Difference Approximation In Time Research Articles

Acoustic Source Localization in CFRP Composite Plate Based on Wave Velocity-Direction Function Fitting.

Composite materials are widely used, but they are often subjected to impacts from foreign objects, causing structural damage. To ensure the safety of use, it is necessary to locate the impact point. This paper investigates impact sensing and localization technology for composite plates and proposes a method of acoustic source localization for CFRP composite plates based on wave velocity-direction function fitting. This method divides the grid of composite plates, constructs the theoretical time difference matrix of the grid points, and compares it with the actual time difference to form an error matching matrix to localize the impact source. In this paper, finite element simulation combined with a lead-break experiment is used to explore the wave velocity-angle function relationship of Lamb waves in composite materials. The simulation experiment is used to verify the feasibility of the localization method, and the lead-break experimental system is built to locate the actual impact source. The results show that the acoustic emission time-difference approximation method can effectively solve the problem of impact source localization in composite structures, and the average localization error is 1.44 cm and the maximum localization error is 3.35 cm in 49 experimental points with good stability and accuracy.

A high-efficiency wavefield decomposition method based on the Hilbert transform

Wavefield decomposition can be used to extract effective information in reverse time migration and full-waveform inversion. The wavefield decomposition methods based on the Hilbert transform (HTWD) and the Poynting vector (PVWD) are the most commonly used. The HTWD needs to save the wavefields at all time steps or introduce additional numerical simulation, which increases the computational cost. PVWD cannot handle multiwave arrivals, and its performance is poor in complex situations. We have developed an efficient wavefield decomposition method based on the Hilbert transform (EHTWD). EHTWD constructs two wavefields to replace the original wavefield and the wavefield after the Hilbert transform. The first wavefield is obtained by using the dispersion relation to modify the frequency components. The other wavefield is obtained by time difference approximation. Therefore, there is a 90° phase change between the two wavefields. In EHTWD, we only need two wavefields at different moments, which avoids the need for additional numerical simulation. EHTWD is also suitable for wavefield decomposition in arbitrary directions. Compared to HTWD, the computational complexity can be greatly reduced with the decrease of the number of imaging time slices. The numerical examples of wavefield decomposition demonstrate that our method can realize wavefield decomposition in any direction. The examples of imaging decomposition and real data also indicate that EHTWD suppresses imaging noise effectively.

A dissipative finite difference Fourier pseudo-spectral method for the symmetric regularized long wave equation with damping mechanism

In this paper, an energy stable time-stepping method using the finite difference approximation in time and Fourier pseudo-spectral method in space is developed for the symmetric regularized long wave equation with damping mechanism. Based upon a careful treatment of nonlinear term, the suggested numerical scheme is proved to preserve energy dissipation at discrete time levels. The energy stable property implies the priori estimates of the numerical solution. The maximum norm error estimate shows that the proposed numerical scheme is of second-order accuracy in time and spectral accuracy in space. Several numerical experiments are presented to show the effectiveness of our numerical method and to confirm our theoretical analysis.

A dissipative finite difference Fourier pseudo-spectral method for the Klein-Gordon-Schrödinger equations with damping mechanism

We develop a semi-linearized, decoupled time-stepping method for solving the Klein-Gordon-Schrödinger equations with damping mechanism. The finite difference approximation in time and Fourier pseudo-spectral discretization in space provide an elegant platform to deal with the physical properties of the original model. We prove that the proposed numerical algorithm preserves the discrete invariant or dissipative properties of system exactly depending on the choices of the damping parameter values. We establish the maximum norm error estimates by virtue of the norm-equivalence between finite difference method and Fourier pseudo-spectral method, the discrete versions of projection and interpolation estimations, and mathematical induction argument. Ample numerical results are presented to show the effectiveness of our numerical method and to confirm our theoretical analysis.

A spline collocation method for a fractional mobile–immobile equation with variable coefficients

The Crank–Nicolson orthogonal spline collocation (OSC) methods are considered for approximate solution of the variable coefficient fractional mobile–immobile equation. The convection, diffusion, and reaction coefficients can depend on both the spatial and temporal variables, simultaneously. Combining with Crank–Nicolson scheme and weighted and shifted Grunwald difference approximation in time, we establish OSC method in space. It is proved that our proposed fully methods are of optimal order in certain $$H_j$$ ($$j=0,1$$) norms. Moreover, we derive $$L^{\infty }$$ estimates in space. Numerical results are also provided to verify our proposed algorithm.

