Third-order Method Research Articles

Parameter Uniform Numerical Method for Third Order Singularly Perturbed Turning Point Problems Exhibiting Boundary Layers

In this paper, a parameter uniform numerical method based on Shishkin mesh is suggested to solve singularly perturbed third order differential equations with a turning point exhibiting boundary layers. An error estimate is derived by using supremum norm. Numerical results are provided to illustrate the theoretical results.

Development of a hybrid black-oil/surfactant enhanced oil recovery reservoir simulator

Abstract The need for improved models to simulate chemical enhanced oil recovery (CEOR) methods involving either free gas or solution gas has increased in recent years as it has become more common to inject surfactants to recover live oils using surfactant-polymer or alkaline-surfactant-polymer flooding, or to inject gas with the surfactant solution (low-tension gas flooding or foam). A four-phase flow and transport formulation is needed to accurately model these EOR methods. New experimental data are now available to help in the development and validation of such models. In particular, the effect of pressure and solution gas on microemulsion phase behavior is now much better understood because of new and much more systematic experimental data. New low-tension gas or alkali surfactant foam experiments have also recently been published. In this paper, we present a four-phase flow formulation implemented in the numerical reservoir simulator UTCHEM, a compositional model developed specifically to simulate both lab-scale and field-scale chemical EOR. UTCHEM includes a third-order TVD finite-difference method to improve numerical accuracy of the extremely non-linear chemical processes. A black-oil model for water/oil/gas is coupled with the surfactant/oil/water (microemulsion) phase behavior model through a new formulation. The new formulation includes water, oil, gas, surfactant, polymer and electrolyte components in the aqueous, oleic, microemulsion and gas phases. The PVT properties are calculated for each phase using this model. The impact of pressure and solution gas on the optimum salinity and interfacial tension will be discussed. The four-phase flow simulator has been used to model low-tension gas flooding taking into account process mechanisms. The results aid in the understanding and interpretation of four-phase flow chemical EOR processes. The simulator can be used to evaluate chemical EOR methods involving solution gas and/or free gas, to scale up lab experiments to the field and to optimize chemical EOR process design for field tests among other applications. The need for a reliable mechanistic simulator has greatly increased due to more field applications of chemical EOR with gas and the expected further increase of this type of EOR.

A study of the excited electronic states of normal and fully deuterated furan by photoabsorption spectroscopy and high-level ab initio calculations

The photoabsorption spectra of C4H4O and C4D4O have been measured between ∼5.5 and 17.7eV using a synchrotron radiation-based Fourier transform spectrometer. In addition to several broad bands due to transitions into valence states, the spectra exhibit numerous sharp bands associated with Rydberg states belonging to series converging onto the X̃2A2 or the Ã2B1 state limits. Vertical excitation energies and oscillator strengths have been computed using the second- and third-order algebraic-diagrammic construction polarisation propagator methods (ADC(2) and ADC(3)), and the equation-of-motion coupled-cluster method at the level of singles and doubles model (EOM-CCSD). Adiabatic excitation energies have been estimated using previously computed corrections. The theoretical predictions have allowed assignments to be proposed for the Rydberg series observed in the present single-photon absorption spectra and for some additional series, mainly of A2 symmetry, reported in previous multiphoton excitation studies. The assignments of some of the Rydberg series converging onto the Ã2B1 state limit have been revised and, guided by our calculations, the principal series is ascribed to the 2b1→nda21B2 and 2b1→ndb11A1 transitions. f-type Rydberg series, previously observed only in the multiphoton absorption spectrum of furan, have been observed and assigned. Such f-type series, converging onto either the X̃2A2 or the Ã2B1 state thresholds, contribute significantly to the single-photon absorption spectrum. Many of the absorption bands associated with Rydberg states display vibrational progressions which resemble those in the corresponding photoelectron band. It appears that some of the structure associated with the 1a2→3pb21B1 and 1a2→3pb11B2 transitions involves excitation of non-totally symmetric vibrational modes.

Iterative Methods of Higher Order for Nonlinear Equations

Two new three-step higher-order iterative methods are developed by modifying two third-order two-step methods for solving nonlinear equations. This is done by introducing Newton’s method as a third step in both methods. We approximate the derivative in the third step by the technique of linear interpolation and divided differences leading to new sixth-order methods with efficiency indices equal to 1.565. The number of iterations and the total number of function evaluations required to get a root correct up to 15 decimal places are taken as performance measures of the methods. A number of numerical examples are worked out to demonstrate the efficacy of the proposed methods. The results obtained are compared with some existing methods. It is observed that the methods give improved results.

