Accurate Numerical Integration Research Articles

A comprehensive comparison of atmospheric mapping functions for GPS measurements in Egypt

AbstractThe principal limiting error source in the Global Positioning System (GPS) is the mismodeling of the delay experienced by radio waves in propagating through the atmosphere. The atmosphere causing the delay in GPS signals consists of two main layers: the ionosphere and the troposphere. The ionospheric delay can be mitigated using dual frequency receivers, but the tropospheric delay is often corrected using a standard tropospheric model. The tropospheric delay can be described as a product of the delay at the zenith and a mapping function, which models the elevation dependence of the propagation delay. A large number of mapping functions have been developed for use in the analysis of space geodetic data. An assessment of most of these mapping functions including those developed by Niell (NMF), Herring (MTT), Davis (CfA-2.2), Ifadis, Chao, Black & Eisner (B & E), Yang & Ping, Moffett, Vienna (VMF), and Isobaric (IMF) have been performed. The behavior of these mapping functions was assessed by comparing their results with highly accurate Numerical Integration based Models (NIM) for three different stations in Egypt (Aswan, Helwan, and Mersa Matrouh) at different times throughout the year. The meteorological data used in this study was taken from the Egyptian Meteorological Authority (EMA) as average values between 1990 and 2005. It can be concluded that the Black & Eisner mapping function is recommended for dry tropospheric delay prediction for low zenith angles, whereas VMF will be the choice for elevation angles up to 10°.

Three dimensional MHD stagnation flow due to a stretchable rotating disk

The three dimensional steady MHD laminar stagnation point flow of an electrically conducting fluid on a radially stretchable rotating disk in the presence of a uniform vertical magnetic field is the main concern of the present study. The problem is an extension of the well-known von Karman stagnation problem to the configuration with a stretchable disk with or without rotation. An exact similarity reduction of the Navier–Stokes equations leads to a system of ordinary differential equations describing the stagnation flow. The system is governed by a stretching and a rotation parameters, based on the wall stretching and angular velocity. Employing a highly accurate spectral numerical integration scheme, the effects of these parameters on the flow are examined. The quantities of particular physical interest, such as the torque and the wall shear stresses are calculated and discussed. Contrary to the classical von Karman flow, for small rotational speeds of the disk, magnetic field is found to thicken the boundary layer when small wall stretching is taken into account.

The Eulerian-Lagrangian Method with Accurate Numerical Integration

AbstractThis paper is devoted to the study of the Eulerian-Lagrangian method (ELM) for convection-diffusion equations on unstructured grids with or without accurate numerical integration. We first propose an efficient and accurate algorithm to calculate the integrals in the Eulerian-Lagrangian method. Our approach is based on an algorithm for finding the intersection of two non-matching grids. It has optimal algorithmic complexity and runs fast enough to make time-dependent velocity fields feasible. The evaluation of the integrals leads to increased precision and the unconditional stability. We demonstrate by numerical examples that the ELM with our proposed algorithm for accurate numerical integration has the following two features: first it is much more accurate and more stable than the ones with traditional numerical integration techniques and secondly the overall cost of the proposed method is comparable with the traditional ones.

Accuracy, Memory, and Speed Strategies in GPU-Based Finite-Element Matrix-Generation

This letter presents strategies on how to optimize graphics processing unit (GPU)-based finite-element matrix-generation that occurs in the finite element method (FEM) using higher-order curvilinear elements. The goal of the optimization is to increase the speed of evaluation and assembly of large finite-element matrices on a single GPU while maintaining the accuracy of numerical integration at the desired level. For this reason, the choice of the optimal Gaussian quadratures for curvilinear finite elements focused on accuracy, memory usage, and runtime of numerical integration is discussed. Moreover, we show how to efficiently utilize symmetry of local mass and stiffness matrices on a GPU in the numerical integration step. The performance results, obtained on a workstation equipped with one Tesla C2075, indicate that the proposed strategies retain the accuracy of computations, allow generation of larger sparse linear systems, and provide 2.5-fold acceleration of GPU-based finite-element matrix-generation.

