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.
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/9780198528692.003.0009
- Subject:
- Mathematics, Numerical Analysis
This chapter discusses numerical simulations of the incompressible Navier-Stokes equations. Exact Navier-Stokes solutions are presented that are used as benchmarks to validate new codes and evaluate ...
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This chapter discusses numerical simulations of the incompressible Navier-Stokes equations. Exact Navier-Stokes solutions are presented that are used as benchmarks to validate new codes and evaluate the accuracy of a particular discretization. Some aspects of direct numerical simulation (DNS) and large-eddy simulation (LES) are discussed. The issue of stabilization at high Reynolds number is then presented using the concepts of dynamic subgrid modelling, over-integration, and spectral vanishing viscosity. A new parallel paradigm based on multi-level parallelism is introduced that can help realize adaptive refinement more easily. The final section includes a heuristic refinement method for Navier-Stokes equations.Less
This chapter discusses numerical simulations of the incompressible Navier-Stokes equations. Exact Navier-Stokes solutions are presented that are used as benchmarks to validate new codes and evaluate the accuracy of a particular discretization. Some aspects of direct numerical simulation (DNS) and large-eddy simulation (LES) are discussed. The issue of stabilization at high Reynolds number is then presented using the concepts of dynamic subgrid modelling, over-integration, and spectral vanishing viscosity. A new parallel paradigm based on multi-level parallelism is introduced that can help realize adaptive refinement more easily. The final section includes a heuristic refinement method for Navier-Stokes equations.
Gary Smith and Jay Cordes
- Published in print:
- 2019
- Published Online:
- September 2019
- ISBN:
- 9780198844396
- eISBN:
- 9780191879937
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198844396.003.0005
- Subject:
- Mathematics, Applied Mathematics, Numerical Analysis
Computer software, particularly deep neural networks and Monte Carlo simulations, are extremely useful for the specific tasks that they have been designed to do, and they will get even better, much ...
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Computer software, particularly deep neural networks and Monte Carlo simulations, are extremely useful for the specific tasks that they have been designed to do, and they will get even better, much better. However, we should not assume that computers are smarter than us just because they can tell us the first 2000 digits of pi or show us a street map of every city in the world. One of the paradoxical things about computers is that they can excel at things that humans consider difficult (like calculating square roots) while failing at things that humans consider easy (like recognizing stop signs). They can’t pass simple tests like the Winograd Schema Challenge because they do not understand the world the way humans do. They have neither common sense nor wisdom. They are our tools, not our masters.Less
Computer software, particularly deep neural networks and Monte Carlo simulations, are extremely useful for the specific tasks that they have been designed to do, and they will get even better, much better. However, we should not assume that computers are smarter than us just because they can tell us the first 2000 digits of pi or show us a street map of every city in the world. One of the paradoxical things about computers is that they can excel at things that humans consider difficult (like calculating square roots) while failing at things that humans consider easy (like recognizing stop signs). They can’t pass simple tests like the Winograd Schema Challenge because they do not understand the world the way humans do. They have neither common sense nor wisdom. They are our tools, not our masters.