High Resolution Numerical Simulation of Detonation Diffraction of Condensed Explosives

In this paper, the specific expression for pressure and sound speed in chemical reaction zone of condensed explosives are theoretically deduced, and a new method for deriving the partial derivative of pressure in respect of every conserved quantity is proposed. Combined with the third-order TVD Runge–Kutta method, we develop a parallel solver using the fifth-order high-resolution weighted essentially non-oscillatory (WENO) finite difference scheme to simulate detonation diffraction for two-dimensional condensed explosives. The numerical simulation results revealed the forming reasons of the low-pressure region, the low-density region, the "vortex" region and the "dead zone" in the vicinity of the corner. Furthermore, it demonstrated that the retonation will generate along the inner wall, and it plays an important role in the process of detonation diffraction.

Virtual flight simulation of a dual rotor micro air vehicle

In this paper, a new computational method is developed based on computational fluid dynamics (CFD) coupled with rigid body dynamics (RBD) and flight control law in an in-house programmed source code. The CFD solver is established based on momentum source method, preconditioning method, lower–upper symmetric Gauss–Seidel iteration method, and moving overset grid method. Two-equation shear–stress transport k − ω turbulence model is employed to close the governing equations. Third-order Adams prediction-correction method is used to couple CFD and RBD in the inner iteration. The wing-rock motion of the delta wing is simulated to validate the capability of the computational method for virtual flight simulation. Finally, the developed computational method is employed to simulate the longitudinal virtual flight of a dual rotor micro air vehicle (MAV). Results show that the computational method can simulate the virtual flight of the dual rotor MAV.

Scalar and electromagnetic quasinormal modes of an extended black hole in F(R) gravity

In this paper, we study the scalar and electromagnetic perturbations of an extended black hole in F(R) gravity. The quasinormal modes in two cases are evaluated and studied their behavior by plotting graphs in each case. To study the quasinormal mode, we use the third-order WKB method. The present study shows that the absolute value of imaginary part of complex quasinormal modes increases in both cases, thus the black hole is stable against these perturbations. As the mass of the scalar field increases the imaginary part of the frequency decreases. Thus, damping slows down with increasing mass of the scalar field.

High-order Taylor series expansion methods for error propagation in geographic information systems

The quality of modeling results in GIS operations depends on how well we can track error propagating from inputs to outputs. Monte Carlo simulation, moment design and Taylor series expansion have been employed to study error propagation over the years. Among them, first-order Taylor series expansion is popular because error propagation can be analytically studied. Because most operations in GIS are nonlinear, first-order Taylor series expansion generally cannot meet practical needs, and higher-order approximation is thus necessary. In this paper, we employ Taylor series expansion methods of different orders to investigate error propagation when the random error vectors are normally and independently or dependently distributed. We also extend these methods to situations involving multi-dimensional output vectors. We employ these methods to examine length measurement of linear segments, perimeter of polygons and intersections of two line segments basic in GIS operations. Simulation experiments indicate that the fifth-order Taylor series expansion method is most accurate compared with the first-order and third-order method. Compared with the third-order expansion; however, it can only slightly improve the accuracy, but on the expense of substantially increasing the number of partial derivatives that need to be calculated. Striking a balance between accuracy and complexity, the third-order Taylor series expansion method appears to be a more appropriate choice for practical applications.

Evaluation of Third-Order Method for the Tests of Variance Component in Linear Mixed Models

Mixed models provide a wide range of applications including hierarchical modeling and longitudinal studies. The tests of variance component in mixed models have long been a methodological challenge because of its boundary conditions. It is well documented in literature that the traditional first-order methods: likelihood ratio statistic, Wald statistic and score statistic, provide an excessively conservative approximation to the null distribution. However, the magnitude of the conservativeness has not been thoroughly explored. In this paper, we propose a likelihood-based third-order method to the mixed models for testing the null hypothesis of zero and non-zero variance component. The proposed method dramatically improved the accuracy of the tests. Extensive simulations were carried out to demonstrate the accuracy of the proposed method in comparison with the standard first-order methods. The results show the conservativeness of the first order methods and the accuracy of the proposed method in approximating the p-values and confidence intervals even when the sample size is small.

