Third-order Error Research Articles

A well-balanced lattice Boltzmann model for the depth-averaged advection–diffusion equation with variable water depth

A well-balanced lattice Boltzmann model for the depth-averaged advection–diffusion equation is developed to solve the solute transport problem in shallow water flows with obvious gradient changes in water depth. The flow field is simulated by the lattice Boltzmann model for shallow water equations with the central moments (CMs) collision operator and D2Q9 lattice pattern. In this model, the diffusion term of the depth-averaged advection–diffusion equation is divided into two parts by the product rule, one part of which is combined into the advection term to produce the pseudo velocity. The relaxation time is independent of the diffusion coefficient, and the remaining value of (3+3)∕6 derived from the Chapmann–Enskog expansion reduces the third-order error. According to the different treatments of the water depth gradient in the equilibrium distribution function, center scheme and linked scheme are developed. Both schemes can ensure the conservation of the solute mass. Numerical tests show that these two schemes give predictions in good agreement, but the linked scheme works better at the breakpoint. Comparisons with the reference indicate that this model is advantageous with respect to both stability and efficiency.

An Observer-Based Neural Adaptive $PID^2$ Controller for Robot Manipulators Including Motor Dynamics With a Prescribed Performance

This article proposes a novel prescribed performance-based neural adaptive control scheme for robot manipulators including motor dynamics under model uncertainties without velocity, acceleration, and input current measurements. The prescribed performance function approach is used to transform a constrained tracking problem of the robot model including motor dynamics into an unconstrained third-order error model in Euler-Lagrange form which inherits all properties of the robot dynamics. Then, a projection-type neural adaptive PID <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> controller (a PID controller with the second-order derivative) in conjunction with a velocity-acceleration observer is proposed. Lyapunov's direct method is used to prove that the tracking and state observation errors are semiglobally uniformly ultimately bounded and converge to a small ball around the origin with a prescribed overshoot/undershoot, convergence rate, and final tracking accuracy. Finally, simulation, experimental results on a SCARA robot and comparative studies verify that the proposed controller is effective for the joint position trajectory tracking of robot manipulators in the industrial automation with minimum measurement and hardware requirements.

Parameters estimation and imaging of moving target for frequency-modulated continuous-wave synthetic-aperture radar system

According to frequency-modulated continuous-wave synthetic-aperture radar, we establish the signal model of a moving target and propose an algorithm for target parameters estimation and imaging. The key procedures of moving target imaging include velocities estimation and range migration correction. Compared with conventional methods, the effect introduced from the along-track velocity is taken into account during the range migration correction to improve the correction accuracy. Moreover, fractional Fourier transform is adopted to estimate the Doppler parameters. To summarize, the proposed method has three advantages. First, by carrying out rough and precise estimation separately, both accurate velocity estimation and better robustness are obtained. Second, this method is available for moving targets with Doppler ambiguity. Third, the third-order phase error derived from moving target is taken into account to improve the azimuth compression result. Finally, a complete flowchart of the proposed method is presented and is verified by both simulation and real data.

Data Fusion for Indoor Mobile Robot Positioning Based on Tightly Coupled INS/UWB

This paper proposes a novel sensor fusion approach using Ultra Wide Band (UWB) wireless radio and an Inertial Navigation System (INS), which aims to reduce the accumulated error of low-cost Micro-Electromechanical Systems (MEMS) Inertial Navigation Systems used for real-time navigation and tracking of mobile robots in a closed environment. A tightly-coupled model of INS/UWB is established within the integrated positioning system. A two-dimensional kinematic model of the mobile robot based on kinematics analysis is then established, and an Auto-Regressive (AR) algorithm is used to establish third-order error equations of the gyroscope and the accelerometer. An Improved Adaptive Kalman Filter (IAKF) algorithm is proposed. The orthogonality judgment method of innovation is used to identify the “outliers”, and a covariance matching technique is introduced to judge the filter state. The simulation results show that the IAKF algorithm has a higher positioning accuracy than the KF algorithm and the UWB system. Finally, static and dynamic experiments are performed using an indoor experimental platform. The results show that the INS/UWB integrated navigation system can achieve a positioning accuracy of within 0·24 m, which meets the requirements for practical conditions and is superior to other independent subsystems.

