Low Order Of Accuracy Research Articles

A comparison study of spatial and temporal schemes for flow and transport problems in fractured media with large parameter contrasts on small length scales

AbstractIn this work, various high-accuracy numerical schemes for transport problems in fractured media are further developed and compared. Specifically, to capture sharp gradients and abrupt changes in time, schemes with low order of accuracy are not always sufficient. To this end, discontinuous Galerkin up to order two, Streamline Upwind Petrov-Galerkin, and finite differences, are formulated. The resulting schemes are solved with sparse direct numerical solvers. Moreover, time discontinuous Galerkin methods of order one and two are solved monolithically and in a decoupled fashion, respectively, employing finite elements in space on locally refined meshes. Our algorithmic developments are substantiated with one regular fracture network and several further configurations in fractured media with large parameter contrasts on small length scales. Therein, the evaluation of the numerical schemes and implementations focuses on three key aspects, namely accuracy, monotonicity, and computational costs.

Discrete LQR and ILQR methods based on high order Runge-Kutta discretizations

In this paper, discrete linear quadratic regulator (DLQR) and iterative linear quadratic regulator (ILQR) methods based on high-order Runge-Kutta (RK) discretizations are proposed for solving linear and nonlinear quadratic optimal control problems respectively. As discovered in Hager (2000) [11], direct approaches with RK discretization are equivalent with indirect approaches based on symplectic partitioned Runge-Kutta (SPRK) integration. In this paper, we will reconstruct this equivalence by the analogue of continuous and discrete dynamic programming. Then, based on the equivalence, we discuss the issue that the internal-stage controls produced by direct approaches may have lower order accuracy than the RK method used. We propose order conditions for internal-stage controls and then demonstrate that third or fourth order explicit RK methods cannot avoid the order reduction phenomenon. To overcome this obstacle, we calculate node control instead of internal-stage controls in DLQR and ILQR methods. Another advantage of our methods is high computational efficiency which comes from the usage of feedback technique. A simple numerical example will illustrate the validity of ILQR in solving nonlinear optimal control problems with high-order RK discretizations. In this paper, we also demonstrate that ILQR is essentially a quasi-Newton method with linear convergence rate.

A parameter uniform numerical scheme for singularly perturbed differential-difference equations with mixed shifts

In this paper, we consider a second-order singularly perturbed differential-difference equations with mixed delay and advance parameters. At first, we approximate the model problem by an upwind finite difference scheme on a Shishkin mesh. We know that the upwind scheme is stable and its solution is oscillation free, but it gives lower order of accuracy. So, to increase the convergence, we propose a hybrid finite difference scheme, in which we use the cubic spline difference method in the fine mesh regions and a midpoint upwind scheme in the coarse mesh regions. We establish a theoretical parameter uniform bound in the discrete maximum norm. To check the efficiency of the proposed methods, we consider test problems with delay, advance and the mixed parameters and the results are in agreement with our theoretical findings.

Distortions of temporal integration and perceived order caused by the interplay between stimulus contrast and duration.

Stimulus contrast and duration effects on visual temporal integration and order judgment were examined in a unified paradigm. Stimulus onset asynchrony was governed by the duration of the first stimulus in Experiment 1, and by the interstimulus interval in Experiment 2. In Experiment 1, integration and order uncertainty increased when a low contrast stimulus followed a high contrast stimulus, but only when the second stimulus was 20 or 30ms. At 10ms duration of the second stimulus, integration and uncertainty decreased. Temporal order judgments at all durations of the second stimulus were better for a low contrast stimulus following a high contrast one. By contrast, in Experiment 2, a low contrast stimulus following a high contrast stimulus consistently produced higher integration rates, order uncertainty, and lower order accuracy. Contrast and duration thus interacted, breaking correspondence between integration and order perception. The results are interpreted in a tentative conceptual framework.