Behaviour of fixed and critical points of the $$\left( \alpha ,c\right) $$ α , c -family of iterative methods

In this paper we study the dynamical behavior of the \((\alpha ,c)\)-family of iterative methods for solving nonlinear equations, when we apply the fixed point operator associated to this family on quadratic polynomials. This is a family of third-order iterative root-finding methods depending on two parameters; so, as we show throughout this paper, its dynamics is really interesting, but complicated. In fact, we have found in the real \((\alpha ,c)\)-plane a line in which the corresponding elements of the family have a lower number of free critical points. As this number is directly related with the quantity of basins of attraction, it is probable to find more stable behavior between the elements of the family in this region.

Convergence Analysis for Three Parareal Solvers

We analyze in this paper the convergence properties of the parareal algorithm for the symmetric positive definite problem $\mathbf{u}'+A\mathbf{u}=g$. The coarse propagator $\mathcal{G}$ is fixed to the backward-Euler method and three time integrators are chosen for the $\mathcal{F}$-propagator: the trapezoidal rule, the third-order diagonal implicit Runge--Kutta (RK) (DIRK) method, and the fourth-order Gauss RK method. It is well known that the Parareal-Euler algorithm using the backward-Euler method for $\mathcal{F}$ and $\mathcal{G}$ converges rapidly, but less is known when one uses for $\mathcal{F}$ the trapezoidal rule, or the fourth-order Gauss RK method, especially when the mesh ratio $J(=\Delta T/\Delta t)$ is small. We show that for a specified $\lambda_{\max}$(the maximal eigenvalue of $A$ or its upper bound), there exists some critical $J_{\rm cri}$ such that the parareal solvers derived from these three choices of $\mathcal{F}$ converge as fast as Parareal-Euler, provided $J\geq J_{\rm cri}$. Precisely, for $\mathcal{F}$ the trapezoidal rule and the fourth-order Gauss RK method, $J_{\rm cri}$ depends on $\Delta T, \Delta t$, and $\lambda_{\max}$ and we present concise formulas to calculate $J_{cri}$, while for $\mathcal{F}$ the third-order DIRK method, $J_{cri}=4$, independently of these parameters. Numerical examples with applications in fractional PDEs and uncertainty quantification are presented to support the theoretical predictions.

High-Order Implicit Time-Marching Methods Based on Generalized Summation-By-Parts Operators

This article extends the theory of classical finite-difference summation-by-parts (FD-SBP) time-marching methods to the generalized summation-by-parts (GSBP) framework. Dual-consistent GSBP time-marching methods are shown to retain: A and L-stability, as well as superconvergence of integral functionals when integrated with the quadrature associated with the discretization. This also implies that the solution approximated at the end of each time step is superconvergent. In addition GSBP time-marching methods constructed with a diagonal norm are BN-stable. This article also formalizes the connection between FD-SBP/GSBP time-marching methods and implicit Runge-Kutta methods. Through this connection, the minimum accuracy of the solution approximated at the end of a time step is extended for nonlinear problems. It is also exploited to derive conditions under which nonlinearly stable GSBP time-marching methods can be constructed. The GSBP approach to time marching can simplify the construction of high-order fully-implicit Runge-Kutta methods with a particular set of properties favourable for stiff initial value problems, such as L-stability. It can facilitate the analysis of fully discrete approximations to PDEs and is amenable to to multi-dimensional spcae-time discretizations, in which case the explicit connection to Runge-Kutta methods is often lost. A few examples of known and novel Runge-Kutta methods associated with GSBP operators are presented. The novel methods, all of which are L-stable and BN-stable, include a four-stage seventh-order fully-implicit method, a three-stage third-order diagonally-implicit method, and a fourth-order four-stage diagonally-implicit method. The relative efficiency of the schemes is investigated and compared with a few popular non-GSBP Runge-Kutta methods.

A new coupled reduced alternating group explicit method for nonlinear singular two-point boundary value problems on a variable mesh

In this paper, we discuss a new coupled reduced alternating group explicit (CRAGE) and Newton-CRAGE iteration methods to solve the nonlinear singular two-point boundary value problems u″ = f(r, u, u′), 0 < r < 1 subject to given natural boundary conditions u(0) = A1, u(1) = A2 where A1 and A2 are finite constants, along with a third-order numerical method on a geometric mesh. The proposed method is applicable to singular and nonsingular problems. We have discussed the convergence of the CRAGE iteration method in detail. The results obtained from the proposed CRAGE iteration method are compared with the results of the corresponding two-parameter alternating group explicit (TAGE) iteration methods to demonstrate computationally the efficiency of the proposed method.

