Classical Finite Difference Method Research Articles

Deep learning method for finding eigenpairs in Sturm-Liouville eigenvalue problems

Solving the eigenvalue problem for differential equations in inhomogeneous media poses a significant challenge across diverse scientific fields. While classical finite difference methods and finite element methods have produced numerous outcomes, they heavily rely on discretizing the computational domain, which can introduce complexities and limitations. In this study, we present an unsupervised neural network approach tailored for finding eigenpairs in Sturm-Liouville eigenvalue problems within inhomogeneous media. Our method introduces eigenvalues as trainable parameters, crafts a novel cost function, incorporates an adaptive hyper-parameter tuning strategy, and sequentially trains the eigenpairs. The simplicity, accuracy, and interpretability of our approach significantly expand its applicability across various domains. The method we present in this paper can easily tackle boundary value conditions with derivatives, resulting in orthogonal eigenfunctions. This is a very important advantage of deep learning methods that has not yet been noticed. Quantitative estimation of eigenpairs is given for the Sturm-Liouville eigenvalue problems. Furthermore, we extend the proposed methodology to tackle two-dimensional cases, periodic scenarios, demonstrating its versatility and broad potential. For more information see https://ejde.math.txstate.edu/Volumes/2024/53/abstr.html

Quantum algorithms for nonlinear partial differential equations

Computations of nonlinear partial differential equations (PDEs) are some of the most important problems in science and engineering, but come with tremendous costs. Quantum algorithms have the potential to speed up these computations, but current methods are bound to weak or polynomial nonlinearity, short time of validity, or display no quantum advantage otherwise. We construct new quantum algorithms–based on level set methods – for Hamilton-Jacobi and scalar hyperbolic PDEs that generate algorithms with advantages with respect to critical numerical parameters, even for computing the physical observables. These algorithms hold for arbitrary nonlinearity for the equations under study and are valid globally in time. In addition, we also introduce a new quantum encoding, called level set encoding, that allows the efficient computation of physical observables. These PDEs are important for many applications like geometric optics, semi-classical limit of the Schrödinger equations, and high-frequency limit of symmetric linear hyperbolic systems (for example the elastic wave and Maxwell equations). It can display up to exponential advantage –depending on the initial conditions and domain of the relevant observables– in both the dimension of the PDE and the error in computing its observables with respect to classical finite difference methods for these PDEs. Compared to classical finite difference methods, our algorithm also has an advantage with respect to the number of initial data, which has potential important applications in Monte Carlo simulations and uncertainty quantification. For more general PDEs, while an advantage with respect to a very large number of initial data is possible, this is mostly unrealistic for problems of practical interest, and new methods are required.

An Efficient Replacement for Schrödinger's Partial Differential Equation

The classical finite difference method for solving time-dependent partial differential equations has become quite tedious and requires the use of off-the-shelf algorithms such as those in MATLAB. The treatment by finite difference method then solution of the n resulting first order algebraic equations is quite difficult since the underlying matrix is singular. We propose an alternative revolutionary statistical technique using Bmatrix chains, which are a product of the Cairo techniques. This technique is valid for both classical macroscopic physics such as the heat diffusion equation and modern microscopic quantum mechanics such as Schrödinger's PDE. He completely neglects partial differential equations as if they never existed. The numerical results of the proposed numerical statistical method show stability, accuracy and superiority over conventional PDE methods.

A Novel Method for Predicting the Behavior of a Sucker Rod Pumping Unit Based on the Polished Rod Velocity

Fault dynamometer cards are the basis of the diagnosis technique for sucker rod pumping systems. Predicting fault cards with a pumping condition model is an economical and effective method. The usual model is described by a mixed function of the pump displacement and pump load, and it is difficult to use in the prediction method based on the analytical solution of the sucker rod string wave equation. In this paper, a normal pumping condition model described by a function of polished rod velocity is proposed. For the analytical solution of the sucker rod wave equation, an iterative prediction algorithm with pumping condition models is proposed, its convergence is analyzed, and then it is validated by classical finite difference method simulated cards and measured surface dynamometer cards. The results show that the proposed algorithm is accurate. The algorithm has a maximum relative error of 0.10% for the classical method simulated card area and 1.45% for the measured card area. The research of this paper provides an effective scheme for the design, prediction, and fault diagnosis of a sucker rod pumping system with an analytical solution.

