Second-order Error Research Articles

Relaxation Limit and Initial-Layers for a Class of Hyperbolic-Parabolic Systems

We consider a class of hyperbolic-parabolic systems with small diffusion terms and stiff sources. Existence of solutions to the Cauchy problem with ill prepared initial data is established by using composite expansions including initial-layer correctors and a convergence-stability lemma. New multitime expansions are introduced and lead to second-order error estimates between the composite expansions and the solution. Reduced equilibrium systems of second-order accuracy are also investigated as well as initial-layers of Chapman-Enskog expansions.

Stitching test of large flats by using two orthogonally arranged wavefront interferometers.

The most challenging problem in the stitching test of large flats with a small-aperture interferometer is the accumulation effect of the second-order error. As it is approximately enlarged by the square of the ratio of full aperture size to subaperture size, a very small amount of the second-order error in the reference surface of a transmission flat can be accumulated and gets far from negligible when the subaperture is far smaller than the full aperture. We present here a solution by using two orthogonally arranged wavefront interferometers. One is responsible for a subaperture test and the other for the simultaneous measurement of relative tilts. Because the accumulation effect originates from the lateral shift of the second-order error, only the tilt along the subaperture scanning direction needs to be measured accurately. It is no longer determined by stitching optimization instead to avoid the error accumulation. Piston and tilt perpendicular to the scanning direction are still determined by stitching optimization. The method is experimentally verified and compared to the stitching test with the reference surface error calibrated out, both referenced to the full aperture test result obtained with a 24-inch interferometer.

An algorithm for real-time forward kinematics of 6-degree-of-freedom parallel mechanisms

This article proposes a forward kinematics algorithm based on closed-loop feedback solution for real-time pose estimation of 6-degree-of-freedom parallel mechanisms. First, a feedback forward kinematics algorithm with second-order error dynamics built in joint space is presented and its convergence is analyzed. Taking a Stewart platform as an example, fast convergence and high solving accuracy of the algorithm are shown in simulation. However, while applying the algorithm to forward kinematics of a 6-degree-of-freedom redundantly actuated parallel mechanisms, solving errors will be growing slowly and non-convergent, which is caused by redundancy character of the parallel mechanisms. Then a modified algorithm, of which second-order error dynamics is built in Cartesian space, is developed. Considering velocity information is usually difficult to measure or estimate accurately in reality, the performance of feedback forward kinematics algorithm without joint velocity feedforward is also evaluated. Finally, the algorithm is implemented into real-time degree of freedom control of the redundant parallel mechanisms which is calibrated by a coordinate measuring machine. Experiment results show that the proposed algorithm can solve the pose with relatively high accuracy under static conditions and is effective and feasible for real-time pose estimation of parallel mechanisms.

Dominant height growth and dynamic site index models for Crimean pine in the Kastamonu–Taşköprü region of Turkey

Some dynamic site index models based on the generalized algebraic difference approach (GADA) were fitted for Crimean pine (Pinus nigra J.F. Arnold subsp. pallasiana (Lamb.) Holmboe) stands in Taşköprü, Turkey. Data were obtained from 132 dominant trees representing the wide range of site quality in the region. Nonlinear regression analysis and a second-order continuous-time autoregressive error structure were applied. After autoregressive modeling, the fitted models were evaluated both statistically and graphically. The best results were obtained with the dynamic site index model derived from the Bertalanffy–Richards base equation, accounting for about the 99% of the total variance in height–age relationships in dominant trees, with an Akaike information criterion (AIC) value of 119.55 and root mean square error (RMSE) of 0.5446. The selected base-age invariant dynamic site index curves provided the polymorphism with multiple asymptotes and other realistic height growth patterns.

