Helical flow arising from the yielded annular flow of a Bingham fluid

The Bingham fluid model was developed to represent viscoplastic materials that change from rigid bodies at low stress to viscous fluids at high stress – a process termed yielding. Such a fluid model is used in the modeling of slurries, which occur frequently in food processing and other engineering applications.     

Alpha effect in helically forced turbulence

The effect on large-scale dynamics of small-scale helicity injection in three-dimensional resistive magnetohydrodynamic turbulence is investigated for a weak initial magnetic field. The magnetic configuration of α² dynamo prototype flows is such that, generally, the energy concentrates on one large-scale mode. However, we obtain that alpha effect is not limited to the dominant mode and that a non-local equation (in Fourier space) is more appropriate to describe it. It gives some insights into the non-local theory of Pouquet et al. [J. Fluid Mech. 77 (1976) 321] where the inverse cascade results from a competition between the helicity and Alfvén effects.     

Modeling fully developed laminar flow in a helical duct with rectangular cross section and finite pitch

The mass and momentum transport equations are written in an orthogonal coordinate system using Germano’s transformation to model a laminar flow in a helical duct with a rectangular cross section and finite pitch. The system of governing equations are discretized and solved by the finite-volume numerical method. The three dimensional domain is reduced to a two dimensional slab of cells, orthogonal to the main flow direction, enforcing the fully developed state for 2π/(τ · d_h) >> 1 where τ and d_h representing the duct’s centerline torsion and its hydraulic diameter. This approximation and the use of an orthogonal grid allow a great simplification on the numerical procedure. Comparisons of the numerical solution against experimental data are drawn to assess the accuracy of the approximation.     

Using coordinate transformation of Navier-Stokes equations to solve flow in multiple helical geometries

Recent research on small amplitude helical pipes for use as bypass grafts and arterio-venous shunts, suggests that mixing may help prevent occlusion by thrombosis. It is proposed here that joining together two helical geometries, of different helical radii, will enhance mixing, with only a small increase in pressure loss. To determine the velocity field, a coordinate transformation of the Navier-Stokes equations is used, which is then solved using a 2-D high-order mesh combined with a Fourier decomposition in the periodic direction. The results show that the velocity fields in each component geometry differ strongly from the corresponding solution for a single helical geometry. The results suggest that, although the mixing behaviour will be weaker than an idealised prediction indicates, it will be improved from that generated in a single helical geometry.     

Hermitian operator methods for reaction-diffusion equations

A variety of time-linearization, quasilinearization, operator-splitting, and implicit techniques which use compact or Hermitian operators has been developed for and applied to one-dimensional reaction-diffusion equations. Compact operators are compared with second-order accurate spatial approximations in order to assess the accuracy and efficiency of Hermitian techniques. It is shown that time-linearization, quasilinearization, and implicit techniques which use compact operators are less accurate than second-order accurate spatial discretizations if first-order approximations are employed to evaluate the time derivatives. This is attributed to first-order accurati temporal truncation errors. Compact operator techniques which use second-order temporal approximations are found to be more accurate and efficient than second-order accurate, in both space and time, algorithms. Quasilinearization methods are found to be more accurate than time-linearization schemes. However, quasilinearization techniques are less efficient because they require the inversion of block tridiagonal matrices at each iteration. Some improvements in accuracy can be obtained by using partial quasilinearization and linearizing each equation with respect to the variable whose equation is being solved. Operator-splitting methods which use compact differences to evaluate the diffusion operator were found to be less accurate than operator-splitting procedures employ second-order accurate spatial approximations. Comparisons among the methods presented in this paper are shown in terms of the L²-norm errors and computed wave speeds for a variety of time steps and grid spacings: The numerical efficiency is assessed in terms of the CPU time required to achieve the same accuracy.     

Additive blending of local approximations into a globally-valid approximation with application to the dilogarithm

We show how local approximations, each accurate on a subinterval, can be blended together to form a global approximation which is accurate over the entire interval. The blending functions are smoothed approximations to a step function, constructed using the error function. The local approximations may be power series, asymptotic expansion, or other more exotic species. As an example, for the dilogarithm function, we construct a one-line analytic approximation which is accurate to one part in 700. This can be generalized to higher order merely by adding more terms in the local approximations. We also show the failure of the alternative strategy of subtracting singularities.     

A staggered-grid numerical algorithm for the extended Boussinesq equations

A second-order accurate numerical scheme is developed to solve Nwogu’s extended Boussinesq equations. A staggered-grid system is introduced with the first-order spatial derivatives being discretized by the fourth-order accurate finite-difference scheme. For the time derivatives, the fourth-order accurate Adams predictor–corrector method is used. The numerical method is validated against available analytical solutions, other numerical results of Navier–Stokes equations, and experimental data for both 1D and 2D nonlinear wave transformation problems. It is shown that the new algorithm has very good conservative characteristics for mass calculation. As a result, the model can provide accurate and stable results for long-term simulation. The model has proven to be a useful modeling tool for a wide range of water wave problems.     

