Accuracy Of Different Approximations Research Articles

Closed-form expressions for monoprotic weak acid aqueous solutions

A general closed-form expression for {[textrm{H}_{3}textrm{O}^+]} of solutions of a weak acid and a weak acid buffer, as well as their titration with a strong base, has been obtained and mathematically analyzed with the aid of computer algebra systems. This expression is used to evaluate, without the use of numerical approximations, the precision and accuracy of different approximations commonly employed in general chemistry and chemical analysis courses. The closed-form expression for {{[textrm{H}_{3}textrm{O}^+]}} of a buffer solution is used to obtain an analytical expression for the pH stability when a strong base is added. Finally, it is shown that the {[textrm{H}_{3}textrm{O}^+]} expressions for all the systems under study can be obtained from a general closed-form expression in terms of an effective weak acid constant and an effective acid concentration. The formulas found in this work do not display numerical rounding errors and do not require the use of numerical or graphical methods for their evaluation.Graphical abstractGraphical TOC: A closed-form expression for the weak acid and weak acid buffersolutions and their titrations with a strong base

Hexic Polynomial Approximation for Spherical Diffusion to Accelerate Accurate Lithium-Ion Battery Simulation

There is a need for proper estimates of various vital internal states of Li-ion batteries in a battery management system like state of charge, state of health, state of power, and available energy for a better lifetime and user experience. These estimates Require electrochemical models that are generally coupled partial differential equations (PDEs) which are computationally expensive. This work proposes hexic polynomial-based models to provide better accuracy than the lower order polynomial-based models (parabolic and quartic models) while providing comparable computational efficiency in solving the PDE-based models for solid phase diffusion. While the constant current discharge behavior of a battery can be captured reasonably well using parabolic profile approximation, for a time-varying current (especially having high-frequency components), such lower order approximations lead to significant errors hence a hexic polynomial approximation is proposed. To compare the accuracy of different approximations for battery simulation, a single-particle model with thermal effects and an isothermal pseudo-2D model are simulated in this paper. The voltage response for the time-varying current profile in SPM shows the superiority of the hexic polynomial approximation over other lower-order approximations in providing a balance between computational efficiency and accuracy.

Population persistence under high mutation rate: From evolutionary rescue to lethal mutagenesis.

Populations may genetically adapt to severe stress that would otherwise cause their extirpation. Recent theoretical work, combining stochastic demography with Fisher's geometric model of adaptation, has shown how evolutionary rescue becomes unlikely beyond some critical intensity of stress. Increasing mutation rates may however allow adaptation to more intense stress, raising concerns about the effectiveness of treatments against pathogens. This previous work assumes that populations are rescued by the rise of a single resistance mutation. However, even in asexual organisms, rescue can also stem from the accumulation of multiple mutations in a single genome. Here, we extend previous work to study the rescue process in an asexual population where the mutation rate is sufficiently high so that such events may be common. We predict both the ultimate extinction probability of the population and the distribution of extinction times. We compare the accuracy of different approximations covering a large range of mutation rates. Moderate increase in mutation rates favors evolutionary rescue. However, larger increase leads to extinction by the accumulation of a large mutation load, a process called lethal mutagenesis. We discuss how these results could help design "evolution-proof" antipathogen treatments that even highly mutable strains could not overcome.

Semi-analytical solution to the second-order wave loads on a vertical cylinder in bi-chromatic bi-directional waves

A complete solution is presented for the second-order wave loads experienced by a uniform vertical cylinder in bi-chromatic bi-directional waves. The solution is obtained based on the introduction of an assisting radiation potential without explicitly evaluating the second-order diffraction potential. The semi-analytical formulation for calculating the wave loads is provided and an efficient numerical technique is developed to treat the oscillatory free-surface integral that appears in the force formulation. After validating the present solution by comparing with the predictions based on other methods, numerical studies are conducted for different combinations of incident wave frequencies and wave headings, and the influence of frequencies and headings of dual waves on the second-order wave loads is investigated. In addition, by expressing the second-order wave loads in a power expansion with respect to the wave frequency difference and wave heading difference which are both assumed to be small, approximations on the calculation of wave loads are developed. The accuracy of different approximations is assessed by comparing the approximate results with those based on the complete solution.

