were quantified by the first difference method and linear mixed model. The results showed that wheat yield increased significantly, while the rising trend was obvious at more than half of the regions. Extreme climate events (ECEs) showed a relatively stable change trend, although the change trend was significant only in a few areas. Compared with frost and drought in the early growth period, ECEs in the middle growth period (spring ECEs) had more negative effects on wheat growth and yield. Wheat yield was negatively correlated with spring ECEs, and yield loss was between 4.6 and 49.8 kg/ha for each 1 d increase of spring ECEs. The effects of spring ECEs on wheat yield were ranked as AFDD > AFD > ADD. The negative effect of ADD on wheat yield in the late growth period was higher than that in the other periods. The negative effects of spring ECEs on yield in southern regions were higher than those in northern regions. Overall, due to the adverse effects of frost and drought on wheat yield in the middle and late growth periods, the mean annual yield loss was 6.4 %, among which spring AFD caused the greatest loss to wheat yield (3.1 %). The results have important guiding significance for formulating climate adaptation management strategies.

Numerical analyses on the spatial-temporal thermal influence range of embankment in the Qinghai-Tibet Engineering Corridor

The density of linear engineering projects within the Qinghai-Tibet Engineering Corridor has increased, and the study of the thermal influence range of a single linear engineering project has become more important. This paper employs the finite difference method to investigate the influence of embankment size, mean annual ground temperature (MAGT), and construction time on the spatial-temporal evolution of the embankment's lateral and vertical thermal influence range. Additionally, a parameter sensitivity analysis of the lateral and vertical thermal influence ranges throughout the entire service life of the embankment is conducted. The results indicate that under the same service time, the lateral-vertical ratio (Lc/D), as well as the lateral thermal influence range (Lc) and the vertical thermal influence range (D), increase linearly with increasing embankment height and pavement width. The Lc/D increases with increasing MAGT, but the Lc and D decrease with increasing MAGT. The Lc/D, the Lc, and the D are not significantly affected by construction time. During the 50-year service life of the embankment, the most significant factor impacting the Lc is the pavement width, and the MAGT is the most significant factor impacting the D. The results are expected to serve as a guide for the design and construction of roads in the permafrost regions of the Qinghai-Tibet Plateau.

Does E-Commerce Construction Boost Farmers’ Incomes? Evidence from China

Elevating farmers’ incomes is crucial for ensuring socio-economic stability, yet the current stagnation in income growth and the expanding divide between urban and rural incomes present formidable challenges. E-commerce offers a transformative solution by bridging these disparities and fostering sustainable growth. Utilizing county-level data from 2000 to 2020 in China, particularly focusing on “The Pilot Counties of Introducing E-commerce to Rural Areas”, this paper explores the impact of e-commerce construction on farmers’ income growth using the multiple difference method. The findings reveal that e-commerce development exerts a significant positive impact on the enhancement of farmers’ incomes. Mechanistic analyses indicate that the driving effect of the comprehensive demonstration county policy of e-commerce in rural areas on farmers’ incomes is mainly realized through the channels of expanding market accessibility, promoting the advanced transformation of industrial structure, and optimizing the conditions of information infrastructure. Moreover, the efficacy of e-commerce policies in income augmentation is more pronounced in regions characterized by higher levels of human capital and substantial scale effects. This research offers valuable insights for continuously and effectively implementing the “Comprehensive Demonstration of E-commerce in Rural Areas” policy, which are crucial for exploring ways and mechanisms to boost farmers’ incomes in developing countries.

A comprehensive analysis of COVID-19 nonlinear mathematical model by incorporating the environment and social distancing

This research conducts a detailed analysis of a nonlinear mathematical model representing COVID-19, incorporating both environmental factors and social distancing measures. It thoroughly analyzes the model’s equilibrium points, computes the basic reproductive rate, and evaluates the stability of the model at disease-free and endemic equilibrium states, both locally and globally. Additionally, sensitivity analysis is carried out. The study develops a sophisticated stability theory, primarily focusing on the characteristics of the Volterra–Lyapunov (V–L) matrices method. To understand the dynamic behavior of COVID-19, numerical simulations are essential. For this purpose, the study employs a robust numerical technique known as the non-standard finite difference (NSFD) method, introduced by Mickens. Various results are visually presented through graphical representations across different parameter values to illustrate the impact of environmental factors and social distancing measures.

Efficient computational hybrid method for the solution of 2D multi-term fractional order advection-diffusion equation

Abstract Multi-term time-fractional advection diffusion equations are vital for simulating a wide range of physical phenomena, including fluid dynamics and environmental transport processes. However, due to their natural complexity, these equations pose challenges for conventional numerical approaches. In this article, we develop a high order accurate method to solve the multi-term time-fractional advection diffusion equations. We combine the Laplace transform (LT) to integrate the considered equations in time, with Chebyshev spectral method (CSM) for spatial terms The proposed method produces highly accurate solutions with remarkably low computational cost as compared to finite difference method. The propose numerical scheme first employs the LT which reduces the considered problem into a finite set of elliptic equations which may be solved in parallel. Then, the CSM is employed for the disctrezation of spatial operators, which makes it possibly to accurately represent the solution chebyshev grid. Finally, numerical inversion of LT is used to convert the obtain solution from the Laplace domain into the real domain. This work utilizes the modified Talbot’s method and Stehfest’s method for numerical inversion of the LT. To measure the performance, efficiency, and accuracy of the suggested approach, numerical approximations of three models are acquired and verified against the exact solution. The outcomes presented in tables and figures demonstrate that the modified Talbot’s method performed better as compared to Stehfest’s method.

Dynamics of the ohmic heating and chemically reactive time-dependent flow of magnetized Casson fluid inside a rectangular pipe

Chemical reaction and ohmic heating can significantly influence the flow behavior and heat transfer characteristics in industrial processes involving non-Newtonian fluids. These processes include polymer processing, food processing, and manufacturing processes where non-Newtonian fluids are commonly encountered. Therefore, the impacts of chemical reaction and ohmic heating on the unsteady flow of an incompressible magnetized non-Newtonian Casson liquid through a rectangular pipe with a variable pressure gradient is scrutinized in this article. The impacts of heat generation/absorption and viscous dissipation are also incorporated through the energy equation. The numerical simulations of partial differential equations (PDEs) are acquired utilizing the finite difference method with the incorporating relaxation (SOR) algorithm. The impacts of dissimilar imperative parameters on various flow fields are scrutinized through graphs. Outcomes indicate that liquid velocity decreases with enhancing values of the Casson parameter. Furthermore, the liquid temperature has an enhancing behavior with the impact of the heat source/sink parameter. Comparisons of numerical results with previously published results show an excellent agreement. This kind of research may find specific industrial and medical utilizations such as glass manufacturing, crude oil purification, lubrication, paper production, blood transport study in cardiovascular design, etc.