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 ...
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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 limited. More recently, the need to find accurate solutions to the viscous flow equations around complex configurations has led to the development of high-order discretization procedures on unstructured meshes, which are also recognized as more efficient for solution of time-dependent oscillatory solutions over long time periods. This book, an updated edition on the original text, presents the recent and significant progress in multi-domain spectral methods at both the fundamental and application level. Containing material on discontinuous Galerkin methods, non-tensorial nodal spectral element methods in simplex domains, and stabilization and filtering techniques, this text introduces the use of spectral/hp element methods with particular emphasis on their application to unstructured meshes. It provides a detailed explanation of the key concepts underlying the methods along with practical examples of their derivation and application.Less
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 limited. More recently, the need to find accurate solutions to the viscous flow equations around complex configurations has led to the development of high-order discretization procedures on unstructured meshes, which are also recognized as more efficient for solution of time-dependent oscillatory solutions over long time periods. This book, an updated edition on the original text, presents the recent and significant progress in multi-domain spectral methods at both the fundamental and application level. Containing material on discontinuous Galerkin methods, non-tensorial nodal spectral element methods in simplex domains, and stabilization and filtering techniques, this text introduces the use of spectral/hp element methods with particular emphasis on their application to unstructured meshes. It provides a detailed explanation of the key concepts underlying the methods along with practical examples of their derivation and application.
Jan Modersitzki
- 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 ...
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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 non-parametric transformation. It sets up a general framework for the consideration of different registration techniques, which is based on a variational formulation of the registration problem; the numerical schemes to be considered are based on the Euler-Lagrange equations which characterize a minimizer.Less
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 non-parametric transformation. It sets up a general framework for the consideration of different registration techniques, which is based on a variational formulation of the registration problem; the numerical schemes to be considered are based on the Euler-Lagrange equations which characterize a minimizer.
George Em Karniadakis and Spencer J. Sherwin
- 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 ...
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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 convergence philosophy of spectral and finite element methods, the combination of which provides a dual path of convergence, is also introduced. Topics covered include the basic equations of fluid dynamics and numerical discretizations.Less
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 convergence philosophy of spectral and finite element methods, the combination of which provides a dual path of convergence, is also introduced. Topics covered include the basic equations of fluid dynamics and numerical discretizations.
George Em Karniadakis and Spencer J. Sherwin
- 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 ...
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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 dictate time-step restrictions are discussed. Appropriate preconditioning techniques for inversion of the stiffness matrix, non-smooth solutions due to geometric singularities, and recent advances in three-dimensional domains are discussed. The exercises at the end of the chapter build on the exercises of Chapters 3 and 4 to implement a two-dimensional standard Galerkin hp solution to the Helmholtz problem.Less
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 dictate time-step restrictions are discussed. Appropriate preconditioning techniques for inversion of the stiffness matrix, non-smooth solutions due to geometric singularities, and recent advances in three-dimensional domains are discussed. The exercises at the end of the chapter build on the exercises of Chapters 3 and 4 to implement a two-dimensional standard Galerkin hp solution to the Helmholtz problem.
George Em Karniadakis and Spencer J. Sherwin
- 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 ...
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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 versions from theoretical, computational, and implementation standpoints.Less
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 versions from theoretical, computational, and implementation standpoints.
Gary A. Glatzmaier
- 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. ...
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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. The approach involves updating for the vorticity first and then solving for the fluid velocity each time step. The chapter continues with a discussion of two very different spatial discretizations, whereby the vertical derivatives are approximated with a finite-difference method and the horizontal derivatives with a spectral method. The nonlinear terms are computed in spectral space. The chapter also considers the Adams-Bashforth time integration scheme and explains how the Poisson equation can be solved at each time step for the updated streamfunction given the updated vorticity.Less
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. The approach involves updating for the vorticity first and then solving for the fluid velocity each time step. The chapter continues with a discussion of two very different spatial discretizations, whereby the vertical derivatives are approximated with a finite-difference method and the horizontal derivatives with a spectral method. The nonlinear terms are computed in spectral space. The chapter also considers the Adams-Bashforth time integration scheme and explains how the Poisson equation can be solved at each time step for the updated streamfunction given the updated vorticity.
