Second-order Finite Difference Method Research Articles

A numerical study of 3D frequency-domain elastic full-waveform inversion

We have developed a 3D elastic full-waveform inversion (FWI) algorithm with forward modeling and inversion performed in the frequency domain. The Helmholtz equation is solved with a second-order finite-difference method using an iterative solver equipped with an efficient complex-shifted incomplete LU-based preconditioner. The inversion is based on the minimization of the data misfit functional and a total variation regularization for the unknown model parameters. We implement the Gauss-Newton method as the optimization engine for the inversions. The codes are parallelized with a message passing interface based on the number of shots and receivers. We examine the performance of this elastic FWI algorithm and workflow on synthetic examples including surface seismic and vertical seismic profile configurations. With various initial models, we manage to obtain high-quality velocity images for 3D earth models.

Optimization of Biomimetic Propulsive Kinematics of a Flexible Foil Using Integrated Computational Fluid Dynamics–Computational Structural Dynamics Simulations

A computational methodology, which combines a computational fluid dynamics (CFD) technique and a computational structural dynamics (CSD) technique, is employed to design a deformable foil whose kinematics is inspired by the propulsive motion of the fin or the tail of a fish or a cetacean. The unsteady incompressible Navier–Stokes equations are solved using a second-order accurate finite difference method and an immersed-boundary method to effectively impose boundary conditions on complex moving boundaries. A finite element-based structural dynamics solver is employed to compute the deformation of the foil due to interaction with fluid. The integrated CFD–CSD simulation capability is coupled with a surrogate management framework (SMF) for nongradient-based multivariable optimization in order to optimize flapping kinematics and flexibility of the foil. The flapping kinematics is manipulated for a rigid nondeforming foil through the pitching amplitude and the phase angle between heaving and pitching motions. The flexibility is additionally controlled for a flexible deforming foil through the selection of material with a range of Young's modulus. A parametric analysis with respect to pitching amplitude, phase angle, and Young's modulus on propulsion efficiency is presented at Reynolds number of 1100 for the NACA 0012 airfoil.

Dispersion and Stability Condition of Seismic Wave Simulation in TTI Media

For seismic waveform simulation in tilted transversely isotropic (TTI) media, we derive explicitly the numerical dispersion relation and the stability condition for the computation of a 2D pseudo-acoustic wave equation. The numerical dispersion relation indicates that the number of sampling points per wavelength has the greatest influence on the dispersion, while the anisotropic parameters of the TTI media and the mesh rotation angle have little influence on the dispersion. Given an appropriate spatial sampling, the stability condition is for the selection of the time step for the implementation of the TTI wave equation. We partition a numerical model using quadrangle grids in Cartesian coordinates, and map it to a computing model in which any non-rectangular meshes in Cartesian coordinates become rectangular meshes. Then we reformulate the pseudo-acoustic wave equation for the TTI media accordingly in the computational space. We implement seismic waveform simulation using the second-order finite-difference method straightforwardly, and show examples with a desirable accuracy using a model with non-rectangular meshes in Cartesian coordinates along a curved surface and fluctuating interfaces in the TTI media.

Alternating-direction implicit finite difference methods for a new two-dimensional two-sided space-fractional diffusion equation

According to the principle of conservation of mass and the fractional Fick’s law, a new two-sided space-fractional diffusion equation was obtained. In this paper, we present two accurate and efficient numerical methods to solve this equation. First we discuss the alternating-direction finite difference method with an implicit Euler method (ADI–implicit Euler method) to obtain an unconditionally stable first-order accurate finite difference method. Second, the other numerical method combines the ADI with a Crank–Nicolson method (ADI–CN method) and a Richardson extrapolation to obtain an unconditionally stable second-order accurate finite difference method. Finally, numerical solutions of two examples demonstrate the effectiveness of the theoretical analysis.

New energy-preserving algorithms for nonlinear Hamiltonian wave equation equipped with Neumann boundary conditions

In this paper, using the blend of spatial discretization by second-order or fourth-order finite difference methods (FDM) and time integration by the generalized Average Vector Field (GAVF) method or the generalized adapted Average Vector Field (GAAVF) method, we propose and analyze novel energy-preserving algorithms for solving the nonlinear Hamiltonian wave equation equipped with homogeneous Neumann boundary conditions. Firstly, two kinds of finite difference methods are considered to discretize the spatial derivative, which can be of order two and order four respectively in all the spatial grid points. The conservation laws of the discrete energy are established after the semi-discretization, a Hamiltonian system of ODEs is derived whose Hamiltonian can be regarded as the approximate energy of the original continuous system. Then, the GAVF formula and the GAAVF formula are developed and applied to the derived Hamiltonian ODEs to yield some novel and efficient algorithms, which can exactly preserve the discrete energy. The numerical simulation is implemented and the numerical results demonstrate the spatial and temporal accuracy and the remarkable energy-preserving property of the new algorithms presented in this paper.

