On Double Chebyshev Series Approximation

The Numerical Solution of Integral Equations on the Half-Line

Numerial Methods for Volterra Integral Equations with Singular Kernels

One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into account weak non–locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non–locality introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes. In particular, we study a non–trivial limit when both components are incompressible, the pressure is kinematic but the velocity field is non–solenoidal (quasi–incompressibility). To demonstrate the effects of quasi–incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show that when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion even if the fluids are inviscid. In the limit of infinitely thin and well–separated interfacial layers, an appropriately scaled quasi–incompressible Euler–Cahn–Hilliard system converges to the classical sharp interface model. In order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity.

Two higher-order compact finite difference approaches, the Hermitian and the Lax-Wendroff are examined by applying them to the viscous Burgers' equation. The difference equations obtained by the two methods were integrated in time through the third-order Strong-Stability-Preserving Runge-Kutta scheme. Absolute errors are computed by using an exact solution. The results are also compared with a second order central difference scheme. The Hermitian approach is far easier to implement. On uniform grids the Lax-Wendroff scheme produces smaller errors during the initial stages, but both methods are equally good for larger durations of integration. The convergence rate of the Hermitian scheme is slightly higher than the Lax-Wendroff scheme although both are of fourth order. Both schemes are unstable beyond a certain step size in time and space. When numerical boundary conditions are imposed, second-order conditions produce the best results whereas linear extrapolation proved to be the worst. It was also observed that large domains were required to implement the numerical boundary conditions properly. There was no detrimental effect on the accuracy of the results obtained through either of the two schemes when the size of the domain was greatly increased. Both schemes showed remarkable improvement in accuracy when clustered grids were employed. However much smaller time steps are required for stable solutions.

A numerical and laboratory investigation on the purging of saline fluid from a cavity within a $60^\circ$~channel bend is presented to characterise the purging process in open channel bends. The numerical simulation uses a three dimensional, direct numerical simulation, fractional-step solver. The geometry replicates the laboratory experiment to define the mixing processes of a cavity with vertical sidewalls occupying the centre~$20^\circ$ of the $60^\circ$~channel bend. The secondary circulation as the flow moves through the bend increases the vertical momentum exchange within the cavity and hence destabilises the saline interface. References J. R. Anderson and A. K. Morrison. Environmental consequences of saline groundwater intrusion into the wimmera river, victoria. BMR. Journal of Australian Geology and Geophysics, 11:233--235, 1989d. S. Armfield and R. Street. An analysis and comparison of the time accuracy of fractional-step methods for the navier-stokes equations on staggered grids. International Journal for Numerical Methods in Fluids, 38(3):255--282, 2002. S. W. Armfield and W. Debler. Purging of density stabilized basins. International Journal of Heat and Mass Transfer, 36(2):519--530, January 1993. J. C. Bathurst, C. R. Thorne, and R. D. Hey. Secondary flow and shear stress at river bends. Journal of the Hydraulics Division, ASCE, 105 (HY10):1277---1295, 1979. H. Christie. Real-life management of an environmental water reserve, a wimmera perspective. In A.L. Wilson, R.L. Dehaan, R.J. Watts, K.J. Page, K.H. Bowmer, and A. Curtis, editors, Proceedings of the 5th Australian Stream Management Conference. Australian rivers: making a difference. Charles Sturt University, Thurgoona, New South Wales., 2007. W. Debler and S. W. Armfield. The purging of saline water from rectangular and trapezoidal cavities by an overflow of turbulent sweet water. Journal of Hydraulic Research, 35:43--62, 1997. P. H. Gaskell and A. K. C. Lau. Curvature-compensated convective transport: Smart, a new boundedness- preserving transport algorithm. International Journal for Numerical Methods in Fluids, 8(6):617--641, 1988. M. P. Kirkpatrick and S. W. Armfield. Experimental and large eddy simulation results for the purging of salt water from a cavity by an overflow of fresh water. International Journal of Heat and Mass Transfer, 48(2):341--359, January 2005. Henk A. van der Vorst. Iterative solution methods for certain sparse linear systems with a non-symmetric matrix arising from pde-problems. Journal of Computational Physics, 44(1):1--19, November 1981.