Analysis of MHD and Heat Transfer Characteristics of Thermally Radiative Upper-Convected Maxwell Fluid Flow between Moving Plates: Semi-Analytical and Numerical Solution

The current paper studies the MHD and heat transfer characteristics of radiative upper-convected Maxwell fluid flow between two plates approaching or receding from each other with injection at the fixed lower porous plate. The governing momentum and energy equations are reduced into non-linear ordinary differential equations employing similarity transformations. With the help of the Homotopy Perturbation Method (HPM), an approximate analytic solution is obtained. This work aims to determine the effects of Reynolds number (R), Deborah number (De), Radiation parameter (Rd), Magnetic parameter (M), and Prandtl number (Pr) on the velocity and temperature profiles. It is observed that the Deborah number has a direct impact on the velocity profile when there is a squeezed flow. It is also observed that the magnetic parameter shows an indirect impact on the temporal distribution for both the upper plate moving away and towards the lower. The variations in the significant physical parameters on the coefficient of skin friction and heat transfer rates are also calculated. The results are then compared with the classical finite difference method and are in excellent agreement. It is found that larger the magnetic parameter, the dominance of viscous forces retards the velocity in the core region, and the increase in radiation parameter suppresses the heat transfer rates. This study is helpful in industrial applications, specifically in polymer processing.

Direct implementation of discontinuous-grid finite-difference method using multiple point sources method and dynamic wavefield injection

SUMMARY The classical finite-difference (FD) method stimulates wave propagation in uniform grids. In models with considerable velocity variations, the computational efficiency is compromised by oversampling in time and space. Although adopting a discontinuous grid in different wave-speed regions can improve computational efficiency, such a technique is typically hindered by low accuracy in transition zones (i.e. in the vicinity of the discontinuous-grid interface) and is typically unstable in the long term. We propose a direct implementation of the discontinuous-grid FD method by performing two simulations simultaneously: one on a coarse grid and the other on a fine grid. The proposed method applies a dynamic injection strategy to manage wavefield communication between coarse and fine submodels. Compared with previous discontinuous-grid FD methods, where the number of layers required for wavefield communication is one-half that the order of the FD scheme, the proposed method only requires one single layer, which significantly reduces the communication overhead and suppresses wavefield errors. Numerical experiments show that the results yielded by our method are consistent with the reference solutions yielded by the uniform-grid FD method via a fine grid. Furthermore, our method does not encounter numerical instability in long-term simulations. Therefore, the proposed discontinuous-grid FD method can accelerate numerical simulations while retaining stable numerical accuracy, even for long-term simulations.

A Family of Transformed Difference Schemes for Nonlinear Time-Fractional Equations

In this paper, we present a class of finite difference methods for numerically solving fractional differential equations. Such numerical schemes are developed based on the change in variable and piecewise interpolations. Error analysis of the numerical schemes is obtained by using a Grönwall-type inequality. Numerical examples are given to confirm the theoretical results.

A numerical method based on barycentric interpolation collocation for nonlinear convection-diffusion optimal control problems

<abstract><p>This paper describes a study of the barycentric interpolation collocation method for the optimal control problem governed by a nonlinear convection-diffusion equation. Using Lagrangian multipliers, we obtain the continuous optimality system which is composed of state equations, adjoint equations and optimality conditions. Then, barycentric interpolation collocation methods are applied to discretize the optimality system and the nonlinear term is treated by Newton's iteration. Furthermore, the corresponding consistency analyses of discrete schemes are demonstrated. Finally, the validity of the proposed schemes is demonstrated through several numerical experiments. Compared with the classical finite difference method, collocation schemes can yield the higher-order accurate solutions with fewer nodes.</p></abstract>

Electroosmotic slip flow of Eyring fluid under high Zeta potential in a circular microchannel