An asymptotic-preserving Monte Carlo method for the Boltzmann equation

In this work, we propose an asymptotic-preserving Monte Carlo method for the Boltzmann equation that is more efficient than the currently available Monte Carlo methods in the fluid dynamic regime. This method is based on the successive penalty method [39], which is an improved BGK-penalization method originally proposed by Filbet and Jin [16]. Here we introduce the Monte Carlo implementation of the method, which, despite its lower order accuracy, is very efficient in higher dimensions or simulating some complicated chemical processes. This method allows the time step independent of the mean free time which is prohibitively small in the fluid dynamic regime. We study some basic properties of this method, and compare it with some other asymptotic-preserving Monte Carlo methods in terms of numerical performance in different regimes, from rarefied to fluid dynamic regimes, and their computational efficiency.

Subgrid-Scale Model for Reducing the Anisotropy of Turbulent Intensity in Channel Flow

Large-eddy simulation (LES) of a turbulent channel flow is carried out using a finite difference method (FDM) with a low order of accuracy. In this method, overestimation of turbulence anisotropy is a typical shortcoming. Since LES with the FDM is usually applied to practical cases, it is desirable to model the subgrid-scale (SGS) stress such that the problem is mitigated. Therefore, in this study two types of eddy-viscosity models are adopted and their influence on numerical results is investigated. Moreover, the redistribution term of turbulent energy is carefully examined through its co-spectrum. It is shown that the value of the high-wavenumber region plays an important role in improving the anisotropy.

A brief history of the geodetic survey of Canada

This is a brief history of how a small group of people carried out its mandate to survey and map Canada in spite of the hostile environment — the second largest country in the world — in a 100 years.The levelling started from Halifax and by 1916 it reached Vancouver completing the coast-to-coast connection, encompassing about 22 000 km distance of which 94% was done on the railway tracks. The adjustment was carried out in 9 steps, it included the tide gauges in both the east and west coast, and was completed in 1928. The final adjustment used about 37 000 km length, from which 30 000 was measured by the GS. The first order re-levelling started after the Second World War and was carried out on the Trans-Canada highway. By 1994 Canada was covered by a levelling network of about 109 747 km. After this date there was no more conventional levelling.The triangulation used cross-braced quadrilaterals, based on the method used in the USA. In 1961 the GS started to use the conventional methods of chains of triangles. The measurements covered mostly the very southern part of Canada. The mapping of middle and northern part of the country had to await until the electronic era, which was practically made for Canada.The lower order accuracy was provided by Shoran, using about the average of 400 km line length for triangulation. Between 1947 and 1957 this method provided 501 measured triangle sides of about 200 000 km in total length and provided 119 base-point covering about 65% of Canada.The other method, the Aerodist — the Tellurometer version of Shoran — provided second order accuracy. The triangulation, using cross-braced quadrilaterals, with 100 km sides gave 219 points and covered about 25% of the land area of Canada. The measurements were done between 1965 and 1973.Recent measurements make use of GPS for geodetic position determination. A 10 year program started in 1987. In 1992 the GSD defined an official geoid to be used in connection with all satellite work.

High-order shock-fitting methods for direct numerical simulation of hypersonic flow with chemical and thermal nonequilibrium

In recent years, much progress has been made in the direct numerical simulation of laminar-turbulent transition of hypersonic boundary layer flow. However, most of the efforts at the direct numerical simulation of transition previously have been focused on the idealized perfect gas flow or “cold” hypersonic flows. For practical problems in hypersonic flows, high-temperature effects of thermal and chemical nonequilibrium are important and cannot be modeled by a perfect gas model. Therefore, it is necessary to include the real gas models in the numerical simulation of hypersonic boundary layer transition in order to accurately predict flow field parameters. Currently most numerical methods for hypersonic flow with thermo-chemical nonequilibrium are based on shock-capturing approach at relatively low order of accuracy. Shock capturing schemes reduce to first-order accuracy near the shock and have been shown to produce spurious oscillations behind curved strong shocks. There is a need to develop new methods capable of simulating nonequilibrium hypersonic flow fields with uniformly high-order accuracy and avoid spurious oscillations near the shock. This paper presents a fifth-order shock-fitting method for numerical simulation of thermal and chemical nonequilibrium in hypersonic flows. The method is developed based on the state-of-the-art real gas models for thermo-chemical nonequilibrium and transport phenomena. Shock-fitting approach is used because it has the advantage of capturing the entire flow field with high-order accuracy and without any oscillations near the shock. The new method has been tested and validated for a number of test cases over a wide span of free stream conditions. The developed method is applied for the study of receptivity of free stream acoustic waves over a blunt cone for hypervelocity flow. Some preliminary results of the computations of the high order shock fitting method for the above mentioned study have also been presented.

