Reinhard B. Neder and Thomas Proffen
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780199233694
- eISBN:
- 9780191715563
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199233694.003.0004
- Subject:
- Physics, Crystallography: Physics
In most cases, simulations of disordered materials are performed to understand experimental observations, in this case diffraction data. This chapter discusses the calculation of several experimental ...
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In most cases, simulations of disordered materials are performed to understand experimental observations, in this case diffraction data. This chapter discusses the calculation of several experimental quantities: single crystal diffuse scattering, powder diffraction, and the atomic pair distribution function (PDF). Since diffraction data are obtained via a Fourier transform, the finite size of the model crystal as well as issues concerning coherence are discussed in detail. The PDF is basically calculated from the atomic structure directly. Different ways to incorporate thermal motion are illustrated.Less
In most cases, simulations of disordered materials are performed to understand experimental observations, in this case diffraction data. This chapter discusses the calculation of several experimental quantities: single crystal diffuse scattering, powder diffraction, and the atomic pair distribution function (PDF). Since diffraction data are obtained via a Fourier transform, the finite size of the model crystal as well as issues concerning coherence are discussed in detail. The PDF is basically calculated from the atomic structure directly. Different ways to incorporate thermal motion are illustrated.
Rob H. Bisseling
- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198529392
- eISBN:
- 9780191712869
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198529392.001.0001
- Subject:
- Mathematics, Applied Mathematics
This book explains the use of the bulk synchronous parallel (BSP) model and the BSPlib communication library in parallel algorithm design and parallel programming. The main topics treated in the book ...
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This book explains the use of the bulk synchronous parallel (BSP) model and the BSPlib communication library in parallel algorithm design and parallel programming. The main topics treated in the book are central to the area of scientific computation: solving dense linear systems by Gaussian elimination, computing fast Fourier transforms, and solving sparse linear systems by iterative methods based on sparse matrix-vector multiplication. Each topic is treated in depth, starting from the problem formulation and a sequential algorithm, through a parallel algorithm and its cost analysis, to a complete parallel program written in C and BSPlib, and experimental results obtained using this program on a parallel computer. Throughout the book, emphasis is placed on analyzing the cost of the parallel algorithms developed, expressed in three terms: computation cost, communication cost, and synchronization cost. The book contains five example programs written in BSPlib, which illustrate the methods taught. These programs are freely available as the package BSPedupack. An appendix on the message-passing interface (MPI) discusses how to program in a structured, bulk synchronous parallel style using the MPI communication library, and presents MPI equivalents of all the programs in the book.Less
This book explains the use of the bulk synchronous parallel (BSP) model and the BSPlib communication library in parallel algorithm design and parallel programming. The main topics treated in the book are central to the area of scientific computation: solving dense linear systems by Gaussian elimination, computing fast Fourier transforms, and solving sparse linear systems by iterative methods based on sparse matrix-vector multiplication. Each topic is treated in depth, starting from the problem formulation and a sequential algorithm, through a parallel algorithm and its cost analysis, to a complete parallel program written in C and BSPlib, and experimental results obtained using this program on a parallel computer. Throughout the book, emphasis is placed on analyzing the cost of the parallel algorithms developed, expressed in three terms: computation cost, communication cost, and synchronization cost. The book contains five example programs written in BSPlib, which illustrate the methods taught. These programs are freely available as the package BSPedupack. An appendix on the message-passing interface (MPI) discusses how to program in a structured, bulk synchronous parallel style using the MPI communication library, and presents MPI equivalents of all the programs in the book.
Eduard Feireisl
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198528388
- eISBN:
- 9780191713590
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528388.001.0001
- Subject:
- Mathematics, Applied Mathematics
The book presents and develops the most recent ideas and concepts of the mathematical theory of viscous, compressible, and heat conducting fluids. Two main objectives are pursued: (i) global ...
