Difference Approximation Research Articles

An Improved Scheme for the Finite Difference Approximation of the Advective Term in the Heat or Solute Transport Equations

AbstractTransport equations are widely used to describe the evolution of scalar quantities subject to advection, dispersion and, possibly, reactions. Numerical methods are required to solve these equations in applications, adopting either the advective or conservative formulations. Conservative formulations are usually preferred in practice because they conserve mass. Advective formulations do not, but have received more mathematical attention and are required for Lagrangian solution methods. To obtain an advective formulation that conserves mass, we subtract the discretized fluid flow equation, multiplied by concentration, from the conservative form of the transport equation. The resulting scheme not only conserves mass, but is also elegant in that it can be interpreted as averaging the advective term at cell interfaces, instead of approximating it at cell centers as in traditional centered schemes. The two schemes are identical when fluid velocity is constant, and both have second-order convergence, but the truncation errors are slightly different. We argue that the error terms appearing in the proposed scheme actually imply an improved representation of subgrid spreading/contraction and acceleration/deceleration caused by variable velocity. We compare the proposed and traditional schemes on several problems with variable velocity caused by recharge, discharge or evaporation, including two newly developed analytical solutions. The proposed method yields results that are slightly, but consistently, better than the traditional scheme, while always conserving mass (i.e., mass at the end equals mass at the beginning plus inputs minus outputs), which the traditional centered finite differences scheme does not. We conclude that this scheme should be preferred in finite difference solutions of transport.

Stability and computational analysis of Influenza-A epidemic model through double time delay

Delay factors demonstration has a significant role in controlling a strain of infectious disease instead of a pharmaceutical strategy. According to the World Health Organization (WHO), 3–5 million cases are reported annually and approximately 290,000 to 650,000 respiratory deaths annually. So, in the present study, we develop a delayed mathematical model based on delay differential equations (DDEs) for the influenza epidemic using a deterministic approach by introducing double delay parameters. The four distinct sub-populations are considered susceptible, exposed, infected, and recovered. For the rigorous analysis, the fundamental properties of the model like positivity, boundedness, existence, and uniqueness, were studied. The influenza-free equilibrium (IFE) and influenza-existing equilibrium (IEE), are the two nonnegative equilibrium points that the model demonstrates. Both locally and globally, the asymptotic stability of the equilibrium points of the model is established and shown under specific situations of reproduction number. Additionally, investigated the model's parameter sensitivity and determined the relative sensitivity of each parameter. Both standard and nonstandard methods—such as Euler, Runge-Kutta, and nonstandard finite difference with a delayed sense—are presented to make computational analysis support a dynamical analysis and the best visualization of results. The stability of the non-standard finite difference scheme is thoroughly analyzed around the steady states of the model. Additionally, the results show that the nonstandard finite difference approximation is an efficient, cost-effective method, independent of time step size, to solve such highly nonlinear and complex real-world problems.

High-Accuracy Positivity-Preserving Finite Difference Approximations of the Chemotaxis Model for Tumor Invasion.

Numerical simulation of the complex evolution process for tumor invasion plays an extremely important role in-depth exploring the bio-taxis phenomena of tumor growth and metastasis. In view of the fact that low-accuracy numerical methods often have large errors and low resolution, very refined grids have to be used if we want to get high-resolution simulating results, which leads to a great deal of computational cost. In this paper, we are committed to developing a class of high-accuracy positivity-preserving finite difference methods to solve the chemotaxis model for tumor invasion. First, two unconditionally stable implicit compact difference schemes for solving the model are proposed; second, the local truncation errors of the new schemes are analyzed, which show that they have second-order accuracy in time and fourth-order accuracy in space; third, based on the proposed schemes, the high-accuracy numerical integration idea of binary functions is employed to structure a linear compact weighting formula that guarantees fourth-order accuracy and nonnegative, and then a positivity-preserving and time-marching algorithm is established; and finally, the accuracy, stability, and positivity-preserving of the proposed methods are verified by several numerical experiments, and the evolution phenomena of tumor invasion over time are numerically simulated and analyzed.

A K-Space-Based Temporal Compensating Scheme for a First-Order Viscoacoustic Wave Equation with Fractional Laplace Operators

Inherent constant Q attenuation can be described using fractional Laplacian operators. Typically, the fractional Laplacian viscoacoustic or viscoelastic wave equations are addressed utilizing the staggered-grid pseudo-spectral (SGPS) method. However, this approach results in time numerical dispersion errors due to the low-order finite difference approximation. In order to address these time-stepping errors, a k-space-based temporal compensating scheme is established to solve the first-order viscoacoustic wave equation. This scheme offers the advantage of being nearly free from grid dispersion for homogeneous media and enhances simulation stability. Numerical examples indicate that the proposed k-space scheme aligns well with analytical solutions for homogeneous media. Additionally, this method demonstrates excellent applicability for complex models and is more efficient due to its ability to adopt a larger time step compared with conventional staggered-grid pseudo-spectral methods.