Finite difference modelling of rupture propagation with strong velocity-weakening friction

SUMMARY We incorporate rate- and state-dependent friction in explicit finite difference (FD) simulations of mode II dynamic ruptures in elastic media, using the Mimetic Operators Split-Node (MOSN) method, with adjustable order of spatial accuracy (second-, fourth- or mixed-order accurate), including an option that is fourth-order accurate at the fault discontinuity as well as in the elastic volume. At fault points, the rate and state equations combined with the spatially discretized momentum conservation equations form a coupled system of ordinary differential equations (ODEs) for slip velocity and state variable. As a consequence of the rapid damping of velocity perturbations due to the direct effect, this system exhibits numerical stiffness that is inversely proportional to velocity squared. Approximate solutions to this velocity-state system are achieved by two different implicit schemes: (i) a fourth-order Rosenbrock integration of the full system using multiple substeps and (ii) low order integrations (backward Euler and trapezoidal) of the velocity equation, time-staggered with analytic integration of the state equation under the approximation of constant slip velocity over the time step. In assessing the numerical schemes, we use three test problems: ruptures with frictional resistance controlled by (i) a slip evolution law with strong velocity-weakening behaviour at high slip rates, representing thermal weakening due to flash heating of microscopic asperity contacts, (ii) the classic (low-velocity) slip evolution law and (iii) the classic aging evolution law. A convergence analysis is carried out using reference solutions from a spectral boundary integral equation method (BIEM) (a method restricted to homogeneous media, with nominal spectral accuracy in space and second-order accuracy in time for smooth solutions). Errors are measured by root-mean-square differences of fault-plane time histories (slip, slip rate, traction and state). MOSN shows essentially the same convergence rates as BIEM: second-order convergence for slip and state-variable misfits, with slower (but at least first-order) convergence for slip rates and tractions. For a given grid spacing, fourth-order MOSN is as accurate as BIEM for all variables except slip-rate. MOSN-Rosenbrock nominally has fourth-order temporal accuracy for the fault-plane velocity-state ODE integration (compared to lower-order accuracy for the other two MOSN schemes) and therefore provides an important theoretical benchmark. However, it is sensitive to details of the elastic calculation scheme and occasionally its adaptive substepping performs poorly, leading to large excursions from the reference solution. In contrast, MOSN-trapezoidal is robust and reliable, much easier to implement than MOSN-Rosenbrock, and in all cases achieves precision as good as the latter without recourse to substepping. MOSN-Euler has the same advantages as MOSN-trapezoidal, except that its nominal first-order temporal accuracy ultimately leads to larger errors in slip and state variable compared with the higher-order MOSN schemes at sufficiently small grid spacings and time steps.

Performance of a New Binocular Wavefront Aberrometer Based on a Self-imaging Diffractive Sensor

To compare the performance of the Ophthonix Z-View diffractive aberrometer with two different Hartmann-Shack aberrometers. The Ophthonix Z-View was compared with the Alcon LADARWave and VISX WaveScan using 4 model eyes and 68 human eyes. Comparisons using three fixed, single-surface model eyes each with a different wavefront pattern were used to determine higher order accuracy. Lower order accuracy, linearity, and higher order repeatability were tested with a fourth model eye with a movable retinal surface. Manifest refraction spherical equivalent and the calculated spherical equivalent refraction of the aberrometers were compared in 68 human eyes. The Z-View was more accurate with lower noise compared to the WaveScan and LADARWave systems for higher order measurements of the fixed model eyes. Total root-mean-square difference from surface topography derived average values for all model eye configurations were 0.48, 0.95, and 0.74 microm for the Z-View, WaveScan, and LADARWave aberrometers, respectively. Average inter-measurement standard deviations for the fixed model eyes were 0.024, 0.025, and 0.034 microm for the Z-View, WaveScan, and LADARWave aberrometers, respectively. Results were similar among the systems for measuring the movable retina surface model eye and comparing manifest refraction spherical equivalent of the patients. Data gathered using one variable and several fixed-parameter model eyes showed good correlation to predicted values for all of the aberrrometers with one exception. A significant difference was found in the measurement of one individual fixed model eye with one of the three aberrometers. The wavefront refraction provided by the Z-View correlated well with the results of manifest refraction.