A Numerical Approximation Structured by Mixed Finite Element and Upwind Fractional Step Difference for Semiconductor Device with Heat Conduction and Its Numerical Analysis

AbstractA coupled mathematical system of four quasi-linear partial differential equations and the initial-boundary value conditions is presented to interpret transient behavior of three dimensional semiconductor device with heat conduction. The electric potential is defined by an elliptic equation, the electron and hole concentrations are determined by convection-dominated diffusion equations and the temperature is interpreted by a heat conduction equation. A mixed finite element approximation is used to get the electric field potential and one order of computational accuracy is improved. Two concentration equations and the heat conduction equation are solved by a fractional step scheme modified by a second-order upwind difference method, which can overcome numerical oscillation, dispersion and computational complexity. This changes the computation of a three dimensional problem into three successive computations of one-dimensional problem where the method of speedup is used and the computational work is greatly shortened. An optimal second-order error estimate in L2 norm is derived by prior estimate theory and other special techniques of partial differential equations. This type of parallel method is important in numerical analysis and is most valuable in numerical application of semiconductor device and it can successfully solve this international famous problem.

Error correction based on micro-scanning preprocessing for an optical micro-scanning thermal microscope imaging system

In recent years, various thermal microscope imaging systems have been developed to meet the demands of micro-thermal analysis for large-scale integrated circuits, biomedical, science, and research fields. However, conventional thermal microscope imaging systems, which use cooled infrared detectors are heavy and expensive. In order to solve this problem, we developed a thermal microscope imaging system based on an uncooled infrared detector. However, the spatial resolution of the thermal microscope imaging system based on an uncooled infrared detector is low. With optical micro-scanning technology, the spatial resolution of the thermal microscope imaging system can be increased without increasing the detector dimension or reducing the detector unit size. In order to improve its spatial resolution, a micro-scanning system based on optical plate rotation was developed, and an optical microscanning thermal microscope imaging system was obtained after the integrated design. Due to environmental factors, mechanical vibration, alignment error and other factors, there is micro-scanning error in the designed micro-scanning thermal microscope imaging system. The four low-resolution images collected by micro-scanning thermal microscope imaging system are not standard down-sampled images. The quality of the image interpolated directly by four collected images is reduced and the performance of the micro-scanning system isn’t fully exploited. Therefore, based on the proposed second-order oversampling reconstruction micro-scanning error correction algorithm and the new edge directed interpolation algorithm, a new micro-scanning error correction technique is proposed. Simulations and experiments show that the proposed technique can effectively reduce optical micro-scanning error, improve the systems spatial resolution and optimize the effect of the imaging system. It can be applied to other electro-optical imaging systems to improve their spatial resolution.

Geometric error modeling and compensation for five-axis CNC gear profile grinding machine tools

The relative position and orientation deviation between grinding wheel and workpiece caused by geometric errors affect the machining accuracy of five-axis CNC gear profile grinding machine tools directly. Therefore, geometric error modeling and compensation are presented according to homogeneous transformation and differential motion matrix based on the multi-body system theory for the accuracy enhancement of the machine tools. Firstly, the open kinematic chain and the ideal homogeneous transformation matrix from workpiece to grinding wheel are established according to the topological structure of the CFXZAY-type, five-axis gear profile grinding machine tools and the basic homogeneous transformation matrix between coordinate frames. Secondly, the homogenous transformation matrices of linear pairs and rotary pairs with geometric errors are calculated, and the relationships of the error propagation from workpiece to grinding wheel are acquired as well on the basis of 37 geometric error components analysis. The geometric error models including position and orientation errors of grinding wheel in workpiece coordinate system are obtained with matrix multiplication by using small-angle approximation and ignoring the second-order and high-order error terms. Then, the Jacobian is obtained by using transforming differential motion matrix of each motion axis to compensate the integrated error components of the grinding wheel, which can make the compensation effective and convenient with using the corresponding homogeneous transformation matrix. Finally, error measurement, error compensation, and machining experiments are carried out on a five-axis CNC gear profile grinding machine tool SKMC-1200W/10 to verify the applicability and effectiveness of the proposed error modeling, error compensation, and research approach.

