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Spectral/hp Element Methods for Computational Fluid Dynamics

George Karniadakis and Spencer Sherwin

Published in print:
2005
Published Online:
September 2007
ISBN:
9780198528692
eISBN:
9780191713491
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198528692.001.0001
Subject:
Mathematics, Numerical Analysis

Spectral methods have long been popular in direct and large eddy simulation of turbulent flows, but their use in areas with complex-geometry computational domains has historically been much more ... More


FUNDAMENTAL CONCEPTS IN ONE DIMENSION

George Em Karniadakis and Spencer J. Sherwin

in Spectral/hp Element Methods for Computational Fluid Dynamics

Published in print:
2005
Published Online:
September 2007
ISBN:
9780198528692
eISBN:
9780191713491
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198528692.003.0002
Subject:
Mathematics, Numerical Analysis

This chapter discusses the fundamental concepts that are further developed in the remainder of the book, in the context of a one-dimensional formulation. In doing so, the principles and underlying ... More


Spatial Discretizations

Gary A. Glatzmaier

in Introduction to Modeling Convection in Planets and Stars: Magnetic Field, Density Stratification, Rotation

Published in print:
2013
Published Online:
October 2017
ISBN:
9780691141725
eISBN:
9781400848904
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691141725.003.0009
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This chapter considers two ways of employing a spatial resolution that varies with position within a finite-difference method: using a nonuniform grid and mapping to a new coordinate variable. It ... More


Numerical Method

Gary A. Glatzmaier

in Introduction to Modeling Convection in Planets and Stars: Magnetic Field, Density Stratification, Rotation

Published in print:
2013
Published Online:
October 2017
ISBN:
9780691141725
eISBN:
9781400848904
Item type:
chapter
Publisher:
Princeton University Press
DOI:
10.23943/princeton/9780691141725.003.0002
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This chapter describes a numerical method for solving equations of thermal convection on a computer. It begins by introducing the vorticity-streamfunction formulation as a means of conserving mass. ... More


Networks: An Introduction

Mark Newman

Published in print:
2010
Published Online:
September 2010
ISBN:
9780199206650
eISBN:
9780191594175
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199206650.001.0001
Subject:
Physics, Theoretical, Computational, and Statistical Physics

The scientific study of networks, including computer networks, social networks, and biological networks, has received an enormous amount of interest in the last few years. The rise of the Internet ... More


Variational methods for wavelets, with S. Dahlke

Klaus Böhmer

in Numerical Methods for Nonlinear Elliptic Differential Equations: A Synopsis

Published in print:
2010
Published Online:
January 2011
ISBN:
9780199577040
eISBN:
9780191595172
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199577040.003.0009
Subject:
Physics, Theoretical, Computational, and Statistical Physics

Chapter 9 studies S. Dahlke studies wavelet methods. This is the first presentation in this generality and is appropriate for nonlinear problems and bifurcation for elliptic PDEs. As a consequence of ... More


Spectral Methods for Dimensionality Reduction

Saul Lawrence K., Weinberger Kilian Q., Sha Fei, Ham Jihun, and Lee Daniel D.

in Semi-Supervised Learning

Published in print:
2006
Published Online:
August 2013
ISBN:
9780262033589
eISBN:
9780262255899
Item type:
chapter
Publisher:
The MIT Press
DOI:
10.7551/mitpress/9780262033589.003.0016
Subject:
Computer Science, Machine Learning

This chapter provides an overview of unsupervised learning algorithms that can be viewed as spectral methods for linear and nonlinear dimensionality reduction. Spectral methods have recently emerged ... More


Spectral Methods

S. G. Rajeev

in Fluid Mechanics: A Geometrical Point of View

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198805021
eISBN:
9780191843136
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198805021.003.0013
Subject:
Physics, Soft Matter / Biological Physics, Condensed Matter Physics / Materials

Thenumerical solution of ordinary differential equations (ODEs)with boundary conditions is studied here. Functions are approximated by polynomials in a Chebychev basis. Sections then cover spectral ... More


Fluid Mechanics: A Geometrical Point of View

S. G. Rajeev

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198805021
eISBN:
9780191843136
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198805021.001.0001
Subject:
Physics, Soft Matter / Biological Physics, Condensed Matter Physics / Materials

Starting with a review of vector fields and their integral curves, the book presents the basic equations of the subject: Euler and Navier–Stokes. Some solutions are studied next: ideal flows using ... More


Hamiltonian Systems Based on a Lie Algebra

S. G. Rajeev

in Fluid Mechanics: A Geometrical Point of View

Published in print:
2018
Published Online:
October 2018
ISBN:
9780198805021
eISBN:
9780191843136
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/oso/9780198805021.003.0010
Subject:
Physics, Soft Matter / Biological Physics, Condensed Matter Physics / Materials

There is a remarkable analogy between Euler’s equations for a rigid body and his equations for an ideal fluid. The unifying idea is that of a Lie algebra with an inner product, which is not ... More


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