A high order accurate numerical method for solving two‐dimensional dual‐phase‐lagging equation with temperature jump boundary condition in nanoheat conduction

Dual‐phase‐lagging (DPL) equation with temperature jump boundary condition (Robin's boundary condition) shows promising for analyzing nanoheat conduction. For solving it, development of higher‐order accurate and unconditionally stable (no restriction on the mesh ratio) numerical schemes is important. Because the grid size may be very small at nanoscale, using a higher‐order accurate scheme will allow us to choose a relative coarse grid and obtain a reasonable solution. For this purpose, recently we have presented a higher‐order accurate and unconditionally stable compact finite difference scheme for solving one‐dimensional DPL equation with temperature jump boundary condition. In this article, we extend our study to a two‐dimensional case and develop a fourth‐order accurate compact finite difference method in space coupled with the Crank–Nicolson method in time, where the Robin's boundary condition is approximated using a third‐order accurate compact method. The overall scheme is proved to be unconditionally stable and convergent with the convergence rate of fourth‐order in space and second‐order in time. Numerical errors and convergence rates of the solution are tested by two examples. Numerical results coincide with the theoretical analysis. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1742–1768, 2015     

New Accurate Expressions in C‐XSC

C‐XSC is a programming library for Extended Scientific Computing which supports many data types with high accuracy. However, it does not provide the accurate expressions which are already available in the older language PASCAL‐XSC. These expressions have to be replaced by long loops that are more difficult to understand. Therefore, we studied possibilities how these accurate expressions could be implemented in C‐XSC. In this paper we present possibilities like operator overloading with special result types avoiding rounding errors, and a small pre‐processor which converts accurate expressions to function calls. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Asymptotic analysis and accurate approximate solutions for strongly nonlinear conservative symmetric oscillators

A new accurate iterative and asymptotic method is introduced to construct analytical approximate solutions to strongly nonlinear conservative symmetric oscillators. The method is based on applying a second-order expansion with the harmonic balance method and it excludes the requirement of solving a set of coupled nonlinear algebraic equations. Newton's iteration or the linearized model may be readily deduced by considering only the first-order terms in the model. According to this iterative approach, only Fourier series expansions of restoring force function, its first- and second-order derivatives for each iteration are required. It is concluded here that by only one single iteration, very brief and yet accurate analytical approximate solutions can be attained. Three physical examples are solved and accurate solutions are presented to illustrate the physics of the system and the effectiveness of the proposed asymptotic method.     

Model reduction by Neumann series expansion

This paper presents a succession-level approximate reduction method to obtain a highly accurate condensation model for large scale structures. Distinct from the previous reduction techniques, all the inertia items can be embodied by using the Neumann series expansion. Then any level approximation can be achieved when the sufficiently accurate modal parameters are obtained. The first-order and the second-order approximate reduction equations are derived in this study. A numerical example is given to verify the proposed method.     

High-order accurate monotone compact running scheme for multidimensional hyperbolic equations

A high-order accurate monotone compact difference scheme proposed earlier by the author for one-dimensional nonstationary hyperbolic equations is extended to multidimensional equations. The resulting scheme is fourth-order accurate in space on a compact stencil and third-order accurate in time. Additionally, the scheme is conservative, absolutely stable, and efficient and can be solved using the running calculation method in space. By computing initial-boundary value problems for the linear advection equation and the nonlinear Hopf equation on refined meshes, it is shown that the orders of grid convergence of the multidimensional scheme are close to the corresponding orders of accuracy in independent variables. For the propagation of a two-dimensional rectangular pulse and the Hopf equation with a discontinuous solution, the multidimensional scheme is shown to inherit the monotonicity of its one-dimensional counterpart.     

The Prevalence and Disadvantage of Min-Counting in Seventh Grade: Problems with Confidence As Well As Accuracy?

In this research, we examined how 200 students in seventh grade (around 12 years old) solved simple addition problems. A cluster approach revealed that less than half of the cohort displayed proficiency with simple addition: 35% predominantly used min-counting and were accurate, and 16% frequently made min-counting errors. Students who frequently used min-counting for simple addition, regardless of accuracy, showed lower achievement in mathematics compared to students who relied on accurate retrieval-based strategies. The findings indicate that for many students at this stage of schooling, performance is characterized by accurate min-counting, which is suggestive of a lack of confidence with retrieval. Further, this pattern of performance appears to be more prominent among girls. Confidence with retrieval may be impeding strategy development and hindering learning. The role of confidence in retrieval development needs to be better understood and possibilities for further research are discussed.     