A Block-Centered Finite Difference Method for Slightly Compressible Darcy–Forchheimer Flow in Porous Media

A block-centered finite difference method is introduced to solve an initial and boundary value problem for a nonlinear parabolic equation to model the slightly compressible flow in porous media, in which the velocity–pressure relation is described by Darcy–Forchheimer’s Law. The method can be thought as the lowest order Raviart–Thomas mixed element method with proper quadrature formulation. By using the method the velocity and pressure can be approximated simultaneously. We established the second-order error estimates for pressure and velocity in proper discrete norms on non-uniform rectangular grid. No time-step restriction is needed for the error estimates. The numerical experiments using the scheme show that the convergence rates of the method are in agreement with the theoretical analysis.

Convergence error estimates of the Crank-Nicolson scheme for solving decoupled FBSDEs

The Crank-Nicolson (short for C-N) scheme for solving {\it backward stochastic differential equation} (BSDE), driven by Brownian motions, was first developed by the authors W. Zhao, L. Chen and S. Peng [SIAM J. Sci. Comput., 28 (2006), 1563--1581], and numerical experiments showed that the accuracy of this C-N scheme was of second order for solving BSDE. This C-N scheme was extended to solve decoupled {\it forward-backward stochastic differential equations} (FBSDEs) by W. Zhao, Y. Li and Y. Fu [Sci. China. Math., 57 (2014), 665--686], and it was numerically shown that the accuracy of the extended C-N scheme was also of second order. To our best knowledge, among all one-step (two-time level) numerical schemes with second-order accuracy for solving BSDE or FBSDEs, such as the ones in the above two papers and the one developed by the authors D. Crisan and K. Manolarakis [Ann. Appl. Probab., 24, 2 (2014), 652--678], the C-N scheme is the simplest one in applications. The theoretical proofs of second-order error estimates reported in the literature for these schemes for solving decoupled FBSDEs did not include the C-N scheme. The purpose of this work is to theoretically analyze the error estimate of the C-N scheme for solving decoupled FBSDEs. Based on the Taylor and It\^o-Taylor expansions, the Malliavin calculus theory (e.g., the multiple Malliavin integration-by-parts formula), and our new truncation error cancelation techniques, we rigorously prove that the strong convergence rate of the C-N scheme is of second order for solving decoupled FBSDEs, which fills the gap between the second-order numerical and theoretical analysis of the C-N scheme.

Cell-Centered Finite Difference Method for the One-Dimensional Forchheimer Laws

Cell-centered finite difference method is introduced to solve the one-dimensional Forchheimer laws modeling incompressible fluids in porous media. Using this method, velocity and pressure can be approximated at the same time. Second-order accuracy error estimates for velocity and pressure are established. Numerical experiments are carried out to validate the convergence rates and show the efficiency.

Fully decoupled monolithic projection method for natural convection problems

To solve time-dependent natural convection problems, we propose a fully decoupled monolithic projection method. The proposed method applies the Crank–Nicolson scheme in time and the second-order central finite difference in space. To obtain a non-iterative monolithic method from the fully discretized nonlinear system, we first adopt linearizations of the nonlinear convection terms and the general buoyancy term with incurring second-order errors in time. Approximate block lower-upper decompositions, along with an approximate factorization technique, are additionally employed to a global linearly coupled system, which leads to several decoupled subsystems, i.e., a fully decoupled monolithic procedure. We establish global error estimates to verify the second-order temporal accuracy of the proposed method for velocity, pressure, and temperature in terms of a discrete l2-norm. Moreover, according to the energy evolution, the proposed method is proved to be stable if the time step is less than or equal to a constant. In addition, we provide numerical simulations of two-dimensional Rayleigh–Bénard convection and periodic forced flow. The results demonstrate that the proposed method significantly mitigates the time step limitation, reduces the computational cost because only one Poisson equation is required to be solved, and preserves the second-order temporal accuracy for velocity, pressure, and temperature. Finally, the proposed method reasonably predicts a three-dimensional Rayleigh–Bénard convection for different Rayleigh numbers.

