Numerical Analysis

Acknowledgements: This article will appear in the forthcoming Princeton Companion to Mathematics, edited by Timothy Gowers with June Barrow-Green, to be published by Princeton University Press.\ud \ud In preparing this essay I have benefitted from the advice of many colleagues who corrected a number of errors of fact and emphasis. I have not always followed their advice, however, preferring as one friend put it, to "put my head above the parapet". So I must take full responsibility for errors and omissions here.\ud \ud With thanks to: Aurelio Arranz, Alexander Barnett, Carl de Boor, David Bindel, Jean-Marc Blanc, Mike Bochev, Folkmar Bornemann, Richard Brent, Martin Campbell-Kelly, Sam Clark, Tim Davis, Iain Duff, Stan Eisenstat, Don Estep, Janice Giudice, Gene Golub, Nick Gould, Tim Gowers, Anne Greenbaum, Leslie Greengard, Martin Gutknecht, Raphael Hauser, Des Higham, Nick Higham, Ilse Ipsen, Arieh Iserles, David Kincaid, Louis Komzsik, David Knezevic, Dirk Laurie, Randy LeVeque, Bill Morton, John C Nash, Michael Overton, Yoshio Oyanagi, Beresford Parlett, Linda Petzold, Bill Phillips, Mike Powell, Alex Prideaux, Siegfried Rump, Thomas Schmelzer, Thomas Sonar, Hans Stetter, Gil Strang, Endre Süli, Defeng Sun, Mike Sussman, Daniel Szyld, Garry Tee, Dmitry Vasilyev, Andy Wathen, Margaret Wright and Steve Wright

Parallel Factorizations in Numerical Analysis

In this paper we review the parallel solution of sparse linear systems, usually deriving by the discretization of ODE-IVPs or ODE-BVPs. The approach is based on the concept of parallel factorization of a (block) tridiagonal matrix. This allows to obtain efficient parallel extensions of many known matrix factorizations, and to derive, as a by-product, a unifying approach to the parallel solution of ODEs.Comment: 15 pages, 5 figure

Theoretical and Analytical Investigation of Electromagnetic Problems Using Dispersive Material and the Kramers-Kronig Transformations

In reality, there is no material with constant permittivity, permeability, and conductivity values over the entire frequency spectrum. The variation in these parameters is well-known as the dispersion phenomenon, which can be analytically interpreted using the Kramers-Kronig relations. Through this thesis, we extensively explain how to take advantage of the dispersion in these parameters to numerically investigate some electromagnetic problems. Among these is a practical problem of separating the electric conductive losses from the dielectric losses of any dispersive lossy material. On the other hand, the performance of particular electrically small antennas is improved by exploiting the frequency dispersion in the dielectric and magnetic material. Electrically small antennas have great attention in many applications due to their drastically reduced size. However, the size reduction comes at the expense of performance degradation, such as increasing the internally stored electromagnetic energy, narrowing the operating bandwidth, decreasing radiation efficiency, and poor matching to surrounding media, particularly when the antenna element has direct contact with a lossy medium like biomedical tissues. Therefore, we try to numerically investigate the performance of these antennas by coating them with dispersive dielectric or magnetic material. In this thesis, some artificially synthesized material with frequency-dependent permittivity, permeability, and electric conductivity values over a wide range of frequencies are suggested to represent the dispersive lossy material. The Kramers-Kronig (KK) relations are employed as a mathematical solution to interrelate the real and imaginary parts of the suggested frequency-dependent relative permittivity and permeability values of the artificial material. Finally, the solution methods are verified by applying them to real-world material found in the literature

Mode analysis of numerical geodynamo models

It has been suggested in Hoyng (2009) that dynamo action can be analysed by expansion of the magnetic field into dynamo modes and statistical evaluation of the mode coefficients. We here validate this method by analysing a numerical geodynamo model and comparing the numerically derived mean mode coefficients with the theoretical predictions. The model belongs to the class of kinematically stable dynamos with a dominating axisymmetric, antisymmetric with respect to the equator and non-periodic fundamental dynamo mode. The analysis requires a number of steps: the computation of the so-called dynamo coefficients, the derivation of the temporally and azimuthally averaged dynamo eigenmodes and the decomposition of the magnetic field of the numerical geodynamo model into the eigenmodes. For the determination of the theoretical mode excitation levels the turbulent velocity field needs to be projected on the dynamo eigenmodes. We compare the theoretically and numerically derived mean mode coefficients and find reasonably good agreement for most of the modes. Some deviation might be attributable to the approximation involved in the theory. Since the dynamo eigenmodes are not self-adjoint a spectral interpretation of the eigenmodes is not possible

Numerical Analysis of Crosss Flow Hydokinetic Turbine by Using Computational Fluid Dynamics

The invention of cross flow turbine industry from straight blades of the Darrieus turbine was modified by Alexander Gorlov into helical shape. There have been several research projects dealing with the design and analysis for tidal applications. This paper deals with the Numerical analysis of a cross flow hydrokinetic turbine (CFHT) with helical blades. Static analysis with optimum blade velocity and constant pressure conditions was performed for the blade with fixed pitch by using Computational Fluid Dynamics (CFD) in Fluent 15. Solidworks was used to carry out 3D modeling of the turbine. The hydrofoil shape of NACA 0018 was created by the airfoil coordinate database. Two different turbulence models Spalart-Allmaras (One-Equation model) and sst-k (Two ndash;Equation model) were employed to compute and compare the results. Pressure profiles, drag and lift coefficients are calculated under a steady flow of 1.5 m/s