Magnetohydrodynamic(MHD) Boundary Layer Flow of Eyring-Powell Nanofluid Past Stretching Cylinder With Cattaneo-Christov Heat Flux Model

AbstractThis study analyzed the MHD boundary layer flow of Eyring-Powell nanofluid past stretching cylinder with Cattaneo-Christov heat flux model. The governing non-linear partial differential equations corresponding to the momentum, energy and concentration have been formulated and transformed into a set of non-linear ordinary differential equations by using similarity transformations. Then the resulting non-linear high order ordinary differential equations of momentum, energy and concentration, subjected to boundary conditions were solved numerically by utilizing the second-order accurate implicit finite difference method known as Keller-Box which is programmed in the MATLABR2017b software. The results indicated that the velocity profile increases as the Eyring-Powell fluid parameterMand the curvature parameterγincrease but it decreases as the magnetic parameterHaincreases. Both the temperature and the concentration profiles have revealed an increment pattern for large values of the magnetic parameterHaand the thermophoresis parameterNtbut a decrement manner with increasing values of the Eyring-Powell fluid parameterM. The Brownian motion parameterNbhas shown an opposite influence on the temperature and the concentration profiles. The results also depicted that the local skin friction coefficient increases with increasing in Eyring-Powell fluid parameterM, magnetic parameterHa. Besides, it is found that both the local Nusselt numberNuxand the local Sherwood numberShxare higher for large vales of Eyring-Powell fluid parameterMand curvature parameterγ. Furthermore, the present results for the local skin friction coefficient, the local Nusselt number and the local Sherwood number are validated with the data of previously published literature for various limiting conditions where a very sound agreement has been attained.

An Explicit-Implicit Splitting Method for a Convection-Diffusion Problem

Abstract We analyze a second-order accurate finite difference method for a spatially periodic convection-diffusion problem. The method is a time stepping method based on the Strang splitting of the spatially semidiscrete solution, in which the diffusion part uses the Crank–Nicolson method and the convection part the explicit forward Euler approximation on a shorter time interval. When the diffusion coefficient is small, the forward Euler method may be used also for the diffusion term.

The Numerical Study of the Ground States of Spin-1 Bose-Einstein Condensates with Spin-Orbit-Coupling

Abstract A projection gradient method for computing the ground state of the spinorbit-coupled spin-1 Bose-Einstein condensate at extremely low temperatures is proposed. The continuous gradient flows are discretised by a second-order finite difference method in space and the Crank-Nicolson method in time. Our discretisation preserves the total mass conservation and the energy diminishing property. Numerical results show the efficiency of the method.

A study of mimetic and finite difference methods for the static diffusion equation

Two second-order finite difference methods in a staggered mesh to solve the static diffusion equation are proposed in this article. These methods were compared with a standard finite difference method and with two numerical schemes naturally established in staggered grids: mimetic method and conservative method. Also, mimetic discretization is presented in a formal manner. The methods were tested using different configurations, including boundary layers and heterogeneous media. The study shows that the two proposed finite difference methods produce numerical solutions that are comparable to those given by mimetic methods, in terms of rates of convergence and magnitude of the approximation error.

Combined Effects of Thermophoresis, Brownian Motion, and Nanofluid Variable Properties on CuO-Water Nanofluid Natural Convection in a Partially Heated Square Cavity

Numerical investigation of natural convective heat transfer and fluid flow in a differentially heated square cavity filled with a CuO-water nanofluid having variable properties is performed. Governing partial differential equations formulated in nondimensional stream function, vorticity, temperature, and nanoparticles volume fraction are solved by the second-order accurate finite difference method and taking into account the Brownian diffusion and thermophoresis. The effects of Rayleigh number (Ra = 104–106), initial nanoparticles volume fraction (C0 = 0–0.09), location of the heater (Δ = 0.0–0.9), and dimensionless time (τ = 0–300) on flow patterns, isotherms, and concentration fields as well as the local and average Nusselt numbers at the heater surface are studied. The isoconcentrations reveal that for most of the cavity domain the nanoparticle concentration is around the initial average concentration of nanoparticles except for a very limited variation in a region close to the cavity walls that experiences minor deviation from the initial concentration. It was found that the flow strength within the cavity (i.e., ψmaxRa⋅Pr) is inversely proportional to the heater location Δ and is directly proportional to the Rayleigh number. Also, it was found that the best location of the heater, from a heat transfer perspective, is placing it entirely at the left wall of the cavity where a maximum average Nusselt number is registered. This study revealed that for all heater locations there is always an adverse impact of nanoparticles on the heat transfer and the worst case is registered for the Δ = 0 and the least deterioration is noticed for Δ = 0.9.