Analysis of force treatment in the pseudopotential lattice Boltzmann equation method.

In this paper, different force treatments are analyzed in detail for a pseudopotential lattice Boltzmann equation (LBE), and the contribution of third-order error terms to pressure tensor with a force scheme is analyzed by a higher-order Chapman-Enskog expansion technique. From the theoretical analysis, the performance of the original force treatment of Shan-Chen (SC), Ladd, Guo etal., and the exact difference method (EDM) are ɛ_{Ladd}<ɛ_{Guo}<ɛ_{EDM}≤ɛ_{SC} with the relaxation time τ≥1, while ɛ_{Ladd}<ɛ_{Guo}<ɛ_{SC}<ɛ_{EDM} with τ<1; here ɛ is a parameter related to the mechanical stability and the subscripts are the corresponding force scheme. To be consistent with the thermodynamic theory, a force term is introduced to modify the coefficients in the pressure tensor. Some numerical simulations are conducted to show that the predictions of modified force treatment of the pseudopotential LBE are all in good agreement with the analytical solution and other predictions.

Influence of Ionospheric Irregularities on GNSS Remote Sensing

We have used numerical simulation to study the effects of ionospheric irregularities on accuracy of global navigation satellite system (GNSS) measurements, using ionosphere-free (in atmospheric research) and geometry-free (in ionospheric research) dual-frequency phase combinations. It is known that elimination of these effects from multifrequency GNSS measurements is handi-capped by diffraction effects during signal propagation through turbulent ionospheric plasma with the inner scale being smaller than the Fresnel radius. We demonstrated the possibility of reducing the residual ionospheric error in dual-frequency GNSS remote sensing in ionosphere-free combination by Fresnel inversion. The inversion parameter, the distance to the virtual screen, may be selected from the minimum of amplitude fluctuations. This suggests the possibility of improving the accuracy of GNSS remote sensing in meteorology. In the study of ionospheric disturbances with the aid of geometry-free combination, the Fresnel inversion eliminates only the third-order error. To eliminate the random TEC component which, like the measured average TEC, is the first-order correction, we should use temporal filtering (averaging).

Trajectory Tracking for Nonholonomic Mobile Robots based on Extended Models

AbstractThis paper offers a novel approach to circumvent some problems derived from the nonholonomic structure of a two-wheeled mobile robot. It is shown that suitably extending a well-known third-order error model, new kinematics are found that allow direct Lyapunov-based control design as well as straightforward PDC control for its Takagi-Sugeno form. In the latter case, controllability issues are overcome thanks to the proposed design which decouples the uncontrollable mode from the rest of the system. Illustrative examples are provided that show the effectiveness of this technique.

Exact algebraization of the signal equation of spoiled gradient echo MRI

The Ernst equation for Fourier transform nuclear magnetic resonance (MR) describes the spoiled steady-state signal created by periodic partial excitation. In MR imaging (MRI), it is commonly applied to spoiled gradient-echo acquisition in the steady state, created by a small flip angle α at a repetition time TR much shorter than the longitudinal relaxation time T1. We describe two parameter transformations of α and TR/T1, which render the Ernst equation as a low-order rational function. Computer algebra can be readily applied for analytically solving protocol optimization, as shown for the dual flip angle experiment. These transformations are based on the half-angle tangent substitution and its hyperbolic analogue. They are monotonic and approach identity for small α and small TR/T1 with a third-order error. Thus, the exact algebraization can be readily applied to fast gradient echo MRI to yield a rational approximation in α and TR/T1. This reveals a fundamental relationship between the square of the flip angle and TR/T1 which characterizes the Ernst angle, constant degree of T1-weighting and the influence of the local radio-frequency field.

The second-order Rytov approximation and residual error in dual-frequency satellite navigation systems

The second-order Rytov approximation has been used to determine ionospheric corrections for the phase path up to third order. We show the transition of the derived expressions to previous results obtained within the ray approximation using the second-order approximation of perturbation theory by solving the eikonal equation. The resulting equation for the phase path is used to determine the residual ionospheric first-, second- and third-order errors of a dual-frequency navigation system, with diffraction effects taken into account. Formulas are derived for the biases and variances of these errors, and these formulas are analyzed and modeled for a turbulent ionosphere. The modeling results show that the third-order error that is determined by random irregularities can be dominant in the residual errors. In particular, the role of random irregularities is enhanced for small elevation angles. Furthermore, in the case of small angles the role of diffraction effects increases. It is pointed out that a need to pass on to diffraction formulas arises when the Fresnel radius exceeds the inner scale of turbulence.

