Generalized Difference Method Research Articles

Construction of a new time-space two-grid method and its solution for the generalized Burgers’ equation

In this paper, we construct a new time-space two-grid method and apply it to the generalized Burgers’ (GB) equation. We prove that our proposed scheme is solvable, energy-preserving and convergent in the discrete L2-norm. Compared with previous studies on the two-grid method, the new time-space two-grid technique allows for a more flexible selection of the coarse grid in both temporal and spatial dimensions, resulting in greater computational efficiency and a simpler mesh selection. Finally, we compare our scheme with the general difference method (Zhang et al., 2021) and the time two-grid method (Shi and Yang, 2024), revealing that the proposed new time-space two-grid method is more efficient in computational time.

European Funding for Sustainable Transport Systems—Influencing Factor of Regional Economic Development in Romania

Sustainable development is a core concept in regional development. Sustainability is characterized by supporting the building of resilient infrastructure and promoting the sustainable industry. In this context, sustainable transport is particularly important as it represents an opportunity for regional development. This research aims to quantify the impact of investments through structural instruments, specifically EU funds, on promoting a sustainable transport system and eliminating barriers from large-scale transport networks. This study focuses on the impact of these investments on regional economic development in Romania. The analysis used data from all eight development regions of the Romanian economy between 2014 and 2020. Panel data regression models, including the generalized difference method of moments (Dif-GMM) and the system GMM method (Sys GMM), were employed. This study confirms the idea that European structural and investment funds (ESIFs) play a positive role in promoting sustainable transport for regional economic development. Additionally, the quality of regional governance is identified as a key factor in economic development. This study, therefore, reveals a convergence effect between regions. Regions with a lower initial GDP per capita develop quicker compared to regions with a higher initial GDP per capita, indicating a “catch-up” effect. From a policy perspective, these issues can guide decision making and resource allocation.

An Empirical Study on the Relationship between Farmers' Income Increase and Primary Industry in Zhanjiang City

This article briefly introduces the 25 themes of the "No. 1 Document" of the CPC Central Committee. Based on the data of farmers' income, agriculture, forestry, animal husbandry and fishery, and agriculture, animal husbandry and fishery service in Zhanjiang City, Guangdong Province from 1978 to 2021, this paper conducted an empirical study on the relationship between the primary industry and farmers' income in Zhanjiang City. In the empirical analysis, first of all, the stationarity test of the data was carried out, the EG co-integration modeling was carried out, the stepwise regression method was used to eliminate multicollinearity, and the generalized difference method was used to eliminate sequence correlation, and the optimized model was obtained. The results show that the total output value of agriculture and fishery has no significant effect on the increase of farmers' income in Zhanjiang City. Forestry, animal husbandry, agriculture, forestry, animal husbandry and fishery services have a significant impact on farmers' income. The total output value of forestry, animal husbandry and agriculture and the service industry of forestry, animal husbandry and fishery increased by 1%, and the income of farmers increased by 0.429%, 0.240% and 0.619%, respectively. The impact of agriculture and fishery on per capita disposable income in Zhanjiang rural areas is lagging, with a maximum lag of 9 years. Accordingly, the corresponding policy suggestions are put forward.

ROBUST IMPLICIT DIFFERENCE APPROACH FOR THE TIME-FRACTIONAL KURAMOTO–SIVASHINSKY EQUATION WITH THE NON-SMOOTH SOLUTION

This paper formulates the L1 implicit difference scheme (L1IDS) for the time-fractional Kuramoto–Sivashinsky equation (TFKSE) with non-smooth solution. The TFKSE is one of useful descriptions for modeling flame-propagation, viscous flow problems, and reaction–diffusion systems. The proposed method approximates the unknown solution by using two main stages. At the first stage, the L1 method with nonuniform meshes and the general centered difference method is adopted to discretize the Caputo fractional derivative and the spatial derivative, respectively. In the second stage, the fully-discrete L1IDS is established with the help of the Galerkin scheme based on piecewise linear test functions. Meanwhile, an iterative algorithm is adopted to solve the nonlinear systems. Furthermore, the convergence and stability of the proposed method are both demonstrated and confirmed numerically. Finally, three numerical examples highlight the accuracy and efficiency of the proposed strategy.