• Under the combined action of applied electric field force and pressure, the electroosmotic flow of Eyring fluid with Navier slip boundary condition at high Zeta potential. • Apply the Chebyshev spectral method to solve the electric potential distribution, and then apply the numerical method (compound trapezoidal formula) to solve the fluid velocity distribution. • The influences of relevant physical parameters on the velocity are analyzed for the Eyring fluid. In this paper, the electroosmotic flow of Eyring fluid with Navier slip boundary condition under high Zeta potential in a circular micropipe under the combined use of applied electric field force and pressure is studied. The finite difference method is traditionally used to solve the electric potential distribution and fluid velocity distribution under high Zeta potential. In this paper, we try to use the Chebyshev spectral method to solve the electric potential distribution and fluid velocity distribution, and the results are compared with the analytical approximate solution obtained by Debye-Hückel (D-H) linear approximation [1] and the numerical solution obtained by the classical finite difference method. The result not only confirms the validity of the Chebyshev spectral method, but also shows that it has the advantages of high precision and a small amount of calculation; In addition, the effects of various physical parameters on electroosmotic flow are discussed in detail, and it is obtained that the velocity distribution of the Eyring fluid increases with the increase of the electric potential under the high zeta potential.

On the Numerical Solution of 1D and 2D KdV Equations Using Variational Homotopy Perturbation and Finite Difference Methods

The KdV equation has special significance as it describes various physical phenomena. In this paper, we use two methods, namely, a variational homotopy perturbation method and a classical finite-difference method, to solve 1D and 2D KdV equations with homogeneous and non-homogeneous source terms by considering five numerical experiments with initial and boundary conditions. The variational homotopy perturbation method is a semi-analytic technique for handling linear as well as non-linear problems. We derive classical finite difference methods to solve the five numerical experiments. We compare the performance of the two classes of methods for these numerical experiments by computing absolute and relative errors at some spatial nodes for short, medium and long time propagation. The logarithm of maximum error vs. time from the numerical methods is also obtained for the experiments undertaken. The stability and consistency of the finite difference scheme is obtained. To the best of our knowledge, a comparison between the variational homotopy perturbation method and the classical finite difference method to solve these five numerical experiments has not been undertaken before. The ideal extension of this work would be an application of the employed methods for fractional and stochastic KdV type equations and their variants.

Isogeometric Analysis-Based Solidification Simulation and an Improved Way to Apply Supercooling on Droplet Boundary

The classic finite difference method (FDM) has been successfully adopted in the simulation of dendritic solidification, which is based on phase-field theory. Nevertheless, special strategies of boundary integral and projection are required for applying a supercooling rate to a droplet surface. In the present study, isogeometric analysis (IGA) is employed to discretize the phase-field equation due to the two advantages of Non-Uniform Rational B-Splines (NURBS) basis functions, namely an arbitrary order of derivatives and exact description of complex geometry. In addition, an improved, easy way to apply the supercooling rate on a melt droplet surface is proposed to avoid the integral and projection of the cellular boundary required in FDM. Firstly, dendrite growth in a square computational domain is simulated to verify the performance of IGA. Then, the influences of latent heat, anisotropic mode and initial angle on the dendrite shapes are studied by the presented IGA, FDM and finite element method (FEM). Finally, dendritic solidification in a droplet under different cooling rates along irregular boundaries is performed by the proposed IGA.

Finite Difference Gradient Approximation: To Randomize or Not?

We discuss two classes of methods of approximating gradients of noisy black box functions—the classical finite difference method and recently popular randomized finite difference methods. Despite of the popularity of the latter, we argue that it is unclear whether the randomized schemes have an advantage over the traditional methods when employed inside an optimization method. We point to theoretical and practical evidence that show that the opposite is true at least in a general optimization setting. We then pose the question of whether a particular setting exists when the advantage of the new method may be clearly shown, at least numerically. The larger underlying challenge is a development of black box optimization methods that scale well with the problem dimension.

Modified Finite Difference Method for Solution of Two-interval Boundary Value Problems with Transition Conditions

In this study, we have proposed a new modification of classical Finite Difference Method (FDM) for the solution of boundary value problems which are defined on two disjoint intervals and involved additional transition conditions at an common end of these intervals. The proposed modification of FDM differs from the classical FDM in calculating the iterative terms of numerical solutions. To illustrate the efficiency and reliability of the proposed modification of FDM some examples are solved. The obtained results are compared with those obtained by the standart FDM and by the analytical method. Corresponding graphical illustration are also presented.

