INTERVAL ACCELERATION OF CONVERGENCE

In this work, we further investigate the application of the well-known Richardson extrapolation (RE) technique to accelerate the convergence of sequences resulting from linear multistep methods (LMMs) for numerically solving initial-value problems of systems of ordinary differential equations. By extending the ideas of our recent work on global Richardson extrapolation, we now utilize some advanced versions of RE in the form of repeated RE (RRE). Assume that the underlying LMM—the base method—has order p and RE is applied $$\ell $$ ℓ times. Then, we prove that the accelerated sequence has convergence order $$p+\ell $$ p + ℓ . The version we present here is global RE (GRE, also known as passive RE), since the terms of the linear combinations are calculated independently. Thus, the resulting higher-order LMM-RGRE methods can be implemented in a parallel fashion and existing LMM codes can directly be used without any modification. We also investigate how the linear stability properties of the base method (e.g., A- or $$A(\alpha )$$ A ( α ) -stability) are preserved by the LMM-RGRE methods.

The epsilon algorithm in a non-commutative algebra

A new method for calculating atomic multiphoton processes is presented here with application to two-photon processes in atomic hydrogen. The ${L}^{2}$-basis-set approach exploits explicit expansion of the Coulomb radial function and resolvent in terms of Pollaczek polynomials and functions to achieve compact expressions for two-photon radial transition amplitudes. This allows efficient calculation of the Bethe logarithm even for highly excited hydrogenic states and two-photon ionization amplitudes near and above the one-photon ionization threshold. Above the one-photon threshold, the complete-basis limit of the highly oscillatory amplitude is computed by applying the epsilon algorithm carefully to a sequence generated from only 10--15 basis functions. The approach is extended below the one-photon threshold by splitting the transition amplitude into a sum of two formally divergent but geometriclike series, whose analytic continuation is realized by the epsilon algorithm to yield a clearly defined and efficient interpolation between the resonances at highly excited Rydberg states. This suggests a new ${L}^{2}$-basis formulation of the quantum-defect method. The extension to complex basis functions and many-electron atoms in strong fields is discussed.

Numerical analysis of the trajectory stability and penetration ability of different lateral-abnormal projectiles for non-normal penetration into soil based on Modified Integrated Force Law method

13 - Numerical Analysis

A retrospective on 60 years of approximation theory and associated fields

The aim of this chapter is to present and study some recent development in summability theory and its applications. More recently, neutrosophic normed space (NNS) and the statistical convergence for ordinary (single) sequences in NNS has been studied. This chapter provides an overview on statistical convergence and statistical Cauchy for double sequences in NNS and specifically focuses on statistical completeness in connection with a NNS. We know that ideal convergence is more general than statistical convergence for sequences. This has motivated us to study the ideal convergence of double sequences in NNS. That is, in this chapter, we study the concept of ideal convergence and ideal Cauchy for double sequences in NNS. The chapter also aims to analyze studies which criticize the concepts of I * 2-convergence, I * 2-Cauchy and I * 2-Cauchy for double sequences in NNS. The fundamental properties of these concepts with regard to NN N are investigated. The results of the chapter are expected to be a source for researchers in the areas of convergence methods for sequences and applications in NNS.

The right-hand sides of a system of ordinary differential equations may be discontinuous on a certain surface. If a trajectory crossing this surface shall be computed by a one-step method, then a particular numerical analysis is necessary in a neighbourhood of the point of intersection. Such an analysis is presented in this paper. It shows that one can obtain any desired order of convergence if the method has an adequate order of consistency. Moreover, an asymptotic error theory is developed to justify Richardson extrapolation. A general one-step method is constructed satisfying the conditions of the preceding theory. Finally, a simplified Newton iteration scheme is used to implement this method.