The dynamic process of two-electron atom irradiated by intense laser pulse using time dependent quantum Monte Carlo method

We investigate the electron dynamic process of a one-dimensional two-electron atom irradiated by strong laser pulse using the improved time-dependent quantum Monte Carlo (TDQMC) scheme. By comparison with the scheme for solving the time-dependent Schrdinger equation by using the accurate numerical integration, the dynamic variation of particle in a quantum ensemble, corresponding to the calculated wave-packet, is consistent with the evolution of time-dependent wavepacket. The computation efficiency of the TDQMC is more largely enhanced than that of the time-dependent integration method. According to the dynamic evolution behaviors of the calculated classical particle ensemble, we analyze the excitation, ionization, recombination of electron and other non-linear processes in a strong laser field.

An Integral Formulation Approach for Numerical Solution of Tubular Reactors Models

In this paper, we derive an integral formulation approach based on Green’s function for the numerical solution of tubular reactor models described by reaction-diffusion-convection (RDC) equations with Danckwerts-type boundary conditions. The integral formulation approach allows the direct incorporation of boundary conditions and leads to a stable and accurate numerical integration with smooth round-off error. Numerical simulations of two of tubular reactors models are presented in order to illustrate the numerical accuracy of the method. The results are compared with those resulting from using standard finite difference method. Our results show that the integral formulation approach improves the performance of classical FD schemes.

Wall Stretching in Magnetohydrodynamics Rotating Flows in Inertial and Rotating Frames

motion are reduced to a set of nonlinear differential equations by means of conventional similarity transformations. An energy equation accounts for the viscous dissipation and joule heating terms. Employing a highly accurate spectral numerical integration scheme, the effects of a parameter based on wall stretching are examined in both frames. The quantities of particular physical interest, such as the torque, the wall shear stresses, the vertical suction velocity, and the rate of heat transfer, are calculated and discussed. It is found that the influence of the frame diminishes in the large limit of disk stretching.

Detailed steady-state simulation of tubular reactors for LDPE production: Influence of numerical integration accuracy

Simulators are widely used for analyzing and optimizing the production of low density polyethylene in tubular reactors under steady state conditions. This steady state is in practice often simulated by chemical engineers using a series of CSTRs type model due to its stable behavior with respect to spatial discretization and smooth convergence for the underlying stiff model equations. Although already a large number of CSTRs is used, this number appears to be too low for the physical reality. Here, this traditional cascaded CSTR approach is compared with a plug flow type approach for a highly detailed reaction model describing the free radical copolymerization. Additionally, the influence of the discretization is rigorously investigated and quantified. It is shown that the discretization does not significantly affect the temperature and the conversion profile, but has a major impact (deviations up to 30%) on the properties which determine the end product. However, this impact of discretization is in practice often overlooked.

MHD fluid flow and heat transfer due to a stretching rotating disk

The steady MHD laminar flow of an electrically conducting fluid on a radially stretchable rotating disk in the presence of a uniform vertical magnetic field is the subject of the present paper. The problem is an extension of the well-known von Karman viscous pump problem to the configuration with a stretchable disk with or without rotation first imposed in [1]. The governing equations of motion are reduced to a set of nonlinear differential equations by means of conventional similarity transformations. Energy equation accounts for the viscous dissipation and Joule heating terms. Employing a highly accurate spectral numerical integration scheme, the effects of a rotation parameter based on the wall stretching and angular velocity are examined. The quantities of particular physical interest, such as the torque, the wall shear stresses, the vertical suction velocity and the rate of heat transfer are calculated and discussed.

Global error estimation and control in linearly-implicit parallel two-step peer W-methods

The class of linearly-implicit parallel two-step peer W-methods has been designed recently for efficient numerical solutions of stiff ordinary differential equations. Those schemes allow for parallelism across the method, that is an important feature for implementation on modern computational devices. Most importantly, all stage values of those methods possess the same properties in terms of stability and accuracy of numerical integration. This property results in the fact that no order reduction occurs when they are applied to very stiff problems. In this paper, we develop parallel local and global error estimation schemes that allow the numerical solution to be computed for a user-supplied accuracy requirement in automatic mode. An algorithm of such global error control and other technical particulars are also discussed here. Numerical examples confirm efficiency of the presented error estimation and stepsize control algorithm on a number of test problems with known exact solutions, including nonstiff, stiff, very stiff and large-scale differential equations. A comparison with the well-known stiff solver RODAS is also shown.