FINITE DIFFERENCE VALUE IN FORMING ONE-DIMENSIONAL GEOMETRIC IMAGES REPRESENTED BY NUMERICAL SEQUENCES OF ELEMENTARY FUNCTIONAL DEPENDENCES

Discrete modeling of continuous images by the static-geometric method in most cases is associated with certain errors. Therefore, it is relevant to study the formation of geometric images with a given accuracy, using a minimum amount of initial information. This will allow to create models with optimal discretization.
 Further development of this modeling method with rational decrease in initial information volume, is topical. The proposed line of research will open up new possibilities for its use in various industries, such as construction, engineering, economics etc.
 One of the objectives of this work is to continue research on the formation of discrete images of curved lines. The study is based on the classical finite difference method, static-geometric modeling method and the geometric apparatus of superpositions.
 The article proposes a method for determining the distribution functions of finite-difference values in the formation of discrete analogues of linear-fractional, exponential and hyperbolic functional dependencies using finite differences of different orders. These studies define a general method for obtaining similar patterns of distribution of the finite difference value in formation of discrete analogues of elementary functional dependencies using finite differences of different orders.
 Establishing a correspondence between the equation of the curve and the distribution function of a finite difference value can be the basis for assessing the accuracy of the formation of discrete frameworks of curves described by elementary functional dependencies.
 In the future, the results of this work will make it possible to determine coordinates of any point of a numerical sequence of the nth order and numerical sequences of elementary functional dependencies as a superposition of the coordinates of adjacent points. It can also be a superposition of arbitrarily given nodal points of these sequences, if the value of the final difference or its distribution function are already known.

The Stefan Problem With Internal Heat Generation in Spherical Coordinates

Abstract A weakly time-dependent equation for the evolution of the solid–liquid interface in spherical coordinates, driven by internal heat generation, is derived for constant surface temperature boundary conditions. The derivation comes by making an assumption that the interface moves slowly compared to the changes in the temperature so that the technique of separation of variables may be applied for Stefan numbers less than one. Under this approximation, we can separate the nonhomogeneous heat diffusion equation into transient and steady-state terms, and then integrate to get the temperature relations. With the temperature equations in hand, the derivatives are inserted into the interface equation giving a first-order differential equation for the location of the solid–liquid interface as a function of time. The results are compared to a previously derived quasi-static solution and a numerical simulation generated using the method of catching of the front. This method allows for direct tracking of a moving boundary via the calculation of the time it takes to move from node to node in a discretized grid characteristic of classical finite difference methods. Plots of the interface evolution show excellent agreement between the three methods, especially for lower Stefan numbers. The quality of the approximation decreases as the Stefan number increases, but the model is more accurate than the previously studied quasi-static model. For the Stefan numbers St = 1.0 and 10.0, the weakly time-dependent solutions are in better agreement with the numerical results than the quasi-static solutions.

Advection–Diffusion Lattice Boltzmann Method With and Without Dynamical Filter

In multi-component flow and/or thermal flows, when the diffusion coefficient of the advection–diffusion equation is relatively small, the relaxation coefficient in the lattice Boltzmann method will be close to 0.5, which will lead to numerical instability. The stability conditions will become more severe, when there are high gradient regions in the computational domain. In order to improve the stability of advection–diffusion lattice Boltzmann method to simulate scalar transport in complex flow, a hybrid regularized collision operators and a dynamic filtering method which is suitable for the convection-diffusion lattice Boltzmann method are proposed in this paper. The advection–diffusion lattice Boltzmann method is first tested in uniform flow with smooth and discontinuous initial conditions. Then the scalar transport in doubly periodic shear layer flow is tested, which is sensitive to numerical stability. The adaptive dynamic filtering method is also tested. The results are compared to the classical finite difference method and to the lattice Boltzmann method using the projection-based regularized and standard Bahtnagar-Gross-Krook collision operator. The results show that the hybrid regularized collision operator has advantages in simulating the scalar advection-diffusion problem with small diffusion coefficient. In addition, the adaptive filtering method can also improve the numerical stability of the lattice Boltzmann method with limited numerical dissipation.