Discrete time stability of augmented Lagrangian formalism based underactuated inverse dynamics control method

Stability problems of robotic systems arise sometimes suddenly, seemingly for no reason. The digital time sampling is often the main cause of these instabilities. Discrete models, which are capable of the prediction of stability, are available for low-degree-of-freedom template models of linear position and force control. However, for the inverse dynamics control of underactuated systems, the literature has a lack of generally applicable results related to the effect of time discretization. Several control approaches are available in the literature out of which a widely used one, the augmented Lagrangian formalism and its stability properties are analysed in this work. Theoretical stability properties are obtained for a generally usable, linear, underactuated, two-degree-of-freedom constrained template model. The actuator dynamics, the finite difference approximation of the feedback velocity and the filtering of the feedback data are considered in the model. These phenomena strongly affects the stability properties. The theoretically obtained stability maps are experimentally validated on an underactuated crane-like indoor robot. The position and orientation accuracy of the robot were assessed: the absolute position error was below 30 mm and the orientation error was below 3°.

Semi-analytical solutions for steady-state bending-bending coupled forced vibrations of a rotating wind turbine blade by means of Green's functions

Wind power has received significant attention in recent years as a renewable energy source. Wind turbine blades, characterized by their long lengths, are prone to damage as a result of aeroelastic instability. This paper aims to derive semi-analytical solutions for steady-state forced vibrations of rotating wind turbine blades, considering the coupling of bending, bending, and unsteady aerodynamic loads. The novelty of this work lies in the use of the Green's function method to solve differential dynamic equations with variable coefficients, which are induced by aerodynamic forces. To model a wind turbine blade, the Euler-Bernoulli beam model is employed, and Greenberg's expressions are taken into consideration. Governing equations for coupled vibrations of the blade are then determined. In order to solve the governing equations, displacement solutions are decomposed into quasi-static displacements and dynamic displacements. The Laplace transformation and Green's function methods are utilized to obtain fundamental solutions of the governing equations. Subsequently, employing the principle of superposition, Fredholm integral equations for steady-state forced vibration of a rotating wind turbine blade are derived. To spatially discretize Fredholm integral equations, compound trapezoid formulae and central finite difference approximations are employed. This discretization process leads to the formation of a system of algebraic equations. Semi-analytical solutions for coupled forced vibrations of the rotating wind turbine blade are obtained by solving the algebraic equations. In the numerical solution part, validation of proposed solutions are verified by comparing them with numerical and finite element solutions in the literature. Influences of some important physical parameters, such as the rotating velocity, the setting angle, the cone angle, the inflow ratio, and damping, on vibration responses of blades are discussed.

A Spatial Reconstruction Method of Ionospheric foF2 Based on High Accuracy Surface Modeling Theory

The ionospheric F2 critical frequency (foF2) is one of the most crucial application parameters in high-frequency communication, detection, and electronic warfare. To improve the accuracy of spatial reconstruction of the ionospheric foF2, we propose a high-accuracy surface (HAS) modeling method. This method converts difficult-to-solve differential equations into more manageable algebraic equations using direct difference approximation, significantly reducing algorithm complexity and computational load while exhibiting excellent convergence properties. We used seven stations in Brisbane, Canberra, Darwin, Hobart, Learmonth, Perth, and Townsville, with one station as a validation station and six as training stations (e.g., Brisbane as a validation station and the other stations—Canberra, Darwin, Hobart, Learmonth, Perth, and Townsville—as training stations). The training stations and the HAS method were used to train and reconstruct the validation stations at different solar activity periods, seasons, and local times. The predicted values of the validation stations were compared with the measured values, and the proposed method was analyzed and validated. The reconstruction results show the following. (1) The relative root mean square errors (RRMSEs) of HAS method prediction in different solar activity epochs were 13.67%, 7.74%, and 9.19%, respectively, which are 13.57%, 7.41%, and 6.41% higher than the prediction accuracy of the Kriging method, respectively. (2) In the four seasons, the RRMSEs of the HAS method prediction are 9.27%, 13.1%, 8.81%, and 8.09%, respectively, which are 10.83%, 11.73%, 4.25%, and 12.00% higher than the prediction accuracy of the Kriging method. (c) During the daytime and nighttime, the RRMSEs of HAS method prediction were 9.23% and 11.17%, which were 5.92% and 11.99% higher than the prediction accuracy of the Kriging method, respectively. (d) Under the validation dataset, the average predictive RRMSE of the HAS method was 10.29%, and the average predictive RRMSE of the IRI prediction model was 12.35%, with a 2.06% improvement in the predictive accuracy of the HAS method. In general, the prediction effect of the HAS method was better than that of the Kriging method, thus verifying the effectiveness and reliability of the proposed method. In summary, the proposed reconstruction method is of great significance for improving usable frequency prediction and enhancing communication performance.