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The book presents and develops the most recent ideas and concepts of the mathematical theory of viscous, compressible, and heat conducting fluids. Two main objectives are pursued: (i) global existence theory within the framework of variational solutions for the full Navier-Stokes-Fourier system supplemented with large data, and (ii) optimal existence results for barotropic flows with respect to the available a priori estimates.Less
The book presents and develops the most recent ideas and concepts of the mathematical theory of viscous, compressible, and heat conducting fluids. Two main objectives are pursued: (i) global existence theory within the framework of variational solutions for the full Navier-Stokes-Fourier system supplemented with large data, and (ii) optimal existence results for barotropic flows with respect to the available a priori estimates.
Sander van Smaalen
- Published in print:
- 2007
- Published Online:
- January 2008
- ISBN:
- 9780198570820
- eISBN:
- 9780191718762
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198570820.001.0001
- Subject:
- Physics, Crystallography: Physics
Aperiodic crystals are crystalline materials with atomic structures that lack translational symmetry. This book gives a comprehensive account of the superspace theory for the description of the ...
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Aperiodic crystals are crystalline materials with atomic structures that lack translational symmetry. This book gives a comprehensive account of the superspace theory for the description of the crystal structures, and symmetries of incommensurately modulated crystals and composite crystals. It also gives a brief introduction to quasicrystals, thus providing the necessary background for understanding the distinctive features of aperiodic crystals, and it provides the tools for the application of quantitative methods from the realms of crystallography, solid state chemistry, and solid state physics to aperiodic crystal structures. The second half of the book is devoted to crystallographic methods of structural analysis of incommensurate crystals. Thorough accounts are given of the diffraction by incommensurate crystals, the choice of parameters in structure refinements, and the use of superspace in analysing crystal structures. The presentation of methods of structure determination includes direct methods, Fourier methods, Patterson function methods, the maximum entropy method (MEM), and charge flipping. So-called t-plots are introduced as a versatile method for the crystal chemical analysis of incommensurately modulated structures and composite crystals.Less
Aperiodic crystals are crystalline materials with atomic structures that lack translational symmetry. This book gives a comprehensive account of the superspace theory for the description of the crystal structures, and symmetries of incommensurately modulated crystals and composite crystals. It also gives a brief introduction to quasicrystals, thus providing the necessary background for understanding the distinctive features of aperiodic crystals, and it provides the tools for the application of quantitative methods from the realms of crystallography, solid state chemistry, and solid state physics to aperiodic crystal structures. The second half of the book is devoted to crystallographic methods of structural analysis of incommensurate crystals. Thorough accounts are given of the diffraction by incommensurate crystals, the choice of parameters in structure refinements, and the use of superspace in analysing crystal structures. The presentation of methods of structure determination includes direct methods, Fourier methods, Patterson function methods, the maximum entropy method (MEM), and charge flipping. So-called t-plots are introduced as a versatile method for the crystal chemical analysis of incommensurately modulated structures and composite crystals.
Helmut Hofmann
- Published in print:
- 2008
- Published Online:
- September 2008
- ISBN:
- 9780198504016
- eISBN:
- 9780191708480
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198504016.003.0026
- Subject:
- Physics, Nuclear and Plasma Physics
This chapter elucidates various mathematical formulas. Based on expressions for Gaussian integrals in one and many dimensions, the methods of stationary phase and steepest descent are deduced, ...
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This chapter elucidates various mathematical formulas. Based on expressions for Gaussian integrals in one and many dimensions, the methods of stationary phase and steepest descent are deduced, representations of the delta-function are given and applied to Fourier and Laplace transformations. For quantal operators, the Mori product is introduced and an important formula for the derivative of exponentials is shown. Elementary properties of spin and isospin are discussed; for fermions, the formalism of second quantization is produced.Less
This chapter elucidates various mathematical formulas. Based on expressions for Gaussian integrals in one and many dimensions, the methods of stationary phase and steepest descent are deduced, representations of the delta-function are given and applied to Fourier and Laplace transformations. For quantal operators, the Mori product is introduced and an important formula for the derivative of exponentials is shown. Elementary properties of spin and isospin are discussed; for fermions, the formalism of second quantization is produced.