ROBUST NUMERICAL METHOD FOR SINGULARLY PERTURBED CONVECTION-DIFFUSION TYPE PROBLEMS WITH NON-LOCAL BOUNDARY CONDITION

This paper presents the study of singularly perturbed differential equations of convection diffusion type with non-local boundary condition. The proposed numerical scheme is a combination of classical finite difference method for the initial boundary condition and nonstandard finite difference method for the differential equations at the interior points. Maximum absolute errors and rates of convergence for different values of perturbation parameter and mesh sizes are tabulated for the numerical examples considered. The method is shown to be first-order convergence independent of the perturbation parameter ε.

Application of the Fast Expansion Method in Space–Related Problems

In the paper, numerical and approximate analytical solutions for the problem of the motion of a spacecraft from a starting point to a final point during a certain time are obtained. The unpowered and powered portions of the flight are considered. For a numerical solution, a finite-difference scheme of the second order of accuracy is constructed. The space-related problem considered in the study is essentially nonlinear, which necessitates the use of trigonometric interpolation methods to replace the task of calculating the Fourier coefficients with the integral formulas by solving the interpolation system. One of the simplest options for trigonometric sine interpolation on a semi-closed segment [–a, a), where the right end is not included in the general system of interpolation points, is considered. In order to maintain the conditions of orthogonality of sines, an even number of 2M calculation points is uniformly applied to the segment. The sine interpolation theorem is proved and a compact formula is given for calculating the interpolation coefficients. A general theory of fast sine expansion is given. It is shown that in this case, the Fourier coefficients decrease much faster with the increase in serial number compared to the Fourier coefficients in the classical case. This property allows reducing the number of terms taken into account in the Fourier series, as well as the amount of computer calculations, and increasing the accuracy of calculations. The analysis of the obtained solutions is carried out, and their comparison with the exact solution of the test problem is proposed. With the same calculation error, the time spent on a computer using the fast expansion method is hundreds of times less than the time spent on classical finite-difference method.

Comparative Study of Some Numerical and Semi-analytical Methods for Some 1D and 2D Dispersive KdV-type Equations

This paper aims to investigate an approximate-analytical and numerical solutions for some 1D and 2D dispersive homogeneous and non-homogeneous KdV equations by employing two reliable methods namely reduced differential transform method (RDTM) and a classical finite-difference method. RDTM provides an analytical approximate solution in the form of a convergent series. The classical finite-difference method (FDM) to solve dispersive KdV equations is employed by primarily checking Von Neumann’s stability criterion. The performance of the mentioned methods for the considered experiments are compared by computing absolute and relative errors at some spatial nodes at a given time; and to the best of our knowledge, the comparison between these two methods for the considered experiments is novel. Knowledge acquired will enable us to build methods for other related PDEs such as KdV-Burgers, stochastic KdV and fractional KdV-type equations.

A new finite difference algorithm for boundary value problems involving transmission conditions

Abstract The finite difference method (FDM) is used to find an approximate solution to ordinary and partial differential equations of various type using finite difference equations to approximate derivatives. The idea is to replace ordinary or partial derivatives appearing in the boundary-value problem by finite differences that approximate them. There is an extensive literature on this topic. But, as a rule, ordinary differential equations or partial differential equations were studied without an internal singular point and without corresponding transmission conditions. It is our main goal here to develop finite difference method to deal with an boundary value problem involving additional transmission conditions at the interior singular point. In this study, we have proposed a new modification of classical FDM for the solution of boundary value problems which are defined on two disjoint intervals and involved additional transmission conditions at an common end of these intervals. The proposed modification of FDM differs from the classical FDM in calculating the iterative terms of numerical solutions. To demonstrate the efficiency and reliability of the proposed modification of FDM an illustrative example is solved b y this method. The obtained results are compared with those obtained by the standard FDM and by the analytical method. Corresponding graphical illustrations are also presented.