## 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