Biospeckle Activity of Highbush Blueberry Fruits Infested by Spotted Wing Drosophila (Drosophila suzukii Matsumura)

In this study, the potential of the biospeckle phenomenon for detecting fruit infestation by Drosophila suzukii was examined. We tested both graphical and analytical approaches to evaluate biospeckle activity of healthy and infested fruits. As a result of testing the qualitative approach, a generalized difference method proved to be better at identifying infested areas than Fujii’s method. Biospeckle activity of healthy fruits was low and increased with infestation development. It was found that the biospeckle activity index calculated from spatial-temporal speckle correlation of THSP was the best discriminant of healthy fruits and fruits in two different stages of infestation development irrespective of window size and pixel selection strategy adopted to create the THSP. Other numerical indicators of biospeckle activity (inertia moment, absolute value of differences, average differences) distinguished only fruits in later stage of infestation. Regular values of differences turned out to be of no use in detecting infested fruits. We found that to provide a good representation of activity it was necessary to use a strategy aimed at random selection of pixels gathered around the global maximum of biospeckle activity localized on the graphical outcome. The potential of biospeckle analysis for identification of highbush blueberry fruits infested by D. suzukii was confirmed.

The formally second-order BDF ADI difference/compact difference scheme for the nonlocal evolution problem in three-dimensional space

This work formulates two kinds of alternating direction implicit (ADI) schemes for the parabolic-type three-dimensional evolution equation with a weakly singular kernel. The second-order backward differentiation formula (BDF2) and the second-order convolution quadrature (CQ) technique are applied to the discretization of the time derivative and the Riemann-Liouville (R-L) integral, respectively. Then, the fully-discrete BDF2 difference scheme and BDF2 compact difference scheme are constructed via the general centered difference and compact difference method, respectively. Meanwhile, the ADI algorithms are designed reasonably for two schemes to reduce the computational cost. The stability and convergence of two ADI schemes are derived via the energy method. Finally, several numerical examples are provided and tested to validate the theoretical analysis.

Vision sensor and dual MEMS gyroscope integrated system for attitude determination on moving base.

To determine the relative attitude between the objects on a moving base and the base reference system by a MEMS (Micro-Electro-Mechanical Systems) gyroscope, the motion information of the base is redundant, which must be removed from the gyroscope. Our strategy is to add an auxiliary gyroscope attached to the reference system. The master gyroscope is to sense the total motion, and the auxiliary gyroscope is to sense the motion of the moving base. By a generalized difference method, relative attitude in a non-inertial frame can be determined by dual gyroscopes. With the vision sensor suppressing accumulative drift of the MEMS gyroscope, the vision and dual MEMS gyroscope integration system is formed. Coordinate system definitions and spatial transform are executed in order to fuse inertial and visual data from different coordinate systems together. And a nonlinear filter algorithm, Cubature Kalman filter, is used to fuse slow visual data and fast inertial data together. A practical experimental setup is built up and used to validate feasibility and effectiveness of our proposed attitude determination system in the non-inertial frame on the moving base.

A general spectral difference method for calculating the minimum safety distance to avoid the pounding of adjacent structures during earthquakes

With the development of cities, buildings have become increasingly concentrated. Therefore, there are serious potential collision dangers between the adjacent buildings or the building partitions under earthquakes, hence buildings need to set the appropriate safety distance to avoid pounding. However, the calculation of the minimum safety distances (MSD) of the adjacent buildings in Chinese Code is rough and arbitrary, while the traditional response spectrum method is derived based on the assumption of proportional damping and it is difficult to consider the non-proportional damping characteristics for large and complex building systems. Based on the above, this paper proposes a new general spectral difference method(gSDM) to calculate the MSD. First, the pseudo-excitation method is used to derive the relative displacement random expression of adjacent buildings during earthquakes. Then, combined with the response spectrum method, the gSDM formula is established, in addition, the explicit expressions of combination coefficient in the gSDM formula are given. Finally, the examples of two adjacent multi-storey shear-buildings are used to illustrate the accuracy and wide adaptability of the gSDM during the El-Centro earthquake. The results compared and discussed in detail with the classical spectral difference method (cSDM), SRSS method and time history analysis.