The reduced effect of a single scattering with a low-mass particle via a point interaction

In this article, we study a second-order expansion for the effect induced on a large quantum particle which undergoes a single scattering with a low-mass particle via a repulsive point interaction. We give an approximation with third-order error in λ to the map G → Tr 2 [ ( I ⊗ ρ ) S λ ∗ ( G ⊗ I ) S λ ] , where G ∈ B ( L 2 ( R n ) ) is a heavy-particle observable, ρ ∈ B 1 ( R n ) is the density matrix corresponding to the state of the light particle, λ = m M is the mass ratio of the light particle to the heavy particle, S λ ∈ B ( L 2 ( R n ) ⊗ L 2 ( R n ) ) is the scattering matrix between the two particles due to a repulsive point interaction, and the trace is over the light-particle Hilbert space. The third-order error is bounded in operator norm for dimensions one and three using a weighted operator norm on G.

Studies on Nodal Integral Methods for the Convection-Diffusion Equation

The computational efficiencies of two nodal integral methods for the numerical solution of linear convection-diffusion equations are studied. Although the first, which leads to a second-order spatial truncation error, has been reported earlier, it is reviewed in order to lead logically to the development here of the second, which has a third-order error. This third-order nodal integral method is developed by introducing an upwind approximation for the linear terms in the “pseudo-sources” that appear in the transverse-averaged equations introduced in the formulation of nodal integral methods. This upwind approximation obviates the need to develop and solve additional equations for the transverse-averaged first moments of the unknown, as would have to be done in a more straightforwardly developed higher-order nodal integral method. The computational efficiencies of the second-order nodal method and the third-order nodal method—of which there are two versions: one, a full third-order method and the other, which uses simpler second-order equations near the boundaries—are compared with those of both a very traditional method and a recently developed state-of-the-art method. Based on the comparisons reported here for a challenging recirculating flow benchmark problem it appears that, among the five methods studied, the second-order nodal integral method has the highest computational efficiency (the lowest CPU computing times for the same accuracy requirements) in the practical 1% error regime.

Compact scheme on a nonuniform grid for the ψ–ω-system of Navier-Stokes equations

- This paper deals with the construction and the study of a compact scheme on a nonuniform rectangular grid for the quasilinear system of Navier-Stokes equations in the stream functionvorticity variables. The scheme is written on the 3 × 3 point stencil. In the general case it has a thirdorder error with respect to the steps on a spatial grid, which becomes a fourth-order error on a uniform grid. We obtain a stability criterion based on the requirement for nonnegativeness of the scheme coefficients. It limits the extent to which mesh cells can be elongated and the ratio of neighbouring step sizes in each coordinate direction.

The Complete Zeldovich Approximation.

We have developed a generalization of the Zeldovich approximation that is exact in a wide variety of situations, including planar, spherical, and cylindrical symmetries. We have shown that this generalization, which we call the complete Zeldovich approximation (CZA), is exact to second order at an arbitrary point within any field. For Gaussian fields, the third-order error has been obtained and shown to be very small. For statistical purposes, the CZA leads to results exact to the third order.

FOUR-POINT INTERPOLATION SCHEMES FOR CONVECTIVE FLUXES APPROXIMATION

Two four-point interpolation schemes, FPTOI and FPFOI, have been proposed for approximating convective fluxes. In these new schemes, three points in the upstream direction and one in the downstream direction are used for interpolation. The FPTOI scheme has third-order accuracy and smaller odd truncation error terms (seventh derivative), whereas the FPFOI scheme has fourth-order accuracy and a larger critical cell Peclet number (16/5) compared with the QUICK scheme. In order to assess their capabilities, the two schemes are applied to three test problems, one linear and two nonlinear cases, all involving the simultaneous presence of convective dominance and flow-to-grid skewness. The two new schemes have high-order accuracy and further have demonstrated, in computations of flow cases, the ability to preserve better numerical stability behaviors in comparison with the QUICK scheme.