Insight into electronic, mechanical and transport properties of quaternary CoVTiAl: Spin-polarized DFT + U approach

We present a preliminary investigation of band structure and thermoelectric properties of new quaternary CoVTiAl Heusler alloy. Structural, magnetic property and 100% spin polarization of equiatomic CoVTiAl predicts ferromagnetic stable ground state. Band profile outlines the indirect semiconducting behavior in spin down channel with band gap of 1.06eV, and the magnetic moment of 3 µB in accordance with Slater-Pauling rule. To evaluate the accuracy of different approximations in predicting thermoelectric properties, the comparison with available experimental data is made which shows fair agreement for the transport coefficients. The high temperature (800K) positive Seebeck coefficient of 73.71µV/K describes the p-type character of the material with high efficiency due to highly influential semiconducting behavior around the Fermi level. Considering the combination of 100% spin-polarization, high Seebeck coefficient and large figure of merit, ferromagnetic semiconducting CoVTiAl may prove as a potential candidate for high temperature thermoelectrics and an ideal spin source material for spintronic applications.

Investigation of depth effects on the wave exciting low frequency drift forces by different approximation methods

The paper presents an experimental and numerical investigation on the slowly varying wave exciting drift forces acting on a body of simple geometry subjected to bichromatic waves and long crested irregular seas. The body is axis-symmetric about the vertical axis, like a vertical cylinder with a rounded bottom and it is restrained from moving. Three different water depths representing shallow, intermediate and deep waters are considered.The objective is, firstly, to investigate the relation between the water depth, the incident wavelength and the resulting slowly varying drift force. The other objective is to assess the accuracy of different approximations to the solution of the second order problem. With this aim in mind, the quadratic transfer functions are calculated with a boundary element method using several different levels of approximation for the second order forces, as described in the following sentences. The most complete approximation solves the boundary value problem completely up to the second order. The first-order approximation is similar to the former; however, the second order boundary value problem is simplified by neglecting the free surface forcing. The simpler method is of zeroth-order with respect to the difference frequency and it is commonly known as Newman's approximation. A fourth approximation is evaluated, which combines Newman's approximation with a contribution from the second order incident wave potential.A fifth approximation is applied to the vertical second order forces only and it consists on, first, calculate the steady vertical second order force in monochromatic waves corrected by an additional setdown, and second, apply Newman's approximation together with the corrected second order steady vertical forces. The additional setdown is derived from the second order incident bichromatic wave potential as the difference frequency tends to zero.Second order WAMIT code is used for the frequency domain hydrodynamic calculations. The numerical results are compared with experimental data in bichromatic waves and in irregular waves.This study shows that the contribution from the second order velocity potential must be considered for shallow waters calculations to achieve accurate results. For small difference frequencies, second order scattering potential effects are small; therefore, in this case, a good practical approximation consists on considering the second order potential is contributed by the incident waves only.

A numerical comparison of approximations to the Stokes equations used in ice sheet and glacier modeling

A computational analysis of the accuracy of different approximations to the Stokes equations for momentum balance used in ice sheet modeling is performed by solving a particular tractable form of the equations appropriate for small perturbations of the ice surface, describing the uniform flow of ice with a Glen rheology on an infinitely long and broad section. The approximants comprise the shallow ice approximation and various schemes for incorporating longitudinal stresses and, in one case, the horizontal gradient of the horizontal plane shear stresses. The simplifications lead to a vertically one‐dimensional numerical problem, whose solution can be computed rapidly. The relaxation rate of perturbations as well as other response descriptors for the stable full system and approximants are compared. Compared with the shallow ice approximation, the inclusion of longitudinal stresses increases accuracy at shorter wavelengths, but accuracy is poor at wavelengths around or less than the ice sheet thickness. Even though analysis shows that the horizontal gradients of the horizontal plane shear stresses are of similar magnitude to longitudinal stress effects, computations show, in agreement with glaciological belief, that longitudinal stress effects are more significant and need to be corrected for first in practice. Two schemes, a multilayer scheme and a one‐layer scheme, are particularly good and should be investigated further in cases where perturbations from uniformity are large. Some other apparently plausible approximations introduce nonphysical instabilities. New schemes need to be assessed in the way described in this paper before being used in real ice sheet models.

Independent anharmonic oscillator approximation in the theory of structural phase transitions in crystals

A new method in the microscopic theory of anharmonic crystal lattice dynamics—the independent anharmonic oscillator (IAO) approximation—is discussed. The method is compared with traditional approaches (such as the mean-field and renormalized self-consistent phonon methods) from the standpoint of the analysis of the main approximations. Different methods are applied to the description of the structural phase transition in a monoatomic anharmonic crystal, the results obtained are analyzed, and the numerical accuracy of different approximations is estimated. It is demonstrated that, for this model in the displacement-type instability range, the new method, unlike the self-consistent phonon method, adequately describes the phase transition as a second-order transition. The phase transition temperature calculated in the displacement limit monotonically tends to the exact value in contrast with the temperature obtained within the mean field approximation.