Fractional integration operator for numerical solution of the integro-partial time fractional diffusion heat equation with weakly singular kernel

In this paper, an approximate solution for solving weakly singular kernel partial integro-differential equations with time fractional order is proposed. The method is based on using a second-order time difference approximation followed by applying the fractional integral operator and piecewise linear interpolation to compute the singularity of the kernel that appear in the discretization process. The stability of the method is also considered in the sense of von Neumann analysis. Numerical examples are solved to demonstrate the validity and applicability of the presented technique.

Galerkin finite element method and error analysis for the fractional cable equation

The cable equation is one of the most fundamental equations for modeling neuronal dynamics. These equations can be derived from the Nernst-Planck equation for electro-diffusion in smooth homogeneous cylinders. Fractional cable equations are introduced to model electrotonic properties of spiny neuronal dendrites. In this paper, a Galerkin finite element method(GFEM) is presented for the numerical simulation of the fractional cable equation(FCE) involving two integro-differential operators. The proposed method is based on a semi-discrete finite difference approximation in time and Galerkin finite element method in space. We prove that the numerical solution converges to the exact solution with order O(?+hl+1) for the lth-order finite element method. Further, a novel Galerkin finite element approximation for improving the order of convergence is also proposed. Finally, some numerical results are given to demonstrate the theoretical analysis. The results show that the numerical solution obtained by the improved Galerkin finite element approximation converges to the exact solution with order O(?2+hl+1).

Explicit Time Integration Scheme Using Krylov Subspace Method for Reactor Kinetics Equation

The spatial discretization form of the space-dependent reactor kinetics equation is a first-order simultaneous ordinary differential equation in time. Conventional numerical methods of the space-dependent kinetics equation, i.e., the generalized Runge-Kutta method, the implicit method (backward Euler method), and the Theta method, are based on the time difference approximation. However, the present study adopts the analytical solution of the space-dependent kinetics equation expressed by the matrix exponential and no time difference approximation is used. In this context, our present approach is classified as an explicit method in which no iteration calculation on space and energy is necessary. The Krylov subspace method is used to evaluate the matrix exponential observed in the solution of the spatially discretized space-dependent kinetics equation. The Krylov subspace method is implemented into a space-dependent kinetics solver. In order to examine the effectiveness of the Krylov subspace method, the TWIGL benchmark problem is analyzed as a verification calculation. The calculation results show the effectiveness of the present method especially in the step reactivity perturbation.

Explicit Time Integration Scheme Using Krylov Subspace Method for Reactor Kinetics Equation

The spatial discretization form of the space-dependent reactor kinetics equation is a first-order simultaneous ordinary differential equation in time. Conventional numerical methods of the space-dependent kinetics equation, i.e., the generalized Runge-Kutta method, the implicit method (backward Euler method), and the Theta method, are based on the time difference approximation. However, the present study adopts the analytical solution of the space-dependent kinetics equation expressed by the matrix exponential and no time difference approximation is used. In this context, our present approach is classified as an explicit method in which no iteration calculation on space and energy is necessary. The Krylov subspace method is used to evaluate the matrix exponential observed in the solution of the spatially discretized space-dependent kinetics equation. The Krylov subspace method is implemented into a space-dependent kinetics solver. In order to examine the effectiveness of the Krylov subspace method, the TWIGL benchmark problem is analyzed as a verification calculation. The calculation results show the effectiveness of the present method especially in the step reactivity perturbation.

Approximation of optimal feedback control: a dynamic programming approach

We consider the general continuous time finite-dimensional deterministic system under a finite horizon cost functional. Our aim is to calculate approximate solutions to the optimal feedback control. First we apply the dynamic programming principle to obtain the evolutive Hamilton–Jacobi–Bellman (HJB) equation satisfied by the value function of the optimal control problem. We then propose two schemes to solve the equation numerically. One is in terms of the time difference approximation and the other the time-space approximation. For each scheme, we prove that (a) the algorithm is convergent, that is, the solution of the discrete scheme converges to the viscosity solution of the HJB equation, and (b) the optimal control of the discrete system determined by the corresponding dynamic programming is a minimizing sequence of the optimal feedback control of the continuous counterpart. An example is presented for the time-space algorithm; the results illustrate that the scheme is effective.

Stability and convergence of Fourier spectral method for the model of wheezes

In this paper a Fourier spectral method to solve the initial periodic boundary values problem for the model of wheezes is proposed. A second order finite difference approximation in time combined with spectral approximation in space for this problem is investigated. The stability and rate of convergence for the scheme are proved.

