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.