The following article from the International Journal for Numerical Methods in Fluids ‘A flow‐condition‐based interpolation finite element procedure for triangular grids’ by Haruhiko Kohno and Klaus‐Jürgen Bathe published online on 3 August 2005 in Wiley InterScience (www.interscience.wiley.com) and in print on 10 October 2005 in the International Journal for Numerical Methods in Fluids Volume 49 Issue 8 pp. 849–875 has been retracted and replaced by agreement between the authors, the journal editors and John Wiley & Sons, Ltd. The article has been retracted due to printing errors which have been corrected in the replacement article.(DOI: 10.1002/fld.1246)

Surface gravity wave interaction with a submerged horizontal flexible porous plate

Wave structure interaction problems in a three-layer fluid having an elastic plate covered free surface are studied in a three-dimensional fluid domain in both the cases of finite and infinite water depths. Wave characteristics are analyzed from the dispersion relation of the associated wave motion, and approximate results are derived in both the cases of deep water and shallow water waves. Further, the expansion formulae and the associated orthogonal mode-coupling relations are derived for the velocity potentials for the wave structure interaction problems in channels of finite and infinite depths. The utility of the expansion formulae is demonstrated by (1) deriving the source potentials associated with the wave structure interaction problems in a three-layer fluid medium of finite and infinite water depths and (2) analyzing the wave scattering by a partially frozen crack in a floating ice sheet in the three-layer fluid medium in a three-dimensional channel of finite water depth. Various results derived can be used to deal with acoustic wave interaction with flexible structures and other wave structure interaction problems of similar nature arising in different branches of physics and engineering.

Periodic waves propagating at a constant velocity at the surface of a fluid with constant vorticity in water of infinite depth are considered. The problem is solved numerically by a boundary-integral-equation method. Simmen & Saffman (Stud. Appl. Maths 75, 35, 1985) showed that there are families of solutions which have limiting configurations with a 120° angle at their crests or a trapped bubble at their troughs. It is shown that there are additional families of solutions. These families have limiting configurations with trapped bubbles at their crests. Each bubble is circular and contains fluid in rigid-body rotation. The results are consistent with previous calculations for solitary waves in water of finite depth.

Numerical simulation of a square cavity was conducted to validate an implementation of an Immersed Boundary Method (IBM). The cavity consists of two differentially heated side walls and adiabatic top and bottom walls. A Cartesian grid is used with a finite volume, fractional step pressure correction method. Simulations use Dirichlet boundaries for vertical walls and Neumann boundaries for horizontal walls. The Immersed Boundary Method involves modifying the Navier--Stokes equations to include a forcing function in the momentum and energy equations that creates a virtual boundary. This method is useful because the boundary does not necessarily have to coincide with grid points; however, it is much less computationally expensive than other similar methods such as the cut cell method. The IBM is commonly used in simulations involving complex objects and can also accommodate moving boundaries. A standard numerical simulation with grid aligned with the boundary is first compared with previous results. The same geometry is then simulated by tilting the grid at various angles and using the IBM for each of the walls, and comparing these results with those initially obtained. We detail the implementation method and common problems associated with this. Velocity and temperature profiles are presented and the IBM is shown to maintain second order spatial accuracy. References E. A. Fadlun, R. Verzicco, P. Orlandi, and J. Mohd-Yusof. Combined immersed-boundary finite-difference methods for three-dimensional complex flow simulations. Journal of Computational Physics, 161(1):35--60, 2000. doi:10.1006/jcph.2000.6484 T. Gao, Y. H. Tseng, and X. Y. Lu. An improved hybrid cartesian/immersed boundary method for fluid-solid flows. International Journal for Numerical Methods in Fluids, 55(12):1189--1211, 2007. doi:10.1002/fld.1522 M. P. Kirkpatrick and S. W. Armfield. Experimental and large eddy simulation results for the purging of salt water from a cavity by an overflow of fresh water. International Journal of Heat and Mass Transfer, 48(2):341--359, 2005. doi:10.1016/j.ijheatmasstransfer.2004.08.016 M. P. Kirkpatrick, S. W. Armfield, and J. H. Kent. A representation of curved boundaries for the solution of the navier-stokes equations on a staggered three-dimensional cartesian grid. Journal of Computational Physics, 184(1):1--36, 2003. doi:10.1016/S0021-9991(02)00013-X B. P. Leonard and Simin Mokhtari. Beyond first-order upwinding: The ultra-sharp alternative for non-oscillatory steady-state simulation of convection. International Journal for Numerical Methods in Engineering, 30(4):729--766, 1990. doi:10.1002/nme.1620300412 S. H. Peng and L. Davidson. Large eddy simulation for turbulent buoyant flow in a confined cavity. volume 22, pages 323--331, 2001. doi:10.1016/S0142-727X(01)00095-9 C. S. Peskin. Numerical-analysis of blood-flow in heart. Journal of Computational Physics, 25(3):220--252, 1977. doi:10.1016/0021-9991(77)90100-0 J. Salat, S. Xin, P. Joubert, A. Sergent, F. Penot, and P. Le Quere. Experimental and numerical investigation of turbulent natural convection in a large air-filled cavity. International Journal of Heat and Fluid Flow, 25:824--832, 2004. doi:10.1016/j.ijheatfluidflow.2004.04.003 Y. H. Tseng and J. H. Ferziger. A ghost-cell immersed boundary method for flow in complex geometry. Journal of Computational Physics, 192(2):593--623, 2003. doi:10.1016/j/jcp.2003.07.024 N. Zhang and Z. C. Zheng. An improved direct-forcing immersed-boundary method for finite difference applications. Journal of Computational Physics, 221(1):250--268, 2007. doi:10.1016/j.jcp.2006.06.012 N. Zhang, Z. C. Zheng, and S. Eckels. Study of heat-transfer on the surface of a circular cylinder in flow using an immersed-boundary method. International Journal of Heat and Fluid Flow, 29(6):1558--1566, 2008. doi:10.1016/j.ijheatfluidflow.2008.08.009