Development and preliminary V&V for advanced neutron transport lattice code KYLIN-2

In order to simulate the complex structure fuel assembly, Nuclear Power Institute of China(NPIC) has developed an advanced neutron transport lattice code, named KYLIN-2. The subgroup method is adopted in KYLIN-2 to treat resonance problems. And method of characteristics(MOC) is used to solve neutron transport equation and generalized coarse mesh finite difference(GCMFD) method is used to accelerate the calculation. In order to solve the depletion equation, CRAM method and PPC method are used. The graphical input and display interface are developed to make sure the KYLIN-2 code can be used easily by engineers. The numerical results show that the KYLIN-2 can calculate the PWR assembly accurately.

Test of Trend Types in Financial and Economic Time Series with Unit Root

Positive analysis concerning unit root usually assumes that deterministic trend exists and is linear. The standard ADF or PP unit root test in various statistics software packages specifies linear trend, too. But if the trend is not linear, test power will worsen acutely, resulting in a false test. Therefore it is imperative to test whether or not the trend is linear before using linear specification. This paper conducts researches in detail on the test of deterministic trend in unit root test, proposes test methods to determine whether the time series has trend or not. If trend exists, the regression coefficient and equivalent mean test methods are proposed to test a linear or non-linear trend. Meanwhile this paper discusses the impact on the test result for series correlations and suggests generalized difference methods as well as sampling child-series method to remove the impact of residual correlation. Simulation results are offered too.

Optimal approximation to a class of nonlinear evolution equations

In this paper, we have developed and discussed a generalized difference method (GDM) for the approximation of a class of nonlinear evolution equations with integral items in one space dimension. The main idea of the paper is to introduce a generalized Volterra projection operator and we design a generalized difference formulation with a new approximate value of initial function, which is of special importance for the error analysis. Then some theories of the generalized Volterra projection operator are analyzed. Finally, we deduce the optimal error estimates. Moreover, some conclusions are obtained and our further research work is proposed.

Numerical methods for a fluid mixture model

SUMMARY In this paper, we firstly apply generalized difference methods to solve a fluid mixture model. The model is usually used to describe the tissue deformations and contains a nonlinear hyperbolic equation and an elliptic equation. Most people have used finite difference methods for solving the elliptic equation and other schemes for solving the hyperbolic equation. It is well known that the accuracy of traditional finite difference method is not high. This may be a serious disadvantage in the fluid mixture model, which describes cell movements and tissue deformations. The numerical methods we propose to improve accuracy are based on generalized Galerkin methods and dual decomposition. By choosing suitable trial function space and test function space, our generalized upwind difference schemes exhibit second-order convergence in space for smooth problems and can eliminate numerical oscillations for discontinuous problems. Some numerical results are presented to demonstrate the advantages of our methods. Copyright © 2012 John Wiley & Sons, Ltd.

Generalized difference simulation for coupled transport models of unsaturated soil water flow and solute

The generalized difference methods (GDM) have been successfully used in numerical simulations of unsaturated soil water flow problems. The object of this paper is to extend applications of the GDM, that is, apply GDM to numerically solve one-dimensional coupled transport models of unsaturated soil water flow and solute. The semi-discrete and fully discrete generalized difference schemes are established, and the detailed algorithm of numerical approximation schemes is provided. Moreover, some numerical examples illustrate that the generalized difference methods are feasible and efficient to find the numerical solutions of one-dimensional coupled transport models in this paper. Finally, satisfactory results and some other significant and valuable conclusions are obtained from studies.

An upwind finite volume element method based on quadrilateral meshes for nonlinear convection‐diffusion problems

AbstractConsidering an upwind finite volume element method based on convex quadrilateral meshes for computing nonlinear convection‐diffusion problems, some techniques, such as calculus of variations, commutating operator, and the theory of prior error estimates and techniques, are adopted. Discrete maximum principle and optimal‐order error estimates in H1 norm for fully discrete method are derived to determine the errors in the approximate solution. Thus, the well‐known problem [(Li et al., Generalized difference methods for differential equations: numerical analysis of finite volume methods, Marcel Dekker, New York, 2000), p 365.] has been solved. Some numerical experiments show that the method is a very effective engineering computing method for avoiding numerical dispersion and nonphysical oscillations. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2009

Numerical simulation and analysis of generalized difference method on triangular networks for electrical impedance tomography

Electrical impedance tomography is an inverse problem of elliptic differential equations. Numerical methods based on combining generalized difference method and Levenberg–Marquardt iteration on a planar domain are proposed. Positive semi-definiteness and existence of solution of the generalized difference scheme are proved. Element geometry matrix is introduced to shortcut calculation and standardize computer program. A series of numerical experiments verify the reliability of its mathematical model and the feasibility of the algorithm. A class of electrical current patterns is proposed to minimize the number of direct problems to be solved in each iteration. These methods have been applied successfully in practical simulation of electrical impedance tomography.