Dennis Sherwood and Jon Cooper
- Published in print:
- 2010
- Published Online:
- January 2011
- ISBN:
- 9780199559046
- eISBN:
- 9780191595028
- Item type:
- book
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199559046.001.0001
- Subject:
- Physics, Crystallography: Physics
This book presents a complete account of the theory of the diffraction of X-rays by crystals with particular reference to the processes of determining the structures of protein molecules. The book ...
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This book presents a complete account of the theory of the diffraction of X-rays by crystals with particular reference to the processes of determining the structures of protein molecules. The book develops from first principles all relevant mathematics, diffraction, and wave theory. The practical aspects of sample preparation and X-ray data collection using both laboratory and synchrotron sources are covered along with data analysis at both the theoretical and practical levels. The important role played by the Patterson function in structure analysis by both molecular replacement and experimental phasing approaches is covered, as are methods for improving the resulting electron density map. The theoretical basis of methods used in refinement of protein crystal structures are then covered in depth along with the crucial task of defining the binding sites of ligands and drug molecules. The complementary roles of other diffraction methods which reveal further detail of great functional importance in a crystal structure are outlined.Less
This book presents a complete account of the theory of the diffraction of X-rays by crystals with particular reference to the processes of determining the structures of protein molecules. The book develops from first principles all relevant mathematics, diffraction, and wave theory. The practical aspects of sample preparation and X-ray data collection using both laboratory and synchrotron sources are covered along with data analysis at both the theoretical and practical levels. The important role played by the Patterson function in structure analysis by both molecular replacement and experimental phasing approaches is covered, as are methods for improving the resulting electron density map. The theoretical basis of methods used in refinement of protein crystal structures are then covered in depth along with the crucial task of defining the binding sites of ligands and drug molecules. The complementary roles of other diffraction methods which reveal further detail of great functional importance in a crystal structure are outlined.
Gary A. Glatzmaier
- Published in print:
- 2013
- Published Online:
- October 2017
- ISBN:
- 9780691141725
- eISBN:
- 9781400848904
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691141725.001.0001
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This book provides readers with the skills they need to write computer codes that simulate convection, internal gravity waves, and magnetic field generation in the interiors and atmospheres of ...
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This book provides readers with the skills they need to write computer codes that simulate convection, internal gravity waves, and magnetic field generation in the interiors and atmospheres of rotating planets and stars. Using a teaching method perfected in the classroom, the book begins by offering a step-by-step guide on how to design codes for simulating nonlinear time-dependent thermal convection in a 2D box using Fourier expansions in the horizontal direction and finite differences in the vertical direction. It then describes how to implement more efficient a nd accurate numerical methods and more realistic geometries in two and three dimensions. The third part of the book demonstrates how to incorporate more sophisticated physics, including the effects of magnetic field, density stratification, and rotation. The book features numerous exercises throughout, and is an ideal textbook for students and an essential resource for researchers. It explains how to create codes that simulate the internal dynamics of planets and stars, and builds on basic concepts and simple methods. The book shows how to improve the efficiency and accuracy of the numerical methods. It considers more relevant geometries and boundary conditions.Less
This book provides readers with the skills they need to write computer codes that simulate convection, internal gravity waves, and magnetic field generation in the interiors and atmospheres of rotating planets and stars. Using a teaching method perfected in the classroom, the book begins by offering a step-by-step guide on how to design codes for simulating nonlinear time-dependent thermal convection in a 2D box using Fourier expansions in the horizontal direction and finite differences in the vertical direction. It then describes how to implement more efficient a nd accurate numerical methods and more realistic geometries in two and three dimensions. The third part of the book demonstrates how to incorporate more sophisticated physics, including the effects of magnetic field, density stratification, and rotation. The book features numerous exercises throughout, and is an ideal textbook for students and an essential resource for researchers. It explains how to create codes that simulate the internal dynamics of planets and stars, and builds on basic concepts and simple methods. The book shows how to improve the efficiency and accuracy of the numerical methods. It considers more relevant geometries and boundary conditions.