Machine-learning aided calibration and analysis of porous media CFD models used for rotating packed beds

The proposed research is an attempt to advance the state-of-the-art of the numerical modelling of RPB by combining Computational Fluid Dynamics and Machine Learning approaches. The latter creates an accurate framework that should help quantify the potential of Rotating Packed Beds (RPB) technology to intensify conventional CO2 capture processes. To this end, a direct sensitivity analysis is detailed to supplement a machine-learning (ML) algorithm built for calibrating resistance coefficients needed for porous media modelling. The algorithm is used to improve CFD predictions of dry pressure drop in rotating packed beds (RPBs) for a wide range of operating conditions. The sensitivity derivatives with respect to the packing resistance coefficients are demonstrated for the first time in RPBs, which is not available in the current CFD open source and commercial codes. In this regard, sensitivity differential equations are derived from three-dimensional Navier-Stokes equations for porous media in a rotating reference frame. These sensitivity equations are discretized using a finite volume scheme and solved to obtain the sensitivity pressure drop differences at the packing edges. The results are validated against the predictions of the analytical sensitivity analysis and the finite difference approximation. After, the Newton – Gauss method that employs the sensitivity pressure drop derivatives, is used to minimize the error (cost function) between the pressure drop obtained from CFD simulations and the available experimental data. This is achieved by tuning the packing resistance coefficients to the RBPs' operating conditions (gas flowrate and rotating speed) and correlate them using an artificial neural network (ANN). The results of the proposed approach show a significant improvement in porous media-based CFD predictions of RPBs' pressure drop across a wide range of operating conditions and this over conventional porous media-based CFD models. This is necessary for CFD models to be reliably used as a tool that can efficiently improve existing RPBs' designs and/or participate in RPBs' design innovation.

Difference Approximation for 2D Time-Fractional Integro-Differential Equation with Given Initial and Boundary Conditions

Difference Approximation for 2D Time-Fractional Integro-Differential Equation with Given Initial and Boundary Conditions

Approximation of one and two dimensional nonlinear generalized Benjamin-Bona-Mahony Burgers' equation with local fractional derivative

This study presents, a numerical method for the solutions of the generalized nonlinear Benjamin-Bona-Mahony-Burgers' equation, with variable order local time fractional derivative. This derivative is expressed as a product of two functions, the usual integer order time derivative, and a function of time having a fractional exponent. Then, forward difference approximation is used for time derivative. The unknown solution of the differential problem and corresponding derivatives are estimated using Haar wavelet approximations (HWA). The collocation procedure is then implemented in HWA, to transform the given model to the system of linear algebraic equations for the determination of unknown constant coefficient of the Haar wavelet series, which update the derivatives and the numerical solutions. The sufficient condition is established for the stability of the proposed technique, and then verified computationally. To check the performance of the scheme, few illustrative examples in one and two dimensions along with l∞ and l2 error norms are also given. Besides this, the computational convergence rate is calculated for both type equations. Additionally, computed solutions are compared with available results in literature. Simulations and graphical data discloses, that suggested scheme works well for such complex problems.

Numerical Finite-Difference Approximations of a Coupled Reaction-Diffusion System with Gradient Terms

This study focuses on the derivation of explicit and implicit finite difference formulas.The objective of this study is to derive an estimation of the blow-up time for a coupled reaction-diffusion system incorporating gradient terms, employing numerical finite difference approximations. Furthermore, an examination is conducted on the consistency, stability, and convergence of the proposed schemes. Additionally, the study presents two numerical experiments. In each instance, the numerical blow-up time is calculated benefit the suggested methodologies, employing varying space steps and non-fixed time-stepping. The numerical findings obtained demonstrate that the blow-up time sequence exhibits convergence as the space step decreases. Moreover, thenumerical orders of convergence for the blow-up time goes well with the theoretical orders observed in the numerical solutions.

