Backward Difference Method Research Articles

Full-Discrete Weak Galerkin Finite Element Method for Solving Diffusion-Convection Problem.

This paper applied and analyzes full discrete weak Galerkin (WG) finite element method for non steady two dimensional convection-diffusion problem on conforming polygon. We approximate the time derivative by backward finite difference method and the elliptic form by WG finite element method. The main idea of WG finite element methods is the use of weak functions and their corresponding discrete weak derivatives in standard weak form of the model problem. The theoretical evidence proved that the error estimate in norm, the properties of the bilinear form, (v-elliptic and continuity), stability, and the energy conservation law.

Fully discrete second-order backward difference method for Kelvin-Voigt fluid flow model

In this article, based on a second-order backward difference method, a completely discrete scheme is discussed for a Kelvin-Voigt viscoelastic fluid flow model with nonzero forcing function, which is either independent of time or in L ∞ (L 2). After deriving some a priori bounds for the solution of a semidiscrete Galerkin finite element scheme, a second-order backward difference method is applied for temporal discretization. Then, a priori estimates in Dirichlet norm are derived, which are valid uniformly in time using a combination of discrete Gronwall’s lemma and Stolz-Cesaro’s classical result on sequences. Moreover, an existence of a discrete global attractor for the discrete problem is established. Further, optimal a priori error estimates are obtained, whose bounds may depend exponentially in time. Under uniqueness condition, these estimates are shown to be uniform in time. Finally, several numerical experiments are conducted to confirm our theoretical findings.

Nonsmooth data error estimates for FEM approximations of the time fractional cable equation

In this paper, the fractional cable equation, involving two Riemann–Liouville fractional derivatives, with initial/boundary condition is considered. Two fully discrete schemes are obtained by employing piecewise linear Galerkin FEM in space, and using convolution quadrature methods based on the first- and second-order backward difference methods in time. Optimal error estimates in terms of the initial data and the inhomogeneity for the semi-discrete scheme and fully discrete schemes are discussed. Numerical results are shown to verify the theoretical results.

Power difference template for action recognition

This paper proposes power difference template as a new spatial-temporal representation for action recognition. Specifically, spatial power features are first extracted according to the transform of Gaussian convolution on gradients between logarithmic and exponential domain. Using the forward–backward frame power difference method, we thus present normalized projection histogram (NPH) to characterize segmented action spatial features by normalizing histogram of the 2D horizontal–vertical projections. Furthermore, from the perspective of energy conservation, motion kinetic velocity (MKV) is introduced as a supplement for representing temporal relationships of power features by supposing that the variation of power is produced by motion in the form of kinetic energy. Our power difference template fusing NPH and MKV is further integrated to a bag of word model for training and testing under a support vector machine framework. Experiments on KTH, UCF Sports, UCF101 and HMDB datasets demonstrate the effectiveness of the proposed algorithm.

Finite element method for nonlinear Riesz space fractional diffusion equations on irregular domains

In this paper, we consider two-dimensional Riesz space fractional diffusion equations with nonlinear source term on convex domains. Applying Galerkin finite element method in space and backward difference method in time, we present a fully discrete scheme to solve Riesz space fractional diffusion equations. Our breakthrough is developing an algorithm to form stiffness matrix on unstructured triangular meshes, which can help us to deal with space fractional terms on any convex domain. The stability and convergence of the scheme are also discussed. Numerical examples are given to verify accuracy and stability of our scheme.

Stability analysis and evaluation of staggered coupled analysis methods for electromagnetic and structural coupled finite element analysis

The numerical stability of electromagnetic and structural coupled analysis methods is examined in terms of the spectral radius for various methods combined with a coupled algorithm, time integration methods for structural analysis and eddy current analysis, and coupled effect evaluation methods. The coupled analysis method with the conventional serial staggered algorithm, generalized–α method for structure, backward difference method for eddy current, and coupled effect evaluation using previous time step results is seen to be the most suitable from the viewpoints of stability and computing time. The stability of the conventional parallel staggered algorithm is found to be much improved if the generalized–α method is used for structural analysis.

A radial basis function based implicit–explicit method for option pricing under jump-diffusion models

In this article, we present a radial basis function based implicit explicit numerical method to solve the partial integro-differential equation which describes the nature of the option price under jump diffusion model. The governing equation is time semi discrtized by using the implicit–explicit backward difference method of order two (IMEX-BDF2) followed by radial basis function based finite difference (RBF-FD) method. The numerical scheme derived for European option is extended for American option by using operator splitting method. Numerical results for put and call option under Merton and Kou models are given to illustrate the efficiency and accuracy of the present method. The stability of time semi discretized scheme is also proved.