D. A. Bini, G. Latouche, and B. Meini
- Published in print:
- 2005
- Published Online:
- September 2007
- ISBN:
- 9780198527688
- eISBN:
- 9780191713286
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198527688.003.0002
- Subject:
- Mathematics, Numerical Analysis
This chapter describes structural and computational properties which are the basis of the design and analysis of fast algorithms for the numerical solution of structured Markov chains. Toeplitz ...
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This chapter describes structural and computational properties which are the basis of the design and analysis of fast algorithms for the numerical solution of structured Markov chains. Toeplitz matrices, circulant and z-circulant matrices with their block analogs are introduced together with the fast algorithms for their manipulation. The concept of discrete Fourier transform is recalled with the FFT algorithm for its computation. Attention is given to the concept of displacement operator and of displacement rank needed to design efficient algorithms for a wide class of matrices related to Toeplitz matrices.Less
This chapter describes structural and computational properties which are the basis of the design and analysis of fast algorithms for the numerical solution of structured Markov chains. Toeplitz matrices, circulant and z-circulant matrices with their block analogs are introduced together with the fast algorithms for their manipulation. The concept of discrete Fourier transform is recalled with the FFT algorithm for its computation. Attention is given to the concept of displacement operator and of displacement rank needed to design efficient algorithms for a wide class of matrices related to Toeplitz matrices.
Peter Main
- Published in print:
- 2009
- Published Online:
- September 2009
- ISBN:
- 9780199219469
- eISBN:
- 9780191722516
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199219469.003.0001
- Subject:
- Physics, Crystallography: Physics
This introductory chapter provides information on some fundamental aspects of crystal structures and their diffraction of X-rays as a basis for the rest of the book. It describes electrons, atoms, ...
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This introductory chapter provides information on some fundamental aspects of crystal structures and their diffraction of X-rays as a basis for the rest of the book. It describes electrons, atoms, molecules, and crystals scatter X-rays, leading to the observed diffraction pattern, and introduces concepts such as the reciprocal lattice, structure factors, Fourier transforms, Bragg's law for the geometry of diffraction, the phase problem encountered in crystallography, and the meaning of resolution and how it is related to the extent of the measured diffraction pattern.Less
This introductory chapter provides information on some fundamental aspects of crystal structures and their diffraction of X-rays as a basis for the rest of the book. It describes electrons, atoms, molecules, and crystals scatter X-rays, leading to the observed diffraction pattern, and introduces concepts such as the reciprocal lattice, structure factors, Fourier transforms, Bragg's law for the geometry of diffraction, the phase problem encountered in crystallography, and the meaning of resolution and how it is related to the extent of the measured diffraction pattern.
D. Huybrechts
- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780199296866
- eISBN:
- 9780191711329
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199296866.003.0005
- Subject:
- Mathematics, Geometry / Topology
This chapter introduces the central notion of a Fourier-Mukai transform between derived categories. It is the derived version of the notion of a correspondence, which has been studied for all kinds ...
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This chapter introduces the central notion of a Fourier-Mukai transform between derived categories. It is the derived version of the notion of a correspondence, which has been studied for all kinds of cohomology theories for many decades. In fact, Orlov's celebrated result, which is stated but not proved, says that any equivalence between derived categories of smooth projective varieties is of Fourier-Mukai type. Fourier-Mukai functors behave well in many respects: they are exact, admit left and right adjoints, can be composed, etc. The cohomological Fourier-Mukai transform behaves with respect to grading, Hodge structure, and Mukai pairing.Less
This chapter introduces the central notion of a Fourier-Mukai transform between derived categories. It is the derived version of the notion of a correspondence, which has been studied for all kinds of cohomology theories for many decades. In fact, Orlov's celebrated result, which is stated but not proved, says that any equivalence between derived categories of smooth projective varieties is of Fourier-Mukai type. Fourier-Mukai functors behave well in many respects: they are exact, admit left and right adjoints, can be composed, etc. The cohomological Fourier-Mukai transform behaves with respect to grading, Hodge structure, and Mukai pairing.