This chapter is concerned with the numerics of MB-regularised forms for the regu-larised value of a divergent series using the Mathematica software package. To accomplish this, the remainders in the asymptotic forms for u(a)given in Ch. 2 are first MB-regularised. Then an explanation of how to evaluate the resulting MB integrals follows. It is shown that when the values for the MB integrals are added to the truncated asymptotic series and the appropriate Stokes dis-continuous terms, they yield exact values ofu(a)over the principal branch. Because the domains of convergence for the MB integrals extend beyond the Stokes sectors, the two different forms for the regularised value also give exact values of u(a)over their common region of(−π/4,π/4). A similar analysis is then undertaken for the error function erf(z), whose Stokes sectors are shifted compared with those foru(a). A major problem arises when Mathematica attempts to evaluate the MB-regularised values for |argz|>π/2 because the factor ofz −2s in the MB integrals lies outside the principal branch. However, with the introduction of the seemingly innocuous factor of exp(2πi jk)into the asymptotic series, different MB-regularised forms are obtained with domains of convergence that encompass the previously inaccessible sectors of the principal branch. Con-sequently, the Stokes multiplier equals -1/2 for j=±1, while it equals 1/2 forj=0 as in Ch. 5. When a numerical analysis is undertaken for|argz|>π/2 with the new forms for the regularised value, exact values of the error function are obtained irrespective of the magnitude ofz.

A method of the neural identification of the moisture content in brick walls of historic buildings on the basis of non-destructive tests

The newly proposed in-plane resonant nano-electro-mechanical (IP R-NEM) sensor, that includes a doubly clamped suspended beam and two side electrodes, achieved a mass sensitivity of less than zepto g/Hz based on analytical and numerical analyses. The high frequency characterization and numerical/analytical studies of the fabricated sensor show that the high vacuum measurement environment will ease the resonance detection using the capacitance detection technique if only the thermoelsatic damping plays a dominant role for the total quality factor of the sensor. The usage of the intrinsic junction-less field-effect-transistor (JL FET) for the resonance detection of the sensor provides a more practical detection method for this sensor. As the second proposed sensor, the introduction of the monolithically integrated in-plane MOSFET with the suspended beam provides another solution for the ease of resonance frequency detection with similar operation to the junction-less transistor in the IP R-NEM sensor. The challenging fabrication technology for the in-plane resonant suspended gate field-effect-transistor (IP RSG-FET) sensor results in some post processing and simulation steps to fully explore and improve the direct current (DC) characteristics of the sensor for the consequent high frequency measurement. The results of modeling and characterization in this research provide a realistic guideline for these potential ultra-sensitive NEM sensors.

In this paper, we trace back the genesis of Aitken’s Δ2 process and Shanks’ sequence transformation. These methods, which are extrapolation methods, are used for accelerating the convergence of sequences of scalars, vectors, matrices, and tensors. They had, and still have, many important applications in numerical analysis and in applied mathematics. They are related to continued fractions and Pade approximants. We go back to the roots of these methods and analyze the original contributions. New and detailed explanations on the building and properties of Shanks’ transformation and its kernel are provided. We then review their historical algebraic and algorithmic developments. We also analyze how they were involved in the solution of systems of linear and nonlinear equations, in particular in the methods of Steffensen, Pulay, and Anderson. Testimonies by various actors of the domain are given. The paper can also serve as an introduction to this domain of numerical analysis.