Numerical and experimental investigation on static electric charge model at stable cone-jet region

In a typical electro-spinning process, the steady stretching process of the jet beyond the Taylor cone has a significant effect on the dimensions of resulting nanofibers. Also, it sets up the conditions for the onset of the bending instability. The focus of this work is the modeling and simulation of the initial stable jet phase seen during the electro-spinning process. The perturbation method was applied to solve hydrodynamic equations, and the electrostatic equation was solved by a boundary integral method. These equations were coupled with the stress boundary conditions derived appropriate at the fluid-fluid interface. Perturbation equations were discretized by the second-order finite difference method, and the Newton method was implemented to solve the discretized nonlinear system. Also, the boundary element method was utilized to solve the electrostatic equation. In the theoretical study, the fluid is described as a leaky dielectric with charges only on the jet surface in dielectric air. In this study, electric charges were modeled as static. Comparison of numerical and experimental results shows that at low flow rates and high electric field, good agreement was achieved because of the superior importance of the charge transport by conduction rather than convection and charge concentration. In addition, the effect of unevenness of the electric field around the nozzle tip was experimentally studied through plate-plate geometry as well as point-plate geometry.

An effective comparison involving a novel spectral approach and finite difference method for the Schrödinger equation involving the Riesz fractional derivative in the quantum field theory

This paper displays the approach of the time-splitting Fourier spectral (TSFS) technique for the linear Riesz fractional Schrodinger equation (RFSE) in the semi-classical regime. The splitting technique is shown to be unconditionally stable. Further a suitable implicit finite difference discretization of second order has been manifested for the RFSE where the Riesz derivative has been discretized via an approach of fractional centered difference. Moreover the stability analysis for the implicit scheme has also been presented here via von Neumann analysis. The L2-norm and $L^{\infty}$ -norm errors are calculated for $\vert u(x,t)\vert^{2}$ , Re $(u(x,t))$ and Im $(u(x,t))$ for various cases. The results obtained by the methods are further tabulated for the absolute errors for $\vert u(x,t)\vert^{2}$ . Furthermore the graphs are depicted showing comparison of $\vert u(x,t)\vert^{2}$ by both techniques. The derivatives are taken here in the context of the Riesz fractional sense. Apart from that, the comparative study put forth in the following section via tables and graphs between the implicit second-order finite difference method (IFDM) and the TSFS method is for the purpose of investigating the efficiency of the results obtained. Moreover the stability analysis of the presented techniques manifesting their unconditional stability makes the proposed approach more competing and accurate.

Modelling the load transfer and tool surface for friction reduction drilling by vibrating drill-string

A unified mathematical model, considering the two mechanisms of static-dynamic friction conversion and friction decomposition, is developed to analysis the effect of different vibration manners and parameters on axial load transfer and tool face of positive displacement motor and evaluate the vibration level and dynamic security of the drill-string. A second-order finite difference method is used to solve this model. The change of weight on bit, rate of penetration and tool face of positive displacement motor under three different slide drilling manners are obtained. Simulation results show that the vibrations applied on drill-string mitigate the stick-slip and adhesion phenomenon and make the load transfer and movement of the drill-string smoother. The increment of weight on bit is the combined effect of the change of axial friction acting on drill-string and the exciting force. Among the three vibration manners, the axial vibration applied at bottom hole has the best friction reduction and tool face stabilization effects and the lowest vibration intensity. The torsional vibration applied at wellhead mainly affects the upper drill-string and has an optimal torsion angle subject to the torsional depth. The axial vibration applied at wellhead has a strong effect on the drill-string of vertical section. Results provide guidance and advice to the research and applications of friction reduction by vibrating drill-string.

An Efficient Second-Order Finite Difference Method for the One-Dimensional Schrödinger Equation with Absorbing Boundary Conditions

A stable and convergent second-order fully discrete finite difference scheme with efficient approximation of the exact absorbing boundary conditions is proposed to solve the Cauchy problem of the o...