Multiscale model reduction for the time fractional thermoporoelasticity problem in fractured and heterogeneous media

In this paper, we consider the time fractional thermoporoelasticity problem in fractured and heterogeneous media. The mathematical model with a time memory formalism is described by a coupled system of equations for pressure, temperature and displacements. We use an implicit finite difference approximation for temporal discretization. We present a fine grid approximation based on the finite element method and Discrete Fracture Model (DFM) for two-dimensional model problems. Further, we use the Generalized Multiscale Finite Element Method (GMsFEM) for coarse grid approximation. The primary concept behind the proposed method is to streamline the complexity inherent in the thermoporoelasticity problem. Given that our model equation incorporates multiple fractional powers, leading to multiple unknowns with memory effects, we aim to address this intricacy by optimizing the problem’s dimensionality. As a result, the solution is sought on a coarse grid, a strategic choice that not only simplifies the computational cost but also contributes to significant time savings. We present numerical results for the two-dimensional model problems in heterogeneous fractured porous media. We derive relative errors between the reference fine grid solution and the multiscale solution for different numbers of multiscale basis functions. The results confirm that the proposed method is able to achieve good accuracy with a few degrees of freedoms on the coarse grid.

Numerical integration method for two-parameter singularly perturbed time delay parabolic problem

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.

Structural Design and Parameter Optimization of Magnetic Gradient Tensor Measurement System.

Magnetic anomaly detection (MAD) technology based on the magnetic gradient tensor (MGT) has broad application prospects in fields such as unexploded ordnance detection and mineral exploration. The difference approximation method currently employed in the MGT measurement system introduces measurement errors. Designing reasonable geometric structures and configuring optimal structural parameters can effectively reduce measurement errors. Based on research into differential MGT measurement, this paper proposes three simplified planar MGT measurement structures and provides the differential measurement matrix. The factors that affect the design of the baseline distance of the MGT measurement system are also theoretically analyzed. Then, using the magnetic dipole model, the error analysis of the MGT measurement structures is carried out. The results demonstrate that the planar cross-shaped structure is optimal, with the smallest measurement error, only 3.15 × 10-10 T/m. Furthermore, employing the control variable method, the impact of sensor resolution constraints, noise level, target magnetic moment, and detection distance on the design of the optimal baseline distance of the MGT measurement system is simulated and verified. The results indicate that the smaller the target magnetic moment, the farther the detection distance, the lower the magnetometer resolution, the greater the noise, and the greater the baseline distance required. These conclusions provide reference and guidance for the construction of the MGT measurement system based on triaxial magnetometers.

Numerical Solution to Poisson’s Equation for Estimating Electrostatic Properties Resulting from an Axially Symmetric Gaussian Charge Density Distribution: Charge in Free Space

Poisson’s equation frequently emerges in many fields, yet its exact solution is rarely feasible, making the numerical approach notably valuable. This study aims to provide a tutorial-level guide to numerically solving Poisson’s equation, focusing on estimating the electrostatic field and potential resulting from an axially symmetric Gaussian charge distribution. The Finite Difference Method is utilized to discretize the desired spatial domain into a grid of points and approximate the derivatives using finite difference approximations. The resulting system of linear equations is then tackled using the Successive Over-Relaxation technique. Our results suggest that the potential obtained from the direct integration of the distance-weighted charge density is a reasonable choice for Dirichlet boundary conditions. We examine a scenario involving a charge in free space; the numerical electrostatic potential is estimated to be within a tolerable error range compared to the exact solution.

Accurate Computational Approach for Singularly Perturbed Burger-Huxley Equations

The main purpose of this work is to present an accurate computational approach for solving the singularly perturbed Burger-Huxley equations. The quasilinearization technique linearizes the nonlinear term of the differential equation. The finite difference approximation is formulated to approximate the derivatives in the differential equations and then accelerate its rate of convergence to improve the accuracy of the solution. Numerical experiments were conducted to sustain the theoretical results and to show that the presented method produces a more correct solution than some surviving methods in the literature.

Numerical Resolution of Differential Equations Using the Finite Difference Method in the Real and Complex Domain

The paper expands the finite difference method to the complex plane, and thus obtains an improvement in the resolution of differential equations with an increase in numerical precision and a generalization in the mathematical modeling of problems. The article begins with a selection of the best techniques for obtaining finite difference coefficients for approximating derivatives in the real domain. Then, the calculation is expanded to the complex domain. The research expands forward, backward, and central difference approximations of the real case by a quadrant approximation in the complex plane, which facilitates the use in boundary conditions of differential equations. The article shows many real and complex finite difference equations with their respective order of error, intended to serve as a basis and reference, which have been tested in practical examples of solving differential equations used in engineering. Finally, a comparison is made between the real and complex techniques of finite difference methods applied in the Theory of Elasticity. As a surprising result, the article shows that the finite difference method has great advantages in numerical precision, diversity of formulas, and modeling generalities in the complex domain when compared to the real domain.