Roger D. Roger and Miles A. Whittington
- Published in print:
- 2010
- Published Online:
- May 2010
- ISBN:
- 9780195342796
- eISBN:
- 9780199776276
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195342796.003.0002
- Subject:
- Neuroscience, Molecular and Cellular Systems, Development
One of the pioneers in the mathematical and conceptual study of oscillations was Norbert Wiener, whose book Cybernetics first appeared in 1948. The authors review the type of data, experimental ...
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One of the pioneers in the mathematical and conceptual study of oscillations was Norbert Wiener, whose book Cybernetics first appeared in 1948. The authors review the type of data, experimental techniques, and computing tools available to Wiener for the study of brain oscillations, and describes some of the vast expansions in data base, techniques, and ideas about function that have occurred since Wiener's time.Less
One of the pioneers in the mathematical and conceptual study of oscillations was Norbert Wiener, whose book Cybernetics first appeared in 1948. The authors review the type of data, experimental techniques, and computing tools available to Wiener for the study of brain oscillations, and describes some of the vast expansions in data base, techniques, and ideas about function that have occurred since Wiener's time.
Russell L. De Valois and Karen K. De Valois
- Published in print:
- 1991
- Published Online:
- January 2008
- ISBN:
- 9780195066579
- eISBN:
- 9780199872220
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195066579.003.0001
- Subject:
- Psychology, Cognitive Neuroscience
This chapter provides a background on linear systems analysis to allow the reader to follow the applications to visual problems discussed in this book. Specifically, it discusses the Fourier theorem ...
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This chapter provides a background on linear systems analysis to allow the reader to follow the applications to visual problems discussed in this book. Specifically, it discusses the Fourier theorem and its applications to vision. Fourier analysis is a powerful tool for studying complex waveforms, allowing one to specify quantitatively the characteristics of any complex waveform or shape. Another major advantage in its application to vision is that it gives a common basis by which one can examine optical, physiological, and psychophysical data. The principal limitation in its application to visual problems is the underlying assumption of linearity, a condition that is only met by the visual system under limited conditions.Less
This chapter provides a background on linear systems analysis to allow the reader to follow the applications to visual problems discussed in this book. Specifically, it discusses the Fourier theorem and its applications to vision. Fourier analysis is a powerful tool for studying complex waveforms, allowing one to specify quantitatively the characteristics of any complex waveform or shape. Another major advantage in its application to vision is that it gives a common basis by which one can examine optical, physiological, and psychophysical data. The principal limitation in its application to visual problems is the underlying assumption of linearity, a condition that is only met by the visual system under limited conditions.
Stephen Handel
- Published in print:
- 2006
- Published Online:
- September 2007
- ISBN:
- 9780195169645
- eISBN:
- 9780199786732
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195169645.003.0005
- Subject:
- Psychology, Cognitive Psychology
Multiple stimulus dimensions and multiple perceptual processes affect the perception of motion and the dominant one is a function of the overall context. In apparent motion configurations, the first ...
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Multiple stimulus dimensions and multiple perceptual processes affect the perception of motion and the dominant one is a function of the overall context. In apparent motion configurations, the first step is to parse the “somethings” in each image, and the second is to find the correspondence between the “somethings” in each image. The perceptions reflect the physical properties of objects, yielding slow continuous motions of rigid objects that do cross paths. For the simplest one-dot configurations, the perception is multistable, alternating between plausible motions. Two kinds of visual motion have been found: first order (Fourier) motion is based on changes in illumination or color, while second order (non-Fourier) motion is based on changes in contrast. Many visual phenomena have auditory equivalents, including induction and space-time and frequency-time stream segregation trading relationships.Less
Multiple stimulus dimensions and multiple perceptual processes affect the perception of motion and the dominant one is a function of the overall context. In apparent motion configurations, the first step is to parse the “somethings” in each image, and the second is to find the correspondence between the “somethings” in each image. The perceptions reflect the physical properties of objects, yielding slow continuous motions of rigid objects that do cross paths. For the simplest one-dot configurations, the perception is multistable, alternating between plausible motions. Two kinds of visual motion have been found: first order (Fourier) motion is based on changes in illumination or color, while second order (non-Fourier) motion is based on changes in contrast. Many visual phenomena have auditory equivalents, including induction and space-time and frequency-time stream segregation trading relationships.