Macromodeling of Nonlinear Digital I/O Drivers

In this paper, a modeling technique using spline functions with finite time difference approximation is discussed for modeling moderately nonlinear digital input/output (I/O) drivers. This method takes into account both the static and the dynamic memory characteristics of the driver during modeling. Spline function with finite time difference approximation includes the previous time instances of the driver output voltage/current to capture the output dynamic characteristics of digital drivers accurately. In this paper, the speed and the accuracy of the proposed method is analyzed and compared with the radial basis function (RBF) modeling technique, for modeling different test cases. For power supply noise analysis, the proposed method has been extended to multiple ports by taking the previous time instances of the power supply voltage/current into account. The method discussed can be used to capture sensitive effects like simultaneous switching noise (SSN) and cross talk accurately when multiple drivers are switching simultaneously. A comparison study between the presented method and the transistor level driver models indicate a computational speed-up in the range of 10-40 with an error of less than 5%. For highly nonlinear drivers, a method based on recurrent artificial neural networks (RNN) is discussed.

A cell boundary element method applied to laminar vortex shedding from circular cylinders

The two-dimensional unsteady incompressible Navier–Stokes equations are solved for flows around arrangements of circular cylinders at Reynolds number 100 and 200. A hybrid boundary element/finite element method is used to discretise the spatial domain together with a second order implicit finite difference approximation in time. The numerical scheme of study is validated for a uniform stream past an isolated circular cylinder by comparing findings with experimental and numerical studies. Both steady state and time dependent solutions were predicted with good agreement. The numerical approach, known here as the cell boundary element method (cell BEM), was also used to solve flows around two cylinders of equal diameter side by side and in tandem. It was found that no modifications to the method were needed to compute the flow field for these connected domains. In-phase and anti-phase vortex shedding modes were found to exist in the flow simulation. These simulations were in excellent agreement with phenomena observed in experiments. Particle simulations, generated from the cell BEM velocity fields, were found to have great similarity with smoke visualisations from experiment.

The 2F3D-Model of Filtration in the Bottom Zones of Horizontal Oil Wells. II

The difference approximation in time and space of the original continuous equations (Part I) was discussed. The existence conditions for the corresponding algebraic equations were presented. The solution of the difference equations was proved to converge to that of the original equations of the continuous model. The problem of identification of the basic parameters of the finite-difference model was discussed.

A priori 𝐿^{𝜌} error estimates for Galerkin approximations to porous medium and fast diffusion equations

Galerkin approximations to solutions of a Cauchy-Dirichlet problem governed by the generalized porous medium equation \[ ∂ u ∂ t − ∑ i = 1 N ∂ ∂ x i ( | u | ρ − 2 ∂ u ∂ x i ) = f ( x , t ) \frac {\partial u}{\partial t}-\sum ^N_{i=1}\frac \partial {\partial x_i}(|u|^{\rho -2}\frac {\partial u}{ \partial x_i})=f(x,t) \] on bounded convex domains are considered. The range of the parameter ρ \rho includes the fast diffusion case 1 > ρ > 2 1>\rho >2 . Using an Euler finite difference approximation in time, the semi-discrete solution is shown to converge to the exact solution in L ∞ ( 0 , T ; L ρ ( Ω ) ) L^\infty (0,T;L^\rho (\Omega )) norm with an error controlled by O ( Δ t 1 4 ) O(\Delta t^{\frac 14}) for 1 > ρ > 2 1>\rho >2 and O ( Δ t 1 2 ρ ) O(\Delta t^{\frac 1{2\rho }}) for 2 ≤ ρ > ∞ 2\le \rho >\infty . For the fully discrete problem, a global convergence rate of O ( Δ t 1 4 ) O(\Delta t^{\frac 14}) in L 2 ( 0 , T ; L ρ ( Ω ) ) L^2(0,T;L^\rho (\Omega )) norm is shown for the range 2 N N + 1 > ρ > 2 \frac {2N}{N+1}>\rho >2 . For 2 ≤ ρ > ∞ 2\le \rho >\infty , a rate of O ( Δ t 1 2 ρ ) O(\Delta t^{\frac 1{2\rho }}) is shown in L ρ ( 0 , T ; L ρ ( Ω ) ) L^\rho (0,T;L^\rho (\Omega )) norm.

Numerical solution of the sideways heat equation by difference approximation in time

We consider a Cauchy problem for the heat equation in the quarter plane, where data are given at x=1 and a solution is sought in the interval 0<x<1. This sideways heat equation is a model of a problem where one wants to determine the temperature on both sides of a thick wall, but where one side is inaccessible to measurements. The heat equation is discretized by a differential-difference equation, where the time derivative has been replaced by a finite difference, and we analyse the approximation properties of time-discrete approximations using Fourier transform techniques. An error estimate is obtained for one such approximation, and it is shown that when the data error tends to zero, the error in the approximate solution tends to zero logarithmically. This error estimate also gives information about how to choose the step length in the time discretization.