From formal solutions to computational methods avoiding passages to the limit

Before the advent of digital computers, the so-called formal solutions were the only available solutions to differential equations. Formal solutions can be closed solutions, or solutions involving infinite algorithms. The latter involve an infinite number of algebraic operations. Truncation becomes thus necessary, and the concepts of truncation error and convergence become vital. Once digital computers became available, other kinds of computational methods could be used and it became convenient to distinguish between computational methods like finite difference and finite element methods, in which numerical analysis starts before integration, and those like classical integral methods and boundary element methods, in which numerical analysis starts after integration. The classical finite difference method, in which a mesh is required, is a particular case of the generalised difference methods, characterised by a local interpolation around each node together with the collocation technique. The generalised difference method may be regarded as a modality of the meshless techniques. The finite element method differs of the finite difference method in that the approximate solution is generated respectively by variational and by collocation techniques. Hybrid and block elements are dual generalisations of the finite element method in which compatibility and equilibrium are respectively allowed within each element. Also in what concerns the methods in which numerical analysis starts after integration, bold steps have been given toward their generalisation, like those avoiding passages to the limit.

An approach to an objective background subtraction for elemental mapping with core-edges down to 50 eV: description, evaluation and application

To image the distribution of a specific element in a specimen with an energy filtering TEM, the element-unspecific background under the core-edge has to be subtracted. The most commonly used procedure is the three-window power-law method leading to considerable systematic errors for low-energy core-edges. Here a new method is described which can be considered as a generalized difference method. Characteristic examples for element detection in biological specimens using this method are shown. The background under the core-edge can be described by one or two pre-edge windows as a polynome of third order. This function can be deduced from specimen areas that are not known to contain the element or from a second specimen used as a standard. Control experiments showed that background subtraction for non-overlapping core-edges in the low-loss region (50–200 eV) needs two pre-edge images, whereas at higher-energy losses (>300 eV) only one pre-edge image is necessary. With the method described, objective elemental mapping becomes possible even for edges at 50–100 eV. This was proven for the M 2,3-edge of iron at 60 eV. The detection of phosphorus was possible with a signal-to-noise ratio five times higher than when using the three-window method. Preliminary data showed that it should be possible to detect calcium with only one image before the edge.

Numerical computation of a damped slewing beam with tip mass

This paper describes an analytical model for a beam system, based on a modified Timoshenko theory, where the beam is pinned to a hub driven by an actuator at one end and is subject to a heavy load at the other end. A new efficient computational algorithm is then proposed for solving the higher-order non-canonical partial differential equation model, which is developed based on the generalized difference method. This allows a suitable selection of different trial and test spaces, so as to improve the computational efficiency while preserving the high convergence rate of the standard finite element method. With the trial space of cubic Hermite finite elements and the test space of piecewise linear functions, the computational scheme reduces to a semi-discretized or even fully discretized computational algorithm. A numerical simulation result is included to visualize the theoretical modelling and computational results. Copyright © 1999 John Wiley & Sons, Ltd.

Numerical computation of a damped slewing beam with tip mass

This paper describes an analytical model for a beam system, based on a modified Timoshenko theory, where the beam is pinned to a hub driven by an actuator at one end and is subject to a heavy load at the other end. A new efficient computational algorithm is then proposed for solving the higher-order non-canonical partial differential equation model, which is developed based on the generalized difference method. This allows a suitable selection of different trial and test spaces, so as to improve the computational efficiency while preserving the high convergence rate of the standard finite element method. With the trial space of cubic Hermite finite elements and the test space of piecewise linear functions, the computational scheme reduces to a semi-discretized or even fully discretized computational algorithm. A numerical simulation result is included to visualize the theoretical modelling and computational results.

Second order generalized difference methods for one dimensional parabolic equations

In this paper, the second order semi-discrete and full discrete generalized difference schemes for one dimensional parabolic equations are constructed and the optimal orderH1, L2 error estimates and superconvergence results inH1 are obtained. The results in this paper perfect the theory of generalized difference methods.