Using Meshless Local Natural Neighbor Interpolation Method to Solve Two-Dimensional Nonlinear Problems

Based on the meshless local natural neighbor interpolation method (MLNNI), a novel solution procedure is developed for the analysis of nonlinear steady and transient heat transfer of two-dimensional structures in this paper. Nonlinearities arising from temperature dependence of material properties and nonlinear boundary conditions have been taken into account. The present method is developed based on the natural neighbor interpolation (NNI) for constructing shape functions at scattered nodes. The three-node triangular FEM shape function is employed as the test function, which reduces the orders of integrands involved in domain integrals. Due to the delta function property of the natural neighbor shape functions, there is no need to employ special techniques to enforce the essential boundary conditions. The backward difference method is employed for the time integration scheme in transient analysis and the Newton–Raphson iterative procedure is required at each time step. Three numerical examples with different geometries and boundary conditions are presented at the end to demonstrate the validity and accuracy of the proposed method for the solution of a wide class of nonlinear steady and transient heat transfer problems.

Reduced-Order Observer Based Sliding Mode Control for a Quad-Rotor Helicopter

[abstFig src='/00280003/05.jpg' width=""230"" text='Quadcopter for repeated control verification' ] The reduced-order observer design we present estimates the velocity states of a quadrotor helicopter, or quadcopter, based on sampled measurements of position and attitude states. This observer is based on the forward-differentiation Euler model. The observer is robust enough against observation noise that the gain of a closed-loop controller is high enough to improve control performance. A sliding-mode controller stabilizes and implements quadcopter tracking control effectively, as is verified experimentally when compared to a conventional backward-difference method.

A two-grid finite element approximation for a nonlinear time-fractional Cable equation

In this article, a nonlinear fractional Cable equation is solved by a two-grid algorithm combined with finite element (FE) method. A temporal second-order fully discrete two-grid FE scheme, in which the spatial direction is approximated by two-grid FE method and the integer and fractional derivatives in time are discretized by second-order two-step backward difference method and second-order weighted and shifted Gr\"unwald difference (WSGD) scheme, is presented to solve nonlinear fractional Cable equation. The studied algorithm in this paper mainly covers two steps: First, the numerical solution of nonlinear FE scheme on the coarse grid is solved, Second, based on the solution of initial iteration on the coarse grid, the linearized FE system on the fine grid is solved by using Newton iteration. Here, the stability based on fully discrete two-grid method is derived. Moreover, the a priori estimates with second-order convergence rate in time is proved in detail, which is higher than the L1-approximation result with $O(\tau^{2-\alpha}+\tau^{2-\beta})$. Finally, the numerical results by using the two-grid method and FE method are calculated, respectively, and the CPU-time is compared to verify our theoretical results.

Two Fully Discrete Schemes for Fractional Diffusion and Diffusion-Wave Equations with Nonsmooth Data

We consider initial/boundary value problems for the subdiffusion and diffusion-wave equations involving a Caputo fractional derivative in time. We develop two fully discrete schemes based on the piecewise linear Galerkin finite element method in space and convolution quadrature in time with the generating function given by the backward Euler method/second-order backward difference method, and establish error estimates optimal with respect to the regularity of problem data. These two schemes are first- and second-order accurate in time for both smooth and nonsmooth data. Extensive numerical experiments for two-dimensional problems confirm the convergence analysis and robustness of the schemes with respect to data regularity.

FINITE DIFFERENCE/<i>H</i><sup>1</sup>-GALERKIN MFE PROCEDURE FOR A FRACTIONAL WATER WAVE MODEL

In this article, an H1-Galerkin mixed finite element (MFE) method for solving the time fractional water wave model is presented. First-order backward Euler difference method and L1 formula are applied to approximate integer derivative and Caputo fractional derivative with order 1/2, respectively, and H1-Galerkin mixed finite element method is used to approximate the spatial direction. The analysis of stability for fully discrete mixed finite element scheme is made and the optimal space-time orders of convergence for two unknown variables in both H1-norm and L2-norm are derived. Further, some computing results for a priori analysis and numerical figures based on four changed parameters in the studied problem are given to illustrate the effectiveness of the current method.

The solution of Riccati type differential equation by means of variable order variable stepsize backward difference method

In this article a variable order variable step size technique in backwards difference form is used to solve nonlinear Riccati differential equations directly. The method proposed requires calculating the integration coefficients only once at the beginning, in contrast to current divided difference methods which calculate integration coefficients at every step change. Numerical results will show that the variable order variable step size technique reduces computational cost in terms of total steps without effecting accuracy.

금속배관 건전성 감시를 위한 배열형 펄스와전류 탐촉자의 설계 및 배열신호 제안

Abstract An array type probe for monitoring metal tubes is proposed in this paper which utilizes peak value and peak time of a pulsed eddy current(PEC) signal. The probe consists of an array of encircling coils along a tube and the outside of coils is shielded by ferrite to prevent source magnetic fields from directly affecting sensor signals since it is the magnetic fields produced by eddy currents that reflect the condition of metal tubes. The positions of both exciter and sensor coils are consecutively moved automatically so that manual scanning is not necessary. At one position of send-receive coils, peak value and peak time are extracted from a sensor PEC signal and these data are accumulated for all positions to form an array type peak value signal and an array type peak time signal. Numerical simulation was performed using the backward difference method in time and the finite element method for spatial analysis. Simulation results showed that peak value increases and the peak appears earlier as the defect depth or length increases. The proposed array signals are shown to be excellent in reflecting the defect location as well as variations of defect depth and length within the array probe.