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/9780691141725.003.0003
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter describes a linear stability analysis (that is, solving for the critical Rayleigh number Ra and mode) that allows readers to check their linear codes against the analytic solution. For ...
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This chapter describes a linear stability analysis (that is, solving for the critical Rayleigh number Ra and mode) that allows readers to check their linear codes against the analytic solution. For this linear analysis, each Fourier mode n can be considered a separate and independent problem. The question that needs to be addressed now is under what conditions—that is, what values of Ra, Prandtl number Pr, and aspect ratio a—will the amplitude of the linear solution grow with time for a given mode n. This is a linear stability problem. The chapter first introduces the linear equations before discussing the linear code and explaining how to find the critical Rayleigh number; in other words, the value of Ra for a and Pr that gives a solution that neither grows nor decays with time. It also shows how the linear stability problem can be solved using an analytic approach.Less
This chapter describes a linear stability analysis (that is, solving for the critical Rayleigh number Ra and mode) that allows readers to check their linear codes against the analytic solution. For this linear analysis, each Fourier mode n can be considered a separate and independent problem. The question that needs to be addressed now is under what conditions—that is, what values of Ra, Prandtl number Pr, and aspect ratio a—will the amplitude of the linear solution grow with time for a given mode n. This is a linear stability problem. The chapter first introduces the linear equations before discussing the linear code and explaining how to find the critical Rayleigh number; in other words, the value of Ra for a and Pr that gives a solution that neither grows nor decays with time. It also shows how the linear stability problem can be solved using an analytic approach.
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/9780691141725.003.0005
- Subject:
- Physics, Particle Physics / Astrophysics / Cosmology
This chapter explains how to write a postprocessing code, and more specifically how to study the nonlinear simulations using computer graphics and analysis. It first considers how to compute and ...
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This chapter explains how to write a postprocessing code, and more specifically how to study the nonlinear simulations using computer graphics and analysis. It first considers how to compute and store results in a file during the computer simulation, assuming the Fourier transforms to x-space are done within the main computational code during the simulation. It then describes the postprocessing code for reading these files and displaying the various fields, along with the use of graphics software packages that provide additional, more sophisticated visualizations of the scalar and vector data. It also discusses the computer analysis of several additional properties of the solution, focusing on measurements of nonlinear convection such as Rayleigh number, Nusselt number, Reynolds number, and kinetic energy spectrum.Less
This chapter explains how to write a postprocessing code, and more specifically how to study the nonlinear simulations using computer graphics and analysis. It first considers how to compute and store results in a file during the computer simulation, assuming the Fourier transforms to x-space are done within the main computational code during the simulation. It then describes the postprocessing code for reading these files and displaying the various fields, along with the use of graphics software packages that provide additional, more sophisticated visualizations of the scalar and vector data. It also discusses the computer analysis of several additional properties of the solution, focusing on measurements of nonlinear convection such as Rayleigh number, Nusselt number, Reynolds number, and kinetic energy spectrum.
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.
I. Grattan-Guinness
- Published in print:
- 2008
- Published Online:
- May 2008
- ISBN:
- 9780199231256
- eISBN:
- 9780191710803
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199231256.003.0003
- Subject:
- Mathematics, History of Mathematics
This chapter reviews the early works of William Thomson. Thomson began to publish exceptionally early, in 1841, while still in his late teens; and a steady stream of papers came from him thereafter. ...
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This chapter reviews the early works of William Thomson. Thomson began to publish exceptionally early, in 1841, while still in his late teens; and a steady stream of papers came from him thereafter. Most of them in the 1840s concerned heat diffusion, electricity, or magnetism, and related mathematical methods such as the Fourier series and potential theory. These papers manifest a procedure of working by analogy in which methods were taken from one topic and adapted for use in another one.Less
This chapter reviews the early works of William Thomson. Thomson began to publish exceptionally early, in 1841, while still in his late teens; and a steady stream of papers came from him thereafter. Most of them in the 1840s concerned heat diffusion, electricity, or magnetism, and related mathematical methods such as the Fourier series and potential theory. These papers manifest a procedure of working by analogy in which methods were taken from one topic and adapted for use in another one.