Efficient and accurate numerical integration of exchange-correlation density functionals

We present a method for efficient calculation of exchange correlation in a density-functional theory calculation with Gaussian basis functions. It seamlessly combines the treatments of smooth (low-energy cutoff) and nonsmooth parts of the electron density such that computational performance can be maximized without approximation. Benchmark calculations show that the scheme can speed up the calculation of the exchange-correlation by as much as ten times or more without error.

Measurement of the eddy diffusion term in chromatographic columns. I. Application to the first generation of 4.6 mm I.D. monolithic columns

Abstract The corrected heights equivalent to a theoretical plate (HETP) of three 4.6 mm I.D. monolithic Onyx-C 18 columns (Onyx, Phenomenex, Torrance, CA) of different lengths (2.5, 5, and 10 cm) are reported for retained (toluene, naphthalene) and non-retained (uracil, caffeine) small molecules. The moments of the peak profiles were measured according to the accurate numerical integration method. Correction for the extra-column contributions was systematically applied. The peak parking method was used in order to measure the bulk diffusion coefficients of the sample molecules, their longitudinal diffusion terms, and the eddy diffusion term of the three monolithic columns. The experimental results demonstrate that the maximum efficiency was 60 000 plates/m for retained compounds. The column length has a large impact on the plate height of non-retained species. These observations were unambiguously explained by a large trans-column eddy diffusion term in the van Deemter HETP equation. This large trans-rod eddy diffusion term is due to the combination of a large trans-rod velocity bias (≃3%), a small radial dispersion coefficient in silica monolithic columns, and a poorly designed distribution and collection of the sample streamlets at the inlet and outlet of the monolithic rod. Improving the performance of large I.D. monolithic columns will require (1) a detailed knowledge of the actual flow distribution across and along these monolithic rod and (2) the design of appropriate inlet and outlet distributors designed to minimize the nefarious impact of the radial flow heterogeneity on band broadening.

The propagation of sound above a hard‐backed porous layer.

In a recent paper [J. F. Allard, J. Acoust. Soc. Am. 125, 1864–1867 (2009)], the inaccuracy of the classical formulation for the acoustic field of a point source above a hard‐back porous layer was addressed. The current study develops a set of closed form analytical solutions for predicting the propagation of sound due to a monopole near a hard backed layer. The current formulation avoids the use of the effective impedance approximation to obtain asymptotic solutions. The asymptotic formulas are derived by applying the method of saddle path integration supplemented by the method of pole subtraction. The analytical solutions are compared with accurate numerical integrations using the fast field formulation and other accurate numerical quadratures. Predicted results suggest that the analytical formulas agree well with exact numerical solutions. Indoor experimental measurements were also conducted to validate the analytical formulas.

Estimation of Turbulent Vertical Velocity from Nonlinear Simulations of Aircraft Response

The objective of this paper is to estimate the unknown vertical velocity component of atmospheric turbulence on a transport aircraft based on data in the flight data recorder with accuracy consistent with the measurement of the alpha sensor. Such vertical velocity component of low frequency (to be called the wind) is not measurable quantitatively with the existing onboardweather radars. The idea is to estimate the vertical wind speed from aircraft response by integrating the nonlinear flight dynamic equations. The motivation of developing this new approach arises from flight trajectory reconstruction of a transport aircraft in an accident with the aerodynamics being unsteady and nonlinear. Because the conventional forward integration with a Runge–Kutta fourth-order scheme tends to diverge, in particular in the predicted altitude, an innovative approach which can maintain the accuracy of numerical integration over a long time period for a system of implicit nonlinear flight dynamic equations is introduced in this paper. The new approach is to add and subtract a linear part of the nonlinear differential equations, with one term being treated implicitly (i.e., unknown) and the other term, together with the other aerodynamic forces and moments, being known function of time. This implicit linear part serves as the numerical damping that reduces or eliminates the growth of errors in numerical forward integrations. Application to a transport aircraft in severe turbulence encounter is illustrated.