Gary A. Glatzmaier
- 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 ...
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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 first provides an overview of nonuniform grids before discussing coordinate mapping as an alternative way of achieving spatial discretization. It then describes an approach for treating both the vertical and horizontal directions with simple finite-difference methods: defining a streamfunction, which automatically satisfies mass conservation, and solving for vorticity via the curl of the momentum conservation equation. It also explains the use of the Chebyshev–Fourier method to simulate the convection or gravity wave problem by employing spectral methods in both the horizontal and vertical directions. Finally, it looks at the basic ideas and some issues that need to be addressed with respect to parallel processing as well as choices that need to be made when designing a parallel code.Less
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 first provides an overview of nonuniform grids before discussing coordinate mapping as an alternative way of achieving spatial discretization. It then describes an approach for treating both the vertical and horizontal directions with simple finite-difference methods: defining a streamfunction, which automatically satisfies mass conservation, and solving for vorticity via the curl of the momentum conservation equation. It also explains the use of the Chebyshev–Fourier method to simulate the convection or gravity wave problem by employing spectral methods in both the horizontal and vertical directions. Finally, it looks at the basic ideas and some issues that need to be addressed with respect to parallel processing as well as choices that need to be made when designing a parallel code.
George Em Karniadakis and Spencer J. Sherwin
- 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 ...
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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 Galerkin formulation for advection equations. Eigenspectra of the advection operators in both two and three dimensions are reviewed, which are relevant for explicit time stepping. Two forms of a semi-Lagrangian method for advection (strong and auxiliary forms) that could potentially prove very effective in enhancing the speed and accuracy of spectral/hp element methods in advection-dominated problems are discussed. A new section on stabilization techniques is introduced that discusses filters, spectral vanishing viscosity, and upwind collocation.Less
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 Galerkin formulation for advection equations. Eigenspectra of the advection operators in both two and three dimensions are reviewed, which are relevant for explicit time stepping. Two forms of a semi-Lagrangian method for advection (strong and auxiliary forms) that could potentially prove very effective in enhancing the speed and accuracy of spectral/hp element methods in advection-dominated problems are discussed. A new section on stabilization techniques is introduced that discusses filters, spectral vanishing viscosity, and upwind collocation.
D. Estep
- 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, ...
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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, we discuss error estimation for a widely used numerical approach for multiphysics, multiscale problems called multiscale operator decomposition. In this approach, a multiphysics model is decomposed into components involving simpler physics over a relatively limited range of scales, and the solution is sought through an iterative procedure involving numerical solutions of the individual components. After describing the ingredients of adjoint-based a posteriori analysis, we describe the extension to multiscale operator decomposition solution methods. While the particulars of the analysis vary considerably with the problem, there are several key ideas underlying a general approach to treat operator decomposition multiscale methods.Less
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, we discuss error estimation for a widely used numerical approach for multiphysics, multiscale problems called multiscale operator decomposition. In this approach, a multiphysics model is decomposed into components involving simpler physics over a relatively limited range of scales, and the solution is sought through an iterative procedure involving numerical solutions of the individual components. After describing the ingredients of adjoint-based a posteriori analysis, we describe the extension to multiscale operator decomposition solution methods. While the particulars of the analysis vary considerably with the problem, there are several key ideas underlying a general approach to treat operator decomposition multiscale methods.
G. Samaey, A. J. Roberts, and I. G. Kevrekidis
- 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 ...