An adaptive mesh generation and higher-order difference approximation for the system of singularly perturbed reaction–diffusion problems

The paper presents a hybrid difference method to solve a coupled system of singularly perturbed reaction–diffusion problems over an adaptive mesh. The solution to the problem manifests a distinctive multi-scale nature, characterized by localized narrow regions where the solution undergoes exponential changes. The mesh-generating procedure relies upon equidistributing a non-negative monitor function. The method utilizes a suitable combination of the cubic and exponential spline difference schemes over a layer adaptive mesh. The proposed method is unconditionally stable and uniformly convergent. The convergence obtained is optimal, being free from logarithmic factors. Moreover, it is free from directional bias. Numerical experiments have been performed and presented for two model problems. The results obtained are in agreement with the theoretical estimates.

Theoretical studies for the detailed electronic structure of organotin(IV) derivatives of 4-fluoroanthranilic acid, X-ray structure of n-dibutyltin bis(4-fluoroanthranilate)

The present study describes the detailed DFT-based electronic structure calculations without any symmetry constraints being performed on di- and triorganotin derivatives of 4-fluoroanthranilic acid (AFA), viz. Me2SnL2 (1), n-Bu2SnL2 (2), Me3SnL (3), and Ph3SnL (4) (where L= monoanion of AFA). The structural and atomic charge analysis have confirmed the previously reported our experimental results, i.e. bicapped tetrahedral and distorted tetrahedral geometry for di- and triorganotin derivatives, respectively, and all the complexes 1–4 contain a positively charged central tin atom surrounded by a negatively charged system. The single crystal X-ray structure of complex 2 (n-Bu2SnL2) also suggests that AFA acts as a monoanion and bidentate ligand resulting a bicapped tetrahedral environment around tin. Various population analysis such as Mulliken (MPA), Hirshfeld (HPA), and natural population analysis (NPA) have been employed to calculate the charges at all the atoms. A finite difference approximation method has been used to explain the charge distribution within the studied complexes 1–4 based on the conceptual global and local reactivity DFT-based descriptions, frontier molecular-orbital analysis (FMOA), and molecular electrostatic potential maps (MEP). Further, the conceptual DFT-based global reactivity descriptors and frontier molecular orbital analysis show that the interactions between CT-DNA and complexes 1–4 may be groove binding or electrostatic. The integral equation formalism-polarizable continuum model (IEF-PCM) has been employed to calculate the UV–visible spectra of 1–4 in the solvent field, whereas the same in the gas phase has been obtained with the help of time-dependent DFT (TD-DFT) at the same level of theory. The intramolecular charge distribution in 1–4 has been investigated with the help of natural bond orbital (NBO) analysis. The topological and energetic parameters at the selected bond critical points in the coordination sphere of complexes 1–4 have been calculated using atoms-in-molecules (AIM) theory. The analysis of different kinds of non-covalent interactions (NCI) in complexes 1–4 has also been investigated.

Estimation of Multiple Contact Conductances in a Silicon-Indium-Silicon Stack

Abstract This report documents evaluation of simultaneous estimation of multiple interfacial heat transfer coefficients (HTCs) using transient measurements from an experiment designed for steady-state operation. The design of a mirror system for directing X-rays under cryogenic conditions requires knowledge of the interfacial HTC (contact conductance) between silicon and indium. An experimental apparatus was constructed to measure temperatures in a stack of five 7.62-mm thick pucks of silicon separated by 0.1-mm thick sheets of indium, which is operated under cryogenic temperatures in vacuum. Multiple pucks and interfaces are incorporated into the apparatus to allow evaluation of HTCs for surfaces of different roughness from a single experiment. Analysis of the sensitivity of each of the measured temperatures to each of the unknown HTCs reveals lack of linear independence of these sensitivities and suggests the recovery of the HTCs will be challenging. Artificially noised “data” were created from two different computational models by solving for temperatures and adding random Gaussian noise with a specified standard deviation. These data are subsequently analyzed using two different iterative parameter estimation methods: a Levenberg scheme and a Tikhonov iterative scheme. The required sensitivity matrix is computed using forward finite difference approximations. The results for the heat transfer coefficients for this model problem suggest that coefficients cannot be estimated independently, but the ratios relative to one of the unknowns can be recovered.