Divergence-free discontinuous Galerkin (DG) finite element methods offer a suitable discretization for the pointwise divergence-free numerical solution of Borrvall and Petersson's model for the topology optimization of fluids in Stokes flow [Topology optimization of fluids in Stokes flow, International Journal for Numerical Methods in Fluids 41 (1) (2003) 77--107]. The convergence results currently found in literature only consider $H^1$-conforming discretizations for the velocity. In this work, we extend the numerical analysis of Papadopoulos and Suli to divergence-free DG methods with an interior penalty [I. P. A. Papadopoulos and E. Suli, Numerical analysis of a topology optimization problem for Stokes flow, arXiv preprint arXiv:2102.10408, (2021)]. We show that, given an isolated minimizer to the analytical problem, there exists a sequence of DG finite element solutions, satisfying necessary first-order optimality conditions, that strongly converges to the minimizer.

In this paper we use a contour integral method to derive a bilateral generating function in the form of a double series involving Chebyshev polynomials expressed in terms of the incomplete gamma function. Generating functions for the Chebyshev polynomial are also derived and summarized. Special cases are evaluated in terms of composite forms of both Chebyshev polynomials and the incomplete gamma function.

Active power in a given frequency band is one of the most important parameters of electrical power grids. To measure active power, the most popular methods are based on averaging of instantaneous power or low-pass filtration of instantaneous power. Regardless of the measurement method, ADC nonlinearity is one of the most important causes of measurement error. In paper the Chebyshev polynomials are applied to evaluate the ADC nonlinearity and its influence on the active power measurement error. A research of influence of the signal parameters (voltage and current amplitudes, frequency, phase shift between voltage and current) and nonlinearity parameters on the active power measurement error is performed. The effect of the Chebyshev polynomial order on the active power measurement error is considered in detail. Analytical relations are obtained to assess this effects. A simplified approach of determining of the approximation polynomial parameters by a typical nonlinearity dependence is proposed. By the application of Matlab, a simulation model was perform that allows to estimate the reliability of obtained analytical expressions. Based on the research results, an estimation of the active power measurement error, caused by ADC nonlinearity with Chebyshev polynomials approximation is proposed.

The transport of one-speed neutrons has been studied in isotropically scattering slab using ultraspherical polynomial approximation whose sub-cases are spherical harmonics approximation and Chebyshev polynomial approximation of first and second kind with reflective boundaries. An integral equation for the total flux has been derived and solved numerically for different values of c and reflection coefficients. In the solution, Marshak type vacuum boundary conditions were used. The critical thicknesses obtained by the Chebyshev polynomial method are in good agreement with literature values. Computations were made by using the TN, UN and PN approximations for comparison.

The development of science and technology is expectedly one of the ultimate goals of every responsible government. It remains the aspiration towards which all science programmes worldwide are geared. Though, the extent to which the target is approached differs from country to country and is a function of several variables including academic planning. The aim of this paper is to examine the challenges of academic planning for effective teaching and learning for science education. All in all, the paper x-rays the role of science education towards ensuring sustainable development and the need to strengthen all the mechanism of integrated science education through proper academic planning.