Supersonic combustion using transverse sonic injection into the supersonic stream is investigated numerically. A threedimensional dual-mode scramjet combustor configuration with significant upstream interaction was selected to provide an understanding of key features of upstream interaction, and to identify physical and numerical issues relating to modeling of dual-mode configurations. The numerical analysis was performed with vitiated air at freestream Mach number of 2.5 using hydrogen as the sonic injectant. The results are presented for a three-dimensional (% scale) geometry. Comparisons are made with experimental results. Results show that numerical modeling using a scaled geometry provides good agreement with experimental results for dual mode combustion of this configuration. This study supports the argument that numerical scaling is usetil in simulating dual-mode scramjet combustor flow-fields and could provide an excellent convergence acceleration technique for dualmode simulations. L INTRODUCTION Recently many studies have been conducted in response to the increased interest in the development of dual mode combustion systems.‘-” It has been found that an optimal combustor design utilizes transverse fuel injection downstream of a rearward-facing step. The step has several purposes: to isolate the inlet boundary layer from the pressure rise generated in the combustion chamber, to improve species mixing, and to create recirculation regions to sustain combustion. Many numerical and experimental studies have been conducted to investigate transverse injection into such a complex flowfield’-“. However, the physics of this flow field are not yet tilly understood. Until recently, the numerical research was limited to two-dimensional models due to the complexities of the physics of the flow and the limitation on computer resources. Recently, Matsuo et al.’ and Mizobuchi et al6 presented numericalexperimental studies to investigate turbulence temperature fluctuations in a scramjet combustion chamber. They * Graduate Research Assistant, Associate Member ASME ** Associate Professor, Member AAIA ’ Manager, Hyper X Program, NASA Langley Research Center, Hampton, VA 23681 ” Eminent Professor/Scholar, AX4 Associate Fellow (c)1999 American Institute of Aeronautics & Astronautics concluded that the treatment of turbulence modeling is the key to an accurate computational scheme. Essential to the design of the scramjet engine is a detailed understanding of the complex flowfield in different regions of the combustion system. Constraints on system size and weight have created a need to improve technology for designing and analyzing such systems. Faster processors and efficient numerical techniques are essential. However, often it is the physical I . dimensions of the problem that, in requiring a fine mesh, consume so much computational time. In such cases, it is the responsibility of ,CFD modelers to understand how physical problems can be scaled to utilize computational space more efficiently. Also, the results from scaled geometry could be used to accelerate the convergence of full,scale geometry simulations. It is well known that for many systems it is possible to scale ~ physical dimensions and governing equations to give non-dimensional results. Unfortunately, few studies have been conducted on the scaling of mixing and reacting flows. The complexities of the 1 turbulence, mixing, and chemical reaction i make scaling of this problem an interesting task. The objective of this computational study is to numerically investigate a threedimensional dual-mode scramjet combustor configuration with significant upstream ’ interaction. This simulation is undertaken to provide an understanding of key features of upstream interaction, physical and numerical issues. In particular, this paper presents a report on the work in progress for an investigation into geometrical scaling of , numerical models where mixing and combustion are of primary concern. The combustion chamber is numerically modeled using a simplified l/d scale three-dimensional model. Numerical results are compared with experimental data. THEORETICAL MODEL The physical model under investigation, transverse hydrogen jet injection into supersonic air flow in a scramjet combustion engine, is shown Fig. 1. The freestream inlet height is 32 mm and the inlet channel is 200 mm long. There are 3.2 mm steps on the upper and lower walls of the chamber. The walls are then parallel over 96 mm. Finally, the chamber expands at 1 .? degree angle. The jets are located 12.8 mm downstream of the steps on the upper and lower walls. The jet injection holes are of 4 mm diameter, five on the lower wall and four on the upper wall. The distance between the injection hole centers is 32mm The governing equations for this problem are Navier-Stokes equations and species continuity equations. AnRNGkepsilon turbulence model is used. A simple one step kinetic model is employed. For the present study, the compressible, unstructured code, RAMPANT, is used to investigate the characteristics of the three-dimensional model of the scramjet combustion chamber. This code uses first/second order finite volume discretization method and a coupled implicit (Block Gauss-Seidel) solver. Further details of the numerical methods used in RAMPANT can be found in Ref. 10. Before using a full-scale model with combustion, it is necessary to simplify the model to investigate certain aspects of the code. As shown in Fig 1, the computational domain is simplified to encompass half of the upper and half of the lower jet (using the symmetry condition for these planes). To conserve computational time the possibility of scaling the problem is also investigated. An analysis of a % scale simplified threedimensional model of the combustion (c)1999 American Institute of Aeronautics & Astronautics chamber is performed. Mesh cells are concentrated around the walls, particularly in the region near the jets.

A new, nonclassical convergence acceleration concept, called \(\mu \)-acceleration of convergence (where \(\mu \) is a positive monotonically increasing sequence), is introduced and compared with the classical convergence acceleration concept. It is shown that this concept allows to compare the speeds of convergence for a larger set of sequences than the classical convergence acceleration concept. Also, regular matrix methods that improve and accelerate the convergence of sequences and series are studied. Some problems related to the speed of convergence of sequences and series with respect to matrix methods are discussed. Several theorems on the improvement and acceleration of the convergence are proved. As an application, the results obtained are used to increase the order of approximation of Fourier expansions and Zygmund means of Fourier expansions in certain Banach spaces.

Fractional Brownian Motions, Fractional Noises and Applications