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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


NON-PARAMETRIC IMAGE REGISTRATION

Jan Modersitzki

in Numerical Methods for Image Registration

Published in print:
2003
Published Online:
September 2007
ISBN:
9780198528418
eISBN:
9780191713583
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198528418.003.0008
Subject:
Mathematics, Applied Mathematics

This chapter introduces non-parametric registrations. The idea behind this type of registration is to come up with an appropriate measure both for the similarity as well as for the likelihood of a ... More


INTRODUCTION

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/9780198528692.003.0001
Subject:
Mathematics, Numerical Analysis

This chapter presents reduced models of the compressible and incompressible Navier-Stokes equations which are used in the various discretization concepts discussed in the rest of the book. The ... More


DIFFUSION EQUATION

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/9780198528692.003.0005
Subject:
Mathematics, Numerical Analysis

This chapter considers the diffusion equation: an implicit in time discretization leads to the Helmholtz equation. Both the temporal discretization and eigenspectra of second-order operators that ... More


NON-CONFORMING ELEMENTS

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/9780198528692.003.0007
Subject:
Mathematics, Numerical Analysis

This chapter discusses the topic of non-conforming elements for second-order operators. It includes a comprehensive presentation of the discontinuous Galerkin method with a comparison of different ... 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


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


ADVECTION AND ADVECTION DIFFUSION

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/9780198528692.003.0006
Subject:
Mathematics, Numerical Analysis

This chapter focuses on the scalar advection equation and develops a Galerkin discretization using the techniques described in Chapter 4. It includes an extended presentation of the discontinuous ... More


Error Estimates for Multiscale Operator Decomposition for Multiphysics Models

D. Estep

in Multiscale Methods: Bridging the Scales in Science and Engineering

Published in print:
2009
Published Online:
February 2010
ISBN:
9780199233854
eISBN:
9780191715532
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199233854.003.0011
Subject:
Mathematics, Applied Mathematics

Multiphysics, multiscale models present significant challenges in terms of computing accurate solutions and for estimating the error in information computed from numerical solutions. In this chapter, ... More


Equation-Free Computation: An Overview of Patch Dynamics

G. Samaey, A. J. Roberts, and I. G. Kevrekidis

in Multiscale Methods: Bridging the Scales in Science and Engineering

Published in print:
2009
Published Online:
February 2010
ISBN:
9780199233854
eISBN:
9780191715532
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199233854.003.0008
Subject:
Mathematics, Applied Mathematics

This chapter overviews recent progress in the development of patch dynamics, an essential ingredient of the equation-free framework. In many applications we have a given detailed microscopic ... More


Back to the Schrödinger equation on a lattice

Efstratios Manousakis

in Practical Quantum Mechanics: Modern Tools and Applications

Published in print:
2015
Published Online:
December 2015
ISBN:
9780198749349
eISBN:
9780191813474
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198749349.003.0003
Subject:
Physics, Atomic, Laser, and Optical Physics

This chapter goes back to the discrete lattice in order to transform the basis to momentum eigenstates. In addition, the chapter takes the continuum limit of space and we recover known relations and ... More


Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics

Howard Elman, David Silvester, and Andy Wathen

Published in print:
2014
Published Online:
September 2014
ISBN:
9780199678792
eISBN:
9780191780745
Item type:
book
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199678792.001.0001
Subject:
Mathematics, Numerical Analysis, Computational Mathematics / Optimization

The subject of this book is the efficient solution of partial differential equations (PDEs) that arise when modelling incompressible fluid flow. The first part (Chapters 1 through 5) covers the ... More


A general discretization theory

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/9780199577040.003.0003
Subject:
Physics, Theoretical, Computational, and Statistical Physics

A new general discretization theory unifies the generalized Petrov-Galerkin method and one of the classical methods. Linearization is a main tool: the derivative of the operator in the exact solution ... More


Non-perturbative heavy quark effective theory

Rainer Sommer

in Modern Perspectives in Lattice QCD: Quantum Field Theory and High Performance Computing: Lecture Notes of the Les Houches Summer School: Volume 93, August 2009

Published in print:
2011
Published Online:
January 2012
ISBN:
9780199691609
eISBN:
9780191731792
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199691609.003.0009
Subject:
Physics, Theoretical, Computational, and Statistical Physics

This chapter describes a heuristic derivation of the HQET Lagrangian from continuum QCD and discusses the renormalization of the effective field theory. A lattice discretization is introduced ... More


The Perspectivation of Simulations

Inge Hinterwaldner

in The Systemic Image: A New Theory of Interactive Real-Time Simulations

Published in print:
2017
Published Online:
September 2017
ISBN:
9780262035040
eISBN:
9780262335546
Item type:
chapter
Publisher:
The MIT Press
DOI:
10.7551/mitpress/9780262035040.003.0003
Subject:
Society and Culture, Technology and Society

Computer simulations are introduced as processes shaped according to the ‘systems perspective’. This expression is a parallel to central perspective. While the latter is a specific medium to form ... More


Solution of Unsteady Navier–Stokes equations

Howard C. Elman, David J. Silvester, and Andrew J. Wathen

in Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics

Published in print:
2014
Published Online:
September 2014
ISBN:
9780199678792
eISBN:
9780191780745
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199678792.003.0011
Subject:
Mathematics, Numerical Analysis, Computational Mathematics / Optimization

This chapter concerns the numerical solution of time-dependent Navier–Stokes equations. The emphasis is on implicit time stepping in conjunction with adaptive time-step control and iterative solution ... More


Solution of Models of Buoyancy-Driven Flow

Howard C. Elman, David J. Silvester, and Andrew J. Wathen

in Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics

Published in print:
2014
Published Online:
September 2014
ISBN:
9780199678792
eISBN:
9780191780745
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199678792.003.0012
Subject:
Mathematics, Numerical Analysis, Computational Mathematics / Optimization

This chapter concerns the numerical solution of the Boussinesq equations, which model buoyancy-driven flow. The focus is on implicit time-stepping methods and time-step control, together with ... More


The Poisson Equation

Howard C. Elman, David J. Silvester, and Andrew J. Wathen

in Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics

Published in print:
2014
Published Online:
September 2014
ISBN:
9780199678792
eISBN:
9780191780745
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199678792.003.0002
Subject:
Mathematics, Numerical Analysis, Computational Mathematics / Optimization

This chapter concerns the statement of the Poisson equation and its weak formulation. This is followed by a description of finite element discretization and properties of the discrete problem.


Optimization with PDE Constraints

Howard C. Elman, David J. Silvester, and Andrew J. Wathen

in Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics

Published in print:
2014
Published Online:
September 2014
ISBN:
9780199678792
eISBN:
9780191780745
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780199678792.003.0006
Subject:
Mathematics, Numerical Analysis, Computational Mathematics / Optimization

This chapter is an introduction to techniques of optimization with constraints defined by partial differential equations. It discusses optimization and discretization methods together with solution ... More


Discrete time

George Jaroszkiewicz

in Images of Time: Mind, Science, Reality

Published in print:
2016
Published Online:
January 2016
ISBN:
9780198718062
eISBN:
9780191787553
Item type:
chapter
Publisher:
Oxford University Press
DOI:
10.1093/acprof:oso/9780198718062.003.0023
Subject:
Physics, Particle Physics / Astrophysics / Cosmology

This chapter discusses difference equations and how they arise from the temporal discretization of differential equations, emphasizing that temporal discretization is not a unique process. It shows ... More


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