Rarefaction Shocks, Shock Errors, and Low Order of Accuracy in ZEUS

We show that there are simple one-dimensional problems for which the MHD code, ZEUS, generates significant errors, whereas upwind conservative schemes perform very well on these problems.

Exponential Fitting for the Solution of Two-Point Boundary Value Problems with Cubic Spline Collocation Tau-Method

This paper concerns the application of the cubic spline collocation tau method by exponential fitting to the solution of two-point boundary value problems. A simple algebraic technique is required in solving the problem of differential equation in which, for lower order accuracy, the computations can be carried out even without the need of a computer. Numerical examples are given to illustrate the effectiveness of the method. Results obtained for problems considered are compared with the results of the methods discussed in [1] .

Evaluating the Accuracy of Laser Levels for Engineering Surveying

A Sokkia LP,3A automatic laser level was tested for the purpose of judging its capability for differential heighting as practiced in engineering surveying works. A test line was first established on flat ground using precise levelling. This was re-measured using the laser level. Two approaches were followed.In the first, several closed loops were levelled and the corresponding misclosures calculated and compared with known levelling standards. In the second, the elevations of the pegs on the test line were reestablished several times from instrument station at the start of the test line. The r.m.s.e. values of height measurement were then calculated for each peg. These were also compared with known levelling standards. The results showed that in the first approach, the test instrument was able to give misclosure values better than ±7 mm for level circuits up to 340 m in length. This is commensurate with the requirements for third order optical levelling. In the second approach, the r.m.s.e. of height measurement of pegs up to 150 m from instrument station is better than ±2 mm. This is within the order of requirements for third order optical levelling. Taking into account the fact that with the laser level observations can be carried out by one person only, it is concluded that the laser level could be effectively used in place of conventional optical levels in localized surveys concerned with preparation of construction sites, drainage works, inner-city road surveys etc. where only lower order accuracy is required.

Fractional runge-kutta methods with application to convection-diffusion equations

When applying the method of lines to partial differential equations and using explicit methods for the time integration, the time step is usually severely restricted by stability conditions. In this paper, we relax the time step condition by applying fractional step (or operator splitting) methods based on Runge-Kutta methods. Typically, fractional step methods have a low order of accuracy. Therefore, we also discuss a variant with increased order. This version has the additional advantage that parallelism can be exploited. The methods are illustrated by application to the Burgers equation and by a comparison with the LOD method.

Angular singularities of elliptic problems

Criticality calculations for nuclear reactors are commonly based [8, 12, 13[ on 5-point difference approximations to the multigroup diffusion equations in plane sections, which have a low order of accuracy: O(V) if the mesh is uniform, and only 0(/r) if the mesh is nonuniform. The results reported here concern. the possibility of developing “finite element” variational methods having higher-order accuracy which could advantageously replace them.l For elliptic problems whose solution is sufficiently smooth (e.g., of class Pm)), variational methods using Hermite interpolation or spline subspaces of piecewise polynomial functions are known2 to give sets of algebraic equations whose solution yields a function approximating the exact solution with order of accuracy O(/Z~~). However, plane sections of nuclear reactors are typically piecewise homogeneous media separated by interfaces with corners, where the “flux” becomevzonanalytic and has what may be called an angular singularity. Analogous “angular singularities” occur in elliptic problems arising in many other branches of physics, such as electromagnetic theory. To develop “finite elements”, some linear combination ,of which will approximate to a high order of accuracy the exact solution of the multigroup diffusion equations in’ such reactors, one must know the analytical’ nature of the angular singularities of the exact solution (flux) near such corners. When difference methods are used, one typically “subtracts out” the leading