Maximum-entropy meshfree method for incompressible media problems

A novel maximum-entropy meshfree method that we recently introduced in Ortiz et al. (2010) [1] is extended to Stokes flow in two dimensions and to three-dimensional incompressible linear elasticity. The numerical procedure is aimed to remedy two outstanding issues in meshfree methods: the development of an optimal and stable formulation for incompressible media, and an accurate cell-based numerical integration scheme to compute the weak form integrals. On using the incompressibility constraint of the standard u– p formulation, a u-based formulation is devised by nodally averaging the hydrostatic pressure around the nodes. A modified Gauss quadrature scheme is employed, which results in a correction to the stiffness matrix that alleviates integration errors in meshfree methods, and satisfies the patch test to machine accuracy. The robustness and versatility of the maximum-entropy meshfree method is demonstrated in three-dimensional computations using tetrahedral background meshes for integration. The meshfree formulation delivers optimal rates of convergence in the energy and L 2-norms. Inf–sup tests are presented to demonstrate the stability of the maximum-entropy meshfree formulation for incompressible media problems.

Numerical Integration for Interactive Terms in Mesh Superposition Method by Auto-Mesh Generation

The interactive terms in the mesh superposition method are calculated approximately because the integrand has a discontinuity along the mesh lines in a superimposed global mesh. In this paper, the accurate numerical integration technique for a discontinuous integrand by the Delaunay triangulation is proposed. Integral ranges including discontinuous integrands are divided into triangles only with a continuum integrand. The exact solution is sum of the integral values calculated in all triangles. Appropriate triangle edges can be regenerated by the swapping algorithm even though generated triangles contain the inappropriate boundary lines which bring about a discontinuity. A comparison of the accuracy of numerical integration between the proposed technique and conventional approximate means and an application to a sensitivity calculation are investigated in the first example. The second example illustrates the swapping algorithm for the adjustment of generated triangles.

Study on Appropriate Decomposition Procedure of Integral Domain for High Accurate Numerical Integration of Interactive Terms in Finite Element Mesh Superposition Method by Auto-Mesh Generation

Study on Appropriate Decomposition Procedure of Integral Domain for High Accurate Numerical Integration of Interactive Terms in Finite Element Mesh Superposition Method by Auto-Mesh Generation

Ab initio study on vibrational dipole moments of XH+ molecular ions: X = 24Mg, 40Ca, 64Zn, 88Sr, 114Cd, 138Ba, 174Yb and 202Hg

The vibrational matrix elements of electric dipole moments were theoretically estimated for the electronic ground state of XH+ molecular ions (X = 24Mg, 40Ca, 64Zn, 88Sr, 114Cd, 138Ba, 174Yb and 202Hg) using the complete active space second-order perturbation theory method. Because of the large rotational constant and zero X-nuclear spin, these molecules are advantageous to be localized to a single (v, J, F) state, where v, J, F are quantum numbers of the vibrational, rotational and hyperfine states, respectively. The information of the dipole moments is very useful to discuss the period to localize the molecular ion to the (v, J, F) = (0, 0, 1/2) state and also the period to remain in this state, which is limited by the interaction with the black body radiation. The agreement of experimental and our theoretical spectroscopic constants ensures the accuracy of our results. Vibrational permanent and transition dipole moments were obtained with special care of accuracy in numerical integration. Spontaneous emission rates were calculated from the vibrational dipole moments and transition energies.

Heat and Mass Transfer on the MHD Fluid Flow Due to a Porous Rotating Disk With Hall Current and Variable Properties

Abstract The steady magnetohydrodynamics (MHD) laminar compressible flow of an electrically conducting fluid on a porous rotating disk is considered in the present paper. The governing equations of motion are reduced to a set of nonlinear differential equations by means of similarity transformations. The fluid properties are taken to be strong functions of temperature and Hall current that also readily accounts for the viscous dissipation and Joule heating terms. Employing a highly accurate spectral numerical integration scheme, the effects of viscosity, thermal conductivity, Hall current, magnetic field, suction/injection, viscous dissipation, and Joule heating on the considered flow are examined. The quantities of particular physical interest, such as the torque, the wall shear stresses, the vertical suction velocity, and the rate of heat transfer are calculated and discussed.

Basis set representation of the electron density at an atomic nucleus

In this paper a detailed investigation of the basis set convergence for the calculation of relativistic electron densities at the position of finite-sized atomic nuclei is presented. The development of Gauss-type basis sets for such electron densities is reported and the effect of different contraction schemes is studied. Results are then presented for picture-change corrected calculations based on the Douglas-Kroll-Hess Hamiltonian. Moreover, the role of electron correlation, the effect of the numerical integration accuracy in density functional calculations, and the convergence with respect to the order of the Douglas-Kroll-Hess Hamiltonian and the picture-change-transformed property operator are studied.