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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 numerical simulator that we wish to use over macroscopic scales. Patch dynamics uses only simulations within a number of small regions (surrounding macroscopic grid points) in the space-time domain to approximate a discretization scheme for an unavailable macroscopic equation. The approach was first presented and analyzed for a standard diffusion problem in one space dimension; here, we will discuss subsequent efforts to generalize the approach and extend its analysis. We show how one can modify the definition of the initial and boundary conditions to allow patch dynamics to mimic any finite difference scheme, and we investigate to what extent (and at what computational cost) one can avoid the need for specifically designed patch boundary conditions. One can surround the patches with buffer regions, where one can impose (to some extent) arbitrary boundary conditions. The convergence analysis shows that the required buffer for consistency depends on the coefficients in the macroscopic equation; in general, for advection dominated problems, smaller buffer regions–as compared to those for diffusion-dominated problems–suffice.Less
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 numerical simulator that we wish to use over macroscopic scales. Patch dynamics uses only simulations within a number of small regions (surrounding macroscopic grid points) in the space-time domain to approximate a discretization scheme for an unavailable macroscopic equation. The approach was first presented and analyzed for a standard diffusion problem in one space dimension; here, we will discuss subsequent efforts to generalize the approach and extend its analysis. We show how one can modify the definition of the initial and boundary conditions to allow patch dynamics to mimic any finite difference scheme, and we investigate to what extent (and at what computational cost) one can avoid the need for specifically designed patch boundary conditions. One can surround the patches with buffer regions, where one can impose (to some extent) arbitrary boundary conditions. The convergence analysis shows that the required buffer for consistency depends on the coefficients in the macroscopic equation; in general, for advection dominated problems, smaller buffer regions–as compared to those for diffusion-dominated problems–suffice.
Efstratios Manousakis
- 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 ...
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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 the delta function.Less
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 the delta function.
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 ...
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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 Poisson equation and the Stokes equations. For each PDE, there is a chapter concerned with finite element discretization and a companion chapter concerned with efficient iterative solution of the algebraic equations obtained from discretization. Chapter 5 describes the basics of PDE-constrained optimization. The second part of the book (Chapters 6 to 11) is a more advanced introduction to the numerical analysis of incompressible flows. It starts with four chapters on the convection–diffusion equation and the steady Navier–Stokes equations, organized by equation with a chapter describing discretization coupled with a companion concerned with iterative solution algorithms. The book concludes with two chapters describing discretization and solution methods for models of unsteady flow and buoyancy-driven flow.Less
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 Poisson equation and the Stokes equations. For each PDE, there is a chapter concerned with finite element discretization and a companion chapter concerned with efficient iterative solution of the algebraic equations obtained from discretization. Chapter 5 describes the basics of PDE-constrained optimization. The second part of the book (Chapters 6 to 11) is a more advanced introduction to the numerical analysis of incompressible flows. It starts with four chapters on the convection–diffusion equation and the steady Navier–Stokes equations, organized by equation with a chapter describing discretization coupled with a companion concerned with iterative solution algorithms. The book concludes with two chapters describing discretization and solution methods for models of unsteady flow and buoyancy-driven flow.
Klaus Böhmer
- 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 ...
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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 has to be boundedly invertible. For quasilinear problems, in Sobolev spaces Wm'p(Ω), with 2 ≤ p 〈 ∞, this chapter obtains stability and convergence results with respect to discrete Hm(Ω) norms. This is complemented by the monotone approach for 1 ≤ p 〈 ∞ with Wm'p(Ω) convergence. Our approach allows a unified proof for stability, convergence and Fredholm results for the discrete solutions and their computation. A few well-known basic concepts from functional analysis and approximation theory are combined: coercive bilinear forms or monotone operators, their compact perturbations, interpolation, best approximation and inverse estimates for approximating spaces yield the classical “consistency and stability imply convergence”. The mesh independence principle is the key for an efficient solution for all discretizations of all nonlinear problems considered here.Less
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 has to be boundedly invertible. For quasilinear problems, in Sobolev spaces Wm'p(Ω), with 2 ≤ p 〈 ∞, this chapter obtains stability and convergence results with respect to discrete Hm(Ω) norms. This is complemented by the monotone approach for 1 ≤ p 〈 ∞ with Wm'p(Ω) convergence. Our approach allows a unified proof for stability, convergence and Fredholm results for the discrete solutions and their computation. A few well-known basic concepts from functional analysis and approximation theory are combined: coercive bilinear forms or monotone operators, their compact perturbations, interpolation, best approximation and inverse estimates for approximating spaces yield the classical “consistency and stability imply convergence”. The mesh independence principle is the key for an efficient solution for all discretizations of all nonlinear problems considered here.