Optimal control computation for integro-differential aerodynamic equations

In order to maintain spectrally accurate solutions, the grids on which a non-linear physical problem is to be solved must also be obtained by spectrally accurate techniques. The purpose of this paper is to describe a pseudospectral computational method of solving integro-differential systems with quadratic performance index. The proposed method is based on the idea of relating grid points to the structure of orthogonal interpolating polynomials. The optimal control and the trajectory are approximated by the m th degree interpolating polynomial. This interpolating polynomial is spectrally constructed using Legendre–Gauss–Lobatto grid points as the collocation points, and Lagrange polynomials as trial functions. The integrals involved in the formulation of the problem are calculated by Gauss–Lobatto integration rule, thereby reducing the problem to a mathematical programming one to which existing well-developed algorithms may be applied. The method is easy to implement and yields very accurate results. An illustrative example is included to confirm the convergence of the pseudospectral Legendre method, and a comparison is made with an existing result in the literature. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.     

Pseudospectral Legendre-based optimal computation of nonlinear constrained variational problems

It is well known that, spectrally accurate solution can be maintained if the grids on which a nonlinear physical problem is to be solved must be obtained by spectrally accurate techniques. In this paper, the pseudospectral Legendre method for general nonlinear smooth and nonsmooth constrained problems of the calculus of variations is studied. The technique is based on spectral collocation methods in which the trajectory, x(t), is approximated by the Nth degree interpolating polynomial, using Legendre-Gauss-Lobatto points as the collocation points, and Lagrange polynomials as trial functions. The integral involved in the formulation of the problem is approximated based on Legendre-Gauss-Lobatto integration rule, thereby reducing the problem to a nonlinear programming one to which existing well-developed algorithms may be applied. The method is easy to implement and yields very accurate results. Illustrative examples are included to confirm the convergence of the pseudospectral Legendre method. Moreover, a numerical experiment (on a nonsmooth problem) indicates that by applying a smoothing filter procedure to the pseudospectral Legendre approximation, one can recover the nonsmooth solution within spectral accuracy.     

Calcolo degli autovalori di matrici tridiagonali non normali

This article is about the accurate computation of the eigenvalues of nonnormal tridiagonal matrices. Results of a comparison we made between several methods and algorithms are reported, and an algorithm derived from the method presented in [15], [16] is shown to be the only one able to give accurate outputs in case of ill-conditioned eigenvalues.

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Conferenza tenuta da L. Pasquini il 29 maggio 1995     

Space‐time spectral method for two‐dimensional semilinear parabolic equations

In this paper, a high‐order accurate numerical method for two‐dimensional semilinear parabolic equations is presented. We apply a Galerkin–Legendre spectral method for discretizing spatial derivatives and a spectral collocation method for the time integration of the resulting nonlinear system of ordinary differential equations. Our formulation can be made arbitrarily high‐order accurate in both space and time. Optimal a priori error bound is derived in the L²‐norm for the semidiscrete formulation. Extensive numerical results are presented to demonstrate the convergence property of the method, show our formulation have spectrally accurate in both space and time. John Wiley & Sons, Ltd.     

Spurious modes in finite-element discretizations of the wave equation may not be all that bad

If higher-order finite elements are used to discretize the wave equation, spurious modes may occur. These modes are classified as unphysical and supposedly make elements of high order useless for accurate computations. This is in conflict with numerical experiments which appear to provide good results. Here Fourier analysis is used to investigate the behaviour of the numerical error for a number of higher-order one-dimensional finite elements. It is shown that the spurious modes have a contribution to the numerical error that behaves in a reasonable manner, and that higher-order elements can be more accurate than lower-order elements. Lumped elements with Gauss–Lobatto nodes appear to be the best choice.     

An analytic solution of unsteady boundary-layer flows caused by an impulsively stretching plate

In this paper, the unsteady boundary-layer flows caused by an impulsively stretching flat plate is solved by means of an analytic approach. Unlike perturbation techniques, this approach gives accurate analytic approximations uniformly valid for all dimensionless time. Besides, a simple but accurate analytic formula for the local skin friction is given, which agrees well with numerical results and thus is useful in the related industries. To the best of our knowledge, this type of analytic solutions has been never reported. Furthermore, the proposed analytic approach has general meaning and therefore may be applied in the similar way to other unsteady boundary-layer flows to get accurate analytic solutions valid for all time.     

Solidification modelling with a control volume method on domains subjected to viscoplastic deformation

The rise in importance of semi-solid based products has created a need for accurate modelling approaches to coupled solidification and deformation. Current approaches to solidification modelling, using the finite element method (FEM), are principally founded on capacitance methods. Unfortunately they suffer from a major drawback in that energy is not correctly transported through elements, so providing a source of inaccuracy. This paper is concerned with the development and application of a control volume capacitance method (CVCM) to problems where viscoplastic deformation and solidification are combined. The approach adopted is founded on the theory that describes energy transfer through a control volume (CV) moving relative to the deforming mass. This essentially arbitrary Lagrangian–Eulerian (ALE) method facilitates the accurate treatment of discontinuities. The CV approach is tested against known analytical solutions and is shown to be accurate, stable and computationally competitive.