Variationally Derived Discontinuity Factors for the Asymptotic Homogenized Diffusion Equation

In this work, we derive and test variational discontinuity factors (DFs) for the asymptotic homogenized diffusion equation. We begin with a functional for optimally estimating the reactor multiplication factor, then introduce asymptotic expressions for the forward and adjoint angular fluxes, and finally require that all first-order error terms vanish. In this way, the reactor multiplication factor can be calculated with second-order error. The analysis leads to (1) an alternate derivation of the asymptotic homogenized diffusion equation, (2) variational boundary conditions for large periodic systems, and (3) variational DFs to be applied between adjacent periodic regions (e.g., fuel assemblies). Numerical tests show that applying the variational DFs to the asymptotic homogenized diffusion equation yields the most accurate estimates of the reactor multiplication factor compared to other DFs for a wide range of problems. However, the resulting assembly powers are less accurate than those obtained using other DFs for many realistic problems.

Numerical method of mixed finite volume-modified upwind fractional step difference for three-dimensional semiconductor device transient behavior problems

Transient behavior of three-dimensional semiconductor device with heat conduction is described by a coupled mathematical system of four quasi-linear partial differential equations with initial-boundary value conditions. The electric potential is defined by an elliptic equation and it appears in the following three equations via the electric field intensity. The electron concentration and the hole concentration are determined by convection-dominated diffusion equations and the temperature is interpreted by a heat conduction equation. A mixed finite volume element approximation, keeping physical conservation law, is used to get numerical values of the electric potential and the accuracy is improved one order. Two concentrations and the heat conduction are computed by a fractional step method combined with second-order upwind differences. This method can overcome numerical oscillation, dispersion and decreases computational complexity. Then a three-dimensional problem is solved by computing three successive one-dimensional problems where the method of speedup is used and the computational work is greatly shortened. An optimal second-order error estimate in L2 norm is derived by using prior estimate theory and other special techniques of partial differential equations. This type of mass-conservative parallel method is important and is most valuable in numerical analysis and application of semiconductor device.

Efficient dimension-by-dimension higher order finite-volume methods for a Cartesian grid with cell-based refinement

Abstract A new higher-order scheme for hyperbolic systems of conservation law on Cartesian grids with cell-based refinement is proposed. The scheme is based on the finite volume methods, which admits a simple formulation around the hanging-nodes. Dimension-by-dimension reconstruction is applied to utilize the advantage of Cartesian grids. A quadrature modification flux (QMF) is introduced, which modifies the second-order error term of the flux integration over the cell-interface. The QMF can be evaluated using the state variables and the first derivatives stored in the face-left and right cells. There is no need to calculate the point-value or to use multi-points flux quadrature. In addition, a two-step reconstruction method is introduced to achieve a higher-order reconstructed value. The scheme achieves fourth-order accuracy where the grids are locally uniform, and retains second-order accuracy around the hanging-nodes. The accuracy and efficiency of the scheme is demonstrated in some example problems: inviscid vortex advection, Shu–Osher problem, double Mach reflection, and transonic flow around the NACA 0012 airfoil. The present scheme is demonstrated to function stably and consistently.

Symplectic fourth-order maps for the collisionalN-body problem

We study analytically and experimentally certain symplectic and time-reversible N-body integrators which employ a Kepler solver for each pair-wise interaction, including the method of Hernandez & Bertschinger (2015). Owing to the Kepler solver, these methods treat close two-body interactions correctly, while close three-body encounters contribute to the truncation error at second order and above. The second-order errors can be corrected to obtain a fourth-order scheme with little computational overhead. We generalise this map to an integrator which employs a Kepler solver only for selected interactions and yet retains fourth-order accuracy without backward steps. In this case, however, two-body encounters not treated via a Kepler solver contribute to the truncation error.

A Bidirectional Fiber-Wireless and Fiber-IVLLC Convergence System with a Dual-Polarization Modulation Scheme and an MZM-OEO-Based BLS