Reactivity ratios for the copolymerization of acrylates and methacrylates by nuclear magnetic resonance spectroscopy

Nuclear magnetic resonance spectroscopy has been applied to the analysis of copolymers of methyl methacrylate with ethyl acrylate, propyl acrylate, butyl acrylate, ethyl methacrylate and methyl acrylate and the reactivity ratios calculated. In the first four of these systems accurate analyses were possible by comparing the OCH 2 and OCH 3 absorption bands which are sufficiently resolved. A lower order of accuracy was achieved for the methyl methacrylate-methyl acrylate system as neither incorporates OCH 2 structures, and recourse had to be made in this case to measurement of the total absorption for analytical purposes. Even this method cannot be used, however, when both monomer molecules contain the same number of protons, as for example ethyl methacrylate and propyl acrylate.

Reaction of polymeric radicals with organometallic compounds

AbstractThe reactivities of polystyryl, polymethyl methacrylyl and polyacrylonitryl radicals with thirteen organometallic substrates were measured and correlated at 60°C. The organometallics chosen were saturated, aliphatic derivatives of elements from Groups 2b, 3a, 4a, and 5a of the Periodic Table. Correlation was accomplished by the method proposed by Bamford, Jenkins, and Johnston. The correlation based on the above three monomers allowed the prediction of the chain transfer constant C of four of the tested substrates with an untested radical, the polymethyl acrylyl, with an accuracy of better than 101. Since the correlation was based on absolute values of C which covered the range of 106 and which were accurate to within 100.8, it was concluded that the correlation was a valid one and was useful for predictive purposes. Values of C for untested substrates can be predicted with only a low order of accuracy because data are not available on the polarizability of most compounds. The introduction of oxygen into the vicinity of the metal–carbon bond reduces the values of C drastically and confirms that C is very sensitive to polar factors. There appears to be a correlation between β (the polarizability of the substrate) and β (the intrinsic reactivity of the substrate) within a given group of the Periodic Table. The fact that there is not a general correlation between α and α for the whole Periodic Table suggests that the mechanism of the transfer reaction is different from group to group.

A NOTE ON THE ADJUSTMENT OF MINOR TRAVERSE NETWORKS

From time to time articles are published concerning the adjustment of traverse networks. These seem invariably to be concerned with a rigorous adjustment by least squares in a more or less simplified form; but, however simplified, such a rigorous adjustment can only be justified for a traverse network of a high order of accuracy. Yet there are many occasions on which a traverse network of a much lower order of accuracy must be computed, and thus there is a good case for devising a simplified form of traverse network adjustment as a parallel to the shnplified adjustment of triangulation figures.

A NOTE ON THE ADJUSTMENT OF MINOR TRAVERSE NETWORKS

From time to time articles are published concerning the adjustment of traverse networks. These seem invariably to be concerned with a rigorous adjustment by least squares in a more or less simplified form; but, however simplified, such a rigorous adjustment can only be justified for a traverse network of a high order of accuracy. Yet there are many occasions on which a traverse network of a much lower order of accuracy must be computed, and thus there is a good case for devising a simplified form of traverse network adjustment as a parallel to the shnplified adjustment of triangulation figures.

BASE MEASUREMENT IN THE ANGLO-EGYPTIAN SUDAN

Since 1899 cadastral mapping in the Sudan has been concentrated along the banks of the Nile from the Egyptian frontier to latitude 13°N., in the towns, and in the area of the Gezira, south of Khartoum, where cotton has been developed. These surveys were controlled by theodolite and steel tape either in the form of traverses or rectangulation. The early triangulation was used mainly to control topographical surveys, for it was essential to cover the whole country as rapidly as possible with a series of maps on scale 1: 250,000. As a consequence much of this early triangulation is of a relatively low order of accuracy, indifferently marked on the ground, and unfit for inclusion in a framework for medium and large scale mapping.