Dissipative multistate systems in the scaling limit.

The dynamics of dissipative multistate systems is studied using integral equations that are derived within the framework of the path integral-formulation of quantum mechanics. As an illustrative example we study a 5-state system coupled to a harmonic reservoir with an Ohmic spectral density with a high cutoff frequency. The dominant exchange mechanism determining the dynamics as well as the range of validity of different approximations to the influence functional are investigated. Besides depending on temperature and the strength of the system-bath coupling, both the exchange mechanism and the accuracy of different approximations to the influence functional depend on the state in which the system is initially prepared.

The accuracy of difference approximations and the regularization of optimal control problems for an elliptic equation with controls in the coefficients

In studies of difference approximations of optimal control problems for partial differential equations, the control only appear on the right-hand sides of the equations, or on the right-hand sides of the boundary and initial conditions. In the present paper, which is allied to [1–4], we consider the difference approximations of a controlled process which is described by a boundary value problem for a non-self-adjoint elliptic equation with controls in the coefficients of the equation and of the boundary condition. We consider a more general type of quadratic functional than in [1–4].

On the accuracy of difference approximations and the regularization of optimal control problems on solutions of elliptic equations

Difference approximations and regularization are considered in problems of minimizing a quadratic functional on the solutions of a boundary value problem for an elliptic equation with variable coefficients and inhomogeneous boundary conditions of the 2nd and 3rd kind with a control on the right-hand side of the equation. Error estimates are obtained for the difference boundary value problem and for the approximation of the initial functional by a difference functional on generalized solutions of W 2 2 (Ω) .

Boundary Approximations for Implicit Schemes for One-Dimensional Inviscid Equations of Gasdynamics

The applicability to practical calculations of recent theoretical developments in the stability analysis of difference approximations is examined for initial boundary-value problems of the hyperbolic type. For the experiments the one-dimensional inviscid gasdynamic equations in conservation law form are selected. A class of implicit schemes based on linear multistep methods for ordinary differential equations is chosen and the use of space or space-time extrapolations as implicit or explicit schemes is emphasized. Some examples with various inflow-outflow conditions highlight the commonly discussed issues: explicit vs implicit schemes, and unconditionally stable schemes. HEN finite-difference schemes are used to solve initial boundary-value problems for the equations of fluid dynamics, it is well known that most methods require more conditions than those required by the governing partial differential equations. These additional conditions for the finite-difference equations are often called numerical conditions. The conditions cannot be imposed arbitrarily but are determined, in general, using interior information, for example, by extrapolation or uncentered approximations. In this paper, any procedure used to provide a condition will be called a boundary scheme/' Whatever schemes are used for the conditions, it is a common practice to assume that the scheme has a local effect and will not affect the solution globally. During the early 1970s, Kreiss,1'2 Osher,3 Gustafsson et al.,4 Varah,5 and Gustafsson6 published a series of papers establishing methods for checking the stability and accuracy of difference approximations with schemes included. Since then, further progress has been made in the theory of linear difference approximations for initial boundary-value problems of the hyperbolic and parabolic type.7'12 Because improper treatment of the conditions can lead to instability and inaccuracy, even though we start with a stable interior scheme (i.e., scheme for the interior points), it is appropriate to adopt an approach that includes the stability and accuracy of the combined interior and schemes. Surveys of recent developments and extensive bibliographies are included in papers by Coughran13 and Yee.14 The purpose of this paper is to examine the applicability to practical calculations (for nonlinear gasdynamic problems) of recent theoretical stability analyses of implicit difference approximations for initial boundary-value problems of the hyperbolic type. As computations have progressed, the use of the conservative form of the gasdynamic equations has gained popularity. For physical reasons it is sometimes desirable to specify conditions in the nonconservative variables and to compute with conservative variables in the interior. We will consider the additional complications introduced by this procedure.

Quasi-Chemical Method in the Statistical Theory of Regular Mixtures

The quasi-chemical equations in the statistics of regular mixtures are deduced by assuming non-interference among local configurations. This method is capable of yielding higher and higher approximations by choosing a larger group of lattice sites as the local configuration under consideration. The comparative accuracy of different approximations can be judged by a simple criterion. Further applications to ferromagnetism and to the order-disorder transition in alloys, and their results are discussed. Equations for ternary mixtures or mixtures of even more components are given. The asymmetry of solubility of one solute in two immiscible solvents and the order effect of ${\mathrm{Ag}}_{2}$Hg${\mathrm{I}}_{4}$ serve as the interesting examples of the ternary case.