A bidirectional fiber-wireless and fiber-invisible laser light communication (IVLLC) convergence system that adopts a dual-polarization modulation scheme and a Mach–Zehnder modulator (MZM)-optoelectronic oscillator (OEO)-based broadband light source (BLS) for hybrid cable television (CATV)/microwave (MW)/millimeter-wave (MMW)/baseband signal transmission is proposed and experimentally demonstrated. The MZM employed in the MZM-OEO is operated at the minimum transmission point, which results in the format of optical carrier suppression for the y -axis component of the light. Using a dual-polarization modulation scheme, the optical carrier modulated with CATV signal ( x -polarization) and the optical sidebands modulated with MW and MMW data signals ( y -polarization) are separated and polarized orthogonally automatically. Through an in-depth observation, good carrier-to-noise ratio, composite second-order, composite triple-beat, and bit error rate performances are obtained over a 40-km single-mode fiber and a 10-m RF/100-m optical wireless transport. Such a bidirectional fiber-wireless and fiber-IVLLC convergence system is a notable option; it would be attractive for providing broadband heterogeneous services such as CATV, Internet, and big data services.

Numerical Analysis of Orbital Perturbation Effects on Inclined Geosynchronous SAR

The geosynchronous synthetic aperture radar (GEO SAR) is susceptible to orbit perturbations, leading to orbit drifts and variations. The influences behave very differently from those in low Earth orbit (LEO) SAR. In this paper, the impacts of perturbations on GEO SAR orbital elements are modelled based on the perturbed dynamic equations, and then, the focusing is analyzed theoretically and numerically by using the Systems Tool Kit (STK) software. The accurate GEO SAR slant range histories can be calculated according to the perturbed orbit positions in STK. The perturbed slant range errors are mainly the first and second derivatives, leading to image drifts and defocusing. Simulations of the point target imaging are performed to validate the aforementioned analysis. In the GEO SAR with an inclination of 53° and an argument of perigee of 90°, the Doppler parameters and the integration time are different and dependent on the geometry configurations. Thus, the influences are varying at different orbit positions: at the equator, the first-order phase errors should be mainly considered; at the perigee and apogee, the second-order phase errors should be mainly considered; at other positions, first-order and second-order exist simultaneously.

Taming systematic uncertainties at the LHC with the central limit theorem

We study the simplifications occurring in any likelihood function in the presence of a large number of small systematic uncertainties. We find that the marginalisation of these uncertainties can be done analytically by means of second-order error propagation, error combination, the Lyapunov central limit theorem, and under mild approximations which are typically satisfied for LHC likelihoods. The outcomes of this analysis are i) a very light treatment of systematic uncertainties ii) a convenient way of reporting the main effects of systematic uncertainties, such as the detector effects occurring in LHC measurements.

Bidirectional fiber-wireless and fiber-IVLLC integrated system based on polarization-orthogonal modulation scheme.

A bidirectional fiber-wireless and fiber-invisible laser light communication (IVLLC) integrated system that employs polarization-orthogonal modulation scheme for hybrid cable television (CATV)/microwave (MW)/millimeter-wave (MMW)/baseband (BB) signal transmission is proposed and demonstrated. To our knowledge, it is the first one that adopts a polarization-orthogonal modulation scheme in a bidirectional fiber-wireless and fiber-IVLLC integrated system with hybrid CATV/MW/MMW/BB signal. For downlink transmission, carrier-to-noise ratio (CNR), composite second-order (CSO), composite triple-beat (CTB), and bit error rate (BER) perform well over 40-km single-mode fiber (SMF) and 10-m RF/50-m optical wireless transport scenarios. For uplink transmission, good BER performance is obtained over 40-km SMF and 50-m optical wireless transport scenario. Such a bidirectional fiber-wireless and fiber-IVLLC integrated system for hybrid CATV/MW/MMW/BB signal transmission will be an attractive alternative for providing broadband integrated services, including CATV, Internet, and telecommunication services. It is shown to be a prominent one to present the advancements for the convergence of fiber backbone and RF/optical wireless feeder.

First- and second-order error estimates in Monte Carlo integration

In Monte Carlo integration an accurate and reliable determination of the numerical integration error is essential. We point out the need for an independent estimate of the error on this error, for which we present an unbiased estimator. In contrast to the usual (first-order) error estimator, this second-order estimator can be shown to be not necessarily positive in an actual Monte Carlo computation. We propose an alternative and indicate how this can be computed in linear time without risk of large rounding errors. In addition, we comment on the relatively very slow convergence of the second-order error estimate.