Positive sequence voltage memory filter for numerical digital relaying applications

A positive sequence voltage memory filter for numerical digital relaying applications is proposed. It is designed based on converting an analogue first-order lowpass filter into an infinite impulse response digital filter using the backward difference method. Its combination with both recursive and non-recursive phasor estimation algorithms is discussed. The obtained results reveal the decaying time constant of the proposed filter transient response is easily set, providing a simple and stable mean to adaptively adjust the positive sequence voltage memory polarisation of numerical relays.

A discontinuous Galerkin method with Lagrange multiplier for hyperbolic conservation laws with boundary conditions

We introduce a discontinuous Galerkin method with Lagrange multiplier (DGLM) to approximate the solution to the hyperbolic conservation laws with boundary conditions. Lagrange multipliers are introduced on the edge/face of the element via weak divergence (Wang and Ye, 2014). The final global system has reduced numbers of unknowns of the standard DG methods. Numerical fluxes from finite volume/difference method are not considered. For the time discretization, backward Euler difference method is used. Stability of the approximate solution is proved in energy norm. Discontinuity of the solution is allowed in the error analysis. Local error estimates of O(hr+12+Δt) with Pr(E) elements (r≥d+12) are derived, where h and Δt are the maximum diameter of the elements and time steps, respectively, and d is the dimension of the spatial domain. The high order approximation is obtained under an appropriate condition on the stabilizing parameter. It is shown that the method preserves the property of the local mass conservation. An explanation on algorithmic aspects is given.

Augmented observer approach for high-impedance haptic system with time delay

Among others measures of performance, a haptic device may be characterized by its behavior in the implementation of a virtual wall in terms of the biggest value of the virtual stiffness leaving stable the device. This value depends on many factors including time delays and the virtual world implementation. The paper proposes a new architecture for this latter one based on the use of an augmented state observer which contributes to improve without additional costs the performance of a haptic device with respect to the standard implementation based on the use of the backward finite difference method. In order to compute the stability boundaries of the device in terms of virtual wall parameters, a new LMI criterion for the stability of discrete-time systems with time-varying delay is given. Some numerical simulation results are included to prove the effectiveness of the proposed approach.

High-order implicit time integration for unsteady turbulent flow simulations

Higher order explicit diagonal Runge–Kutta (EDIRK) schemes are investigated for their applicability to realistic unsteady turbulent flow simulations. A second order, two new third order accurate schemes with different stage order accuracy, and a fourth order accurate scheme are tested and compared to the commonly used implicit second order backward difference (BDF2) method. All methods are implemented in a flow solver for unstructured grids where the non-linear equations from the implicit temporal discretizations are formulated as a steady state problem which is solved by standard techniques for steady flow problems. The main objective for the comparison is the accuracy and efficiency of the methods but the influence of different steady state convergence criteria is also addressed. The time integrators are evaluated on van der Pol’s equations and unsteady laminar and turbulent flow cases in two and three dimensions. The design orders for all schemes are verified for all test cases except the fourth order scheme for a turbulent case which is approximately third order with increasing accuracy as the time step is refined. The second order schemes are more efficient for larger time steps. Due to lower truncation error, the second order EDIRK scheme is equally or more efficient than the BDF2 scheme although one additional implicit stage is involved per time step. The higher order (⩾3) schemes become more competitive with smaller time steps and are often more efficient than the second order schemes.

A 2N order compact finite difference method for solving the generalized regularized long wave (GRLW) equation

The generalized regularized long wave (GRLW) equation is solved by fully different numerical scheme. The equation is discretized in space by 2N order compact finite difference method and in time by a backward finite difference method. At the inner and the boundary nodes, the first and the second order derivatives with 2N order of accuracy are obtained. To determine the conservation properties of the GRLW equation three invariants of motion are evaluated. The single solitary wave and the interaction of two and three solitary waves are presented to validate the efficiency and the accuracy of the proposed scheme.

A novel meshless local Petrov–Galerkin method for dynamic coupled thermoelasticity analysis under thermal and mechanical shock loading

A meshless local Petrov–Galerkin method (MLPG) based on the moving Kriging interpolation is further developed for two-dimensional linear dynamic coupled thermoelasticity problems under thermal and mechanical shock loading. In this method the moving Kriging interpolation is employed to construct the shape functions, and the Heaviside step function is applied in the local weak-form as the test functions. The equations of motion and transient heat conduction equations of the coupled thermoelasticity interact on each other. So these equations must be solved simultaneously. In this paper the backward difference method is applied to approximate the time derivatives. The main idea of treating the time-dependent problem is based on the reduction procedure of the equations of motion, in which the displacements, velocities and temperature are taken as the field variables. Then a unified explicit expression of the system of differential algebraic equations is obtained. Numerical examples are presented to illustrate the accuracy of this method.