More efficient fifth-order method for solving systems of nonlinear equations

In this paper, we present a one-parameter family of fifth-order methods by extending Nedzhibov's third-order methods for solving systems of nonlinear equations. For a particular value of parameter, the new fifth-order method is more efficient as compared to the existing methods as its computational cost is less. Further, it requires two function evaluations, two first order Fr´echet derivatives and one matrix inversion per iteration. Numerical examples confirm that the proposed method is highly efficient and useful in solving systems of nonlinear equations.

A new family of iterative methods widening areas of convergence

A new parametric class of third-order iterative methods for solving nonlinear equations and systems is presented. These schemes are showed to be more stable than Newton’, Traub’ or Ostrowski’s procedures (in some specific cases), and it has been proved that the set of starting points that converge to the roots of different nonlinear functions is wider than the one of those respective methods. Moreover, the numerical efficiency has been checked through different numerical tests.

A Class of Iterative Methods for Solving Nonlinear Equations with Optimal Fourth-order Convergence

In this paper we construct a new third-order iterative method for solving nonlinear equations for simple roots by using inverse function theorem. After that a class of optimal fourth-order methods by using one function and two first derivative evaluations per full cycle is given which is obtained by improving the existing third-order method with help of weight function. Some physical examples are given to illustrate the efficiency and performance of our methods.

A Third-Order Newton-Type Method for Finding Polar Decomposition

It is attempted to present an iteration method for finding polar decomposition. The approach is categorized in the scope of Newton-type methods. Error analysis and rate of convergence are studied. Some illustrations are also given to disclose the numerical behavior of the proposed method.

Hydraulic Analysis of Water Supply Networks Using a Modified Hardy Cross Method

There are different methods for the hydraulic analysis of water supply networks. In the solution process of most of these methods, a large system of linear equations is solved in each iteration. This usually requires a high computational effort. Hardy Cross method is one of the approaches that do not need such a process and may converge to the solution through scalar divisions. However, this method has two shortcomings: first, initial discharges should satisfy continuity equation at each node; second a large number of iterations are required to converge to solution. In this article an algorithm is suggested for initial discharges that are close to the final results while the continuity equations are automatically established. This algorithm may be directly implemented in the Hardy Cross method. To reduce the number of iterations the Hardy Cross method is combined with third-order and sixteenth-order methods. The results of some numerical examples demonstrate that the use of the combined approach with the suggested initial guess reduces the number of iterations and hydraulic analysis time and the solutions converge with a high accuracy.

Prediction of standard enthalpy of formation in the solid state by a third-order group contribution method

New group contribution model for the prediction of the standard enthalpy of formation in the solid phase of pure compounds is presented. An extensive set of 1222 experimental enthalpies of formation data was collected from different literature sources, evaluated and used to develop this model (85% as the training set and 15% as the test set). The compounds in the data set are composed of hydrogen, carbon, oxygen and nitrogen. The prediction is based exclusively on the molecular structure of the compound and the model results show a good agreement with the experimental data. Compared to the most accurate models for estimating the standard enthalpy of formation in the solid state, this model demonstrates significant improvements in accuracy and applicability.

The third-order algebraic diagrammatic construction method (ADC(3)) for the polarization propagator for closed-shell molecules: efficient implementation and benchmarking.

The implementation of an efficient program of the algebraic diagrammatic construction method for the polarisation propagator in third-order perturbation theory (ADC(3)) for the computation of excited states is reported. The accuracies of ADC(2) and ADC(3) schemes have been investigated with respect to Thiel's recently established benchmark set for excitation energies and oscillator strengths. The calculation of 141 vertical excited singlet and 71 triplet states of 28 small to medium-sized organic molecules has revealed that ADC(3) exhibits mean error and standard deviation of 0.12 ± 0.28 eV for singlet states and -0.18 ± 0.16 eV for triplet states when the provided theoretical best estimates are used as benchmark. Accordingly, the ADC(2)-s and ADC(2)-x calculations revealed accuracies of 0.22 ± 0.38 eV and -0.70 ± 0.37 eV for singlets and 0.12 ± 0.16 eV and -0.55 ± 0.20 eV for triplets, respectively. For a comparison of CC3 and ADC(3), only non-CC3 benchmark values were considered, which comprise 84 singlet states and 19 triplet states. For these singlet states CC3 exhibits an accuracy of 0.23 ± 0.21 eV and ADC(3) an accuracy of 0.08 ± 0.27 eV, and accordingly for the triplet states of 0.12 ± 0.10 eV and -0.10 ± 0.13 eV, respectively. Hence, based on the quality of the existing benchmark set it is practically not possible to judge whether ADC(3) or CC3 is more accurate, however, ADC(3) has a much larger range of applicability due to its more favourable scaling of O(N(6)) with system size.