A Second-Order Finite-Difference Method for Compressible Fluids in Domains with Moving Boundaries

A Second-Order Finite-Difference Method for Compressible Fluids in Domains with Moving Boundaries

Large eddy simulation of density current on sloping beds

We carry out high-resolution large-eddy simulation (LES) of stratified flows developing as dense underflows over two different bed slopes of 5° and 15° in an laboratory-scale tank. Simulations for the corresponding unstratified flows are also carried out to help elucidate the effects of stratification on the dynamics of the flow and overall evolution of the effluent front. The simulated density current flows have also been investigated experimentally by continuously releasing down a sloping plane a fluid that is slightly denser than the ambient fluid. The Navier Stokes equations with the Boussinesq approximation, to account for the stratification due to density difference, are solved numerically using a second-order accurate finite-difference method (Khosronejad and Sotiropoulos, 2014). The simulated structure of the density currents at quasi-equilibrium are shown to be in reasonable agreement with the experimental observations. The variation of the half-width of the density current plume with downstream length is shown to exhibit different power law regimes on a log scale. Spectral analysis reveal existence of large scale structures in the flow. Comparative analysis of the dynamics of the coherent structures in the stratified and non-stratified flows reveal that stratification has a profound effect on the large-scale features of the flow.

On reliability of chaotic dynamics of two Euler–Bernoulli beams with a small clearance

A methodology to detect true chaos (in terms of non-linear dynamics) is developed on an example of a structure composed of two beams with a small clearance. The Euler–Bernoulli hypothesis is employed, and the contact interaction between beams follows the Kantor model. The complex non-linearity results from the von Kármán geometric non-linearity as well as the non-linearity implied by the contact interaction. The governing PDEs are reduced to ODEs by the second-order Finite Difference Method (FDM). The obtained system of equations is solved by Runge–Kutta methods of different accuracy. To purify the signal from errors introduced by numerical methods, the principal component analysis is employed and the sign of the first Lyapunov exponent is estimated by the Kantz, Wolf, Rosenstein methods and the method of neural networks. In the lattermost case, a spectrum of the Lyapunov exponents is estimated. It is illustrated how the number of nodes in the FDM influences numerical results regarding chaotic vibrations. It is also shown that an increase in the distance between beams implies stronger action of the geometric non-linearity. Convergence of the used numerical algorithm for FDM is demonstrated. The essential influence of initial conditions on the numerical results of the studied contact problem is presented and discussed.

On solitary wave diffraction by multiple, in-line vertical cylinders

The interaction of solitary waves with multiple, in-line vertical cylinders is investigated. The fixed cylinders are of constant circular cross section and extend from the seafloor to the free surface. In general, there are N of them lined in a row parallel to the incoming wave direction. Both the nonlinear, generalized Boussinesq and the Green–Naghdi shallow-water wave equations are used. A boundary-fitted curvilinear coordinate system is employed to facilitate the use of the finite-difference method on curved boundaries. The governing equations and boundary conditions are transformed from the physical plane onto the computational plane. These equations are then solved in time on the computational plane that contains a uniform grid and by use of the successive over-relaxation method and a second-order finite-difference method to determine the horizontal force and overturning moment on the cylinders. Resulting solitary wave forces from the nonlinear Green–Naghdi and the Boussinesq equations are presented, and the forces are compared with the experimental data when available.

Flow and Heat Transfer Inside Eccentric Curved Annulus with Arbitrary Radius of Curvature and Inner Tube Position

ABSTRACTThe current bipolar-toroidal coordinate system is restricted by its geometrical limitations in which the curvature is a function of eccentricity and radius ratio. In addition, the inner tube is located just on symmetry line in inner bend. The aim of present study is to modify the current coordinate system to take into account any radius of curvature and inner tube angular position independently and then investigate the fluid flow and its thermal characteristics. To do so, metric is obtained and governing equations including the continuity, full Navier–Stokes and energy equations are derived. Using a second-order finite difference method, these equations are solved on the basis of the projection algorithm under hydrodynamic and thermal developing assumptions. The inner and outer walls are maintained at constant and ambient temperatures, respectively. Finally, the effect of non-dimensional parameters such as the Dean, Reynolds and Prandtl numbers, eccentricity and inner tube angular position are investigated. The obtained results show that eccentricity and Dean number increasing significantly deforms the velocity field and the secondary flow field. Moreover, it has been found that increasing eccentricity intensifies the Nusselt number. Furthermore, changing the angular position of inner tube can increase the heat transfer rate.

A second-order finite difference method for the resolution of a boundary value problem associated to a modified Poisson equation in spherical coordinates

We propose a second-order finite difference method which may be used for a self-contained and efficient numerical resolution of a boundary value problem involving a modified Poisson equation written in spherical coordinates and a Robin type boundary condition. We compare on two realistic examples issued from the combustion of a coal char particle the efficiency of this second-order finite difference method with that of the classical finite difference method and of the finite element method.