Rainer Sommer
- 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 ...
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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 including its Symanzik improvement and renormalization. Matching of the effective theory to QCD is first explained in perturbation theory in the QCD coupling as well as subsequently on a non-perturbative level. The inclusion of 1/M corrections, their renormalization and practical implementation is discussed in detail. The chapter finishes with the description of a practical method for non-perturbative matching of HQET to QCD and examples of results.Less
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 including its Symanzik improvement and renormalization. Matching of the effective theory to QCD is first explained in perturbation theory in the QCD coupling as well as subsequently on a non-perturbative level. The inclusion of 1/M corrections, their renormalization and practical implementation is discussed in detail. The chapter finishes with the description of a practical method for non-perturbative matching of HQET to QCD and examples of results.
Inge Hinterwaldner
- 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 ...
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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 space, the former organizes dynamics. There exist various attempts to reformulate a kind of perspectivation in the domain of new media. They differ from the proposed as they are not elaborated with respect to the rules of its construction logic. In order to assimilate the new, for the ‘systems perspective’, the eye point is compared with parameters or variables, the vanishing point with the aspect of the real time, and the cut through the visual pyramid with the phase space. A first critique of the simulation dynamic is conveyed with the argumentative help of Jean Baudrillard. It touches the generalization through laws, the discretization of dynamic, and the specific mode of communication.Less
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 space, the former organizes dynamics. There exist various attempts to reformulate a kind of perspectivation in the domain of new media. They differ from the proposed as they are not elaborated with respect to the rules of its construction logic. In order to assimilate the new, for the ‘systems perspective’, the eye point is compared with parameters or variables, the vanishing point with the aspect of the real time, and the cut through the visual pyramid with the phase space. A first critique of the simulation dynamic is conveyed with the argumentative help of Jean Baudrillard. It touches the generalization through laws, the discretization of dynamic, and the specific mode of communication.
Howard C. Elman, David J. Silvester, and Andrew J. Wathen
- 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 ...
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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 of the linearized systems.Less
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 of the linearized systems.
Howard C. Elman, David J. Silvester, and Andrew J. Wathen
- 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 ...
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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 iterative solution algorithms of the linearized systems.Less
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 iterative solution algorithms of the linearized systems.
Howard C. Elman, David J. Silvester, and Andrew J. Wathen
- 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.
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.
Howard C. Elman, David J. Silvester, and Andrew J. Wathen
- 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 ...
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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 algorithms for the resulting saddle-point problems.Less
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 algorithms for the resulting saddle-point problems.
George Jaroszkiewicz
- 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 ...
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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 how discrete dynamical systems can be described by difference equations. To do this the chapter first explains how discrete time mechanics action sums are related to continuous time action integrals, introducing the concept of system function. System functions are the discrete time analogues of Lagrangians in continuous time mechanics. The chapter discusses their relationship to Hamilton’s principal function. It uses a discrete time action principle to derive the discrete time equations of motion associated with a given system function. The chapter discusses Caldirola’s microverse model of the electron and show how Maxwell’s continuous time partial differential equations for electrodynamics can be discretized in terms of discrete-time difference equations.Less
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 how discrete dynamical systems can be described by difference equations. To do this the chapter first explains how discrete time mechanics action sums are related to continuous time action integrals, introducing the concept of system function. System functions are the discrete time analogues of Lagrangians in continuous time mechanics. The chapter discusses their relationship to Hamilton’s principal function. It uses a discrete time action principle to derive the discrete time equations of motion associated with a given system function. The chapter discusses Caldirola’s microverse model of the electron and show how Maxwell’s continuous time partial differential equations for electrodynamics can be discretized in terms of discrete-time difference equations.