Accuracy analysis of the split-step Fourier propagator: Implications for seismic modeling and migration

AbstractThe split-step Fourier propagator is a one-way wave propagation method that has been widely used to simulate primary forward and backward (reflected) deterministic/random wave propagation due to its fast computational speed and limited computer memory requirement. The method is useful for rapid modeling of seismic-wave propagation in heterogeneous media where forward scattered waveforms can be considered to be dominant or reverberations can be ignored. The method is based on a solution to the one-way wave equation that requires expanding the square root of an operator and splitting of the resulting noncommutative operators to allow calculation by transferring wave fields between the space and wavenumber domains. Previous analysis of the accuracy of the method has focused on the error related to only a portion of the approximations involved in the propagator. To better understand the accuracy of the propagator, we present a complete formal and numerical accuracy analyses. Our formal analysis indicates that the dominant error of the propagator increases as the first order in the marching interval. We show that nonsymmetrically and symmetrically split-step marching solutions have the same first-order error term. Their second- and third-order error terms are similar. Therefore, the differences between the accuracy of different split-step marching solutions are insignificant. This conclusion is confirmed by our numerical tests. The relation among the phase error of the split-step Fourier propagator, relative velocity perturbation, and propagation angle is numerically studied. The results suggest that the propagator is accurate for up to a 60° propagation angle from the main propagation direction for media with small relative velocity perturbations (10%).

Local second-order boundary methods for lattice Boltzmann models

A new way to implement solid obstacles in lattice Boltzmann models is presented. The unknown populations at the boundary nodes are derived from the locally known populations with the help of a second-order Chapman-Enskog expansion and Dirichlet boundary conditions with a given momentum. Steady flows near a flat wall, arbitrarily inclined with respect to the lattice links, are then obtained with a third-order error. In particular, Couette and Poiseuille flows are exactly recovered without the Knudsen layers produced for inclined walls by the bounce back condition.

A new equation for calculating wave loads on offshore structures

This paper derives an equation for the potential-flow wave loading on a lattice-type offshore structure moving partially immersed in waves. It is for the limiting case of small lattice-member diameter, and deals entirely in member-centreline fluid properties, so that it can be applied computationally by a simple ‘stick model’ computer program. This field is currently served by a simple two-term semiempirical formula ‘Morison's equation’: the new equation is effectively a replacement for the Morison inertial term, allowing the Morison drag term (or some refinement of it) to describe exclusively the effects of vorticity, which can in principle be calculated to greater accuracy when isolated in this way.The new equation calculates the potential-flow wave load accurate to second order in wave height, which is a great improvement on ‘Morison's equation’: such results can currently only be sought by very much more complicated and computationally intensive methods, of currently uncertain repeatability. Moreover the third-order error is localized at the free-surface intersection, so the equation remains attractive for fully nonlinear problems involving intermittent immersion of lattice members, which are currently beyond even the most sophisticated of these computationally intensive methods. It is shown that the primary reason for this large contrast in computational efficiency is that the loads are derived from energy considerations rather than direct integration of surface pressures, which requires a lower level of flow detail for a given level of load-calculation accuracy.These improvements must of course be seen against the current levels of uncertainty over the calculation of vorticity-induced loads, which in many applications completely dwarf inaccuracies in potential-flow load calculation. The conditions are accordingly established under which the improvements are comparable to the total wave load predicted by the Morison drag and inertia terms in combination. They are that the lattice member diameter is greater than its length/10, or the relative fluid motion/5, or the structure's motion radius/20, or the wavelength/30: if any one of these conditions is satisfied, the new equation is worthwhile even when used in combination with simple vorticity-induced load calculations from a Morison drag term.

A note on the Petzval field curvature in electron-optical systems

1. The Petzval field curvature produced in a compound glass optical system of axial symmetry is given by a well-known formula. If the system consists of a number of media, of refractive indices n0, n1, n2, …, having spherical faces whose radii of curvature are r0, r1, r2, …, the formula is [see for example, (1) or (2)]and this is in common use amongst those concerned with the design of optical equipment. The analogous integral expression in the electron optical case has not, however, received the attention it deserves, in spite of the developments in electron optics during the past decade. In 1935 Glaser(3) presented the third-order error theory of an axially symmetric electron optical system.