Leon Ehrenpreis
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198509783
- eISBN:
- 9780191709166
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198509783.003.0005
- Subject:
- Mathematics, Mathematical Physics
In order to deal with the nonlinear Radon transform, this chapter introduces a new type of Fourier transform: the nonlinear Fourier transform. This can be used to analyze various nonlinear Radon ...
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In order to deal with the nonlinear Radon transform, this chapter introduces a new type of Fourier transform: the nonlinear Fourier transform. This can be used to analyze various nonlinear Radon transforms. It is also useful in giving a new approach to wave front sets. (The quadratic case is similar to the FBI transform.)Less
In order to deal with the nonlinear Radon transform, this chapter introduces a new type of Fourier transform: the nonlinear Fourier transform. This can be used to analyze various nonlinear Radon transforms. It is also useful in giving a new approach to wave front sets. (The quadratic case is similar to the FBI transform.)
Eduard Feireisl
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198528388
- eISBN:
- 9780191713590
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528388.003.0004
- Subject:
- Mathematics, Applied Mathematics
The concept of a variational solution to the full Navier-Stokes-Fourier system is introduced in this chapter. Each equation of the complete system, that means, the equation of continuity, the ...
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The concept of a variational solution to the full Navier-Stokes-Fourier system is introduced in this chapter. Each equation of the complete system, that means, the equation of continuity, the momentum equation, and the energy equation, is treated separately. The basic facts of the theory of renormalized solutions of the continuity equation are reviewed. Moreover, the renormalized solutions of the thermal energy are introduced as a suitable tool to deal with possible concentrations of temperature.Less
The concept of a variational solution to the full Navier-Stokes-Fourier system is introduced in this chapter. Each equation of the complete system, that means, the equation of continuity, the momentum equation, and the energy equation, is treated separately. The basic facts of the theory of renormalized solutions of the continuity equation are reviewed. Moreover, the renormalized solutions of the thermal energy are introduced as a suitable tool to deal with possible concentrations of temperature.
Eduard Feireisl
- Published in print:
- 2003
- Published Online:
- September 2007
- ISBN:
- 9780198528388
- eISBN:
- 9780191713590
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198528388.003.0007
- Subject:
- Mathematics, Applied Mathematics
This chapter contains a complete proof of the existence of global-in-time variational solutions for the full system of the Navier-Stokes equations of a viscous compressible and heat conducting fluid ...
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This chapter contains a complete proof of the existence of global-in-time variational solutions for the full system of the Navier-Stokes equations of a viscous compressible and heat conducting fluid under suitable restrictions imposed on the constitutive laws. These restrictions are by no means optimal but, on the other hand, they seem to be in a good agreement with the underlying physical theory. Probably the most questionable hypothesis seems to be the necessity of the viscosity coefficients to be constant in all temperature regimes. On the other hand, as a by-product of our approach, we derive ‘optimal’ existence results for the barotropic flows with respect to the available (known) a priori estimates. Another novelty allowed by the present method is the possibility to consider general, not necessarily monotone, pressure-density constitutive equations arising in applications.Less
This chapter contains a complete proof of the existence of global-in-time variational solutions for the full system of the Navier-Stokes equations of a viscous compressible and heat conducting fluid under suitable restrictions imposed on the constitutive laws. These restrictions are by no means optimal but, on the other hand, they seem to be in a good agreement with the underlying physical theory. Probably the most questionable hypothesis seems to be the necessity of the viscosity coefficients to be constant in all temperature regimes. On the other hand, as a by-product of our approach, we derive ‘optimal’ existence results for the barotropic flows with respect to the available (known) a priori estimates. Another novelty allowed by the present method is the possibility to consider general, not necessarily monotone, pressure-density constitutive equations arising in applications.
