Weakly nonlinear theory of shear-banding instability in a granular plane Couette flow: analytical solution, comparison with numerics and bifurcation
TL;DR: In this article, a weakly nonlinear theory, in terms of the well-known Landau equation, has been developed to describe the nonlinear saturation of the shearbanding instability in a rapid granular plane Couette flow using the amplitude expansion method.
Abstract: A weakly nonlinear theory, in terms of the well-known Landau equation, has been developed to describe the nonlinear saturation of the shear-banding instability in a rapid granular plane Couette flow using the amplitude expansion method. The nonlinear modes are found to follow certain symmetries of the base flow and the fundamental mode, which helped to identify analytical solutions for the base-flow distortion and the second harmonic, leading to an exact calculation of the first Landau coefficient. The present analytical solutions are used to validate a spectral-based numerical method for the nonlinear stability calculation. The regimes of supercritical and subcritical bifurcations for the shear-banding instability have been identified, leading to the prediction that the lower branch of the neutral stability contour in the (H, φ0)-plane, where H is the scaled Couette gap (the ratio between the Couette gap and the particle diameter) and φ0 is the mean density or the volume fraction of particles, is subcritically unstable. The predicted finite-amplitude solutions represent shear localization and density segregation along the gradient direction. Our analysis suggests that there is a sequence of transitions among three types of pitchfork bifurcations with increasing mean density: from (i) the bifurcation from infinity in the Boltzmann limit to (ii) subcritical bifurcation at moderate densities to (iii) supercritical bifurcation at larger densities to (iv) subcritical bifurcation in the dense limit and finally again to (v) supercritical bifurcation near the close packing density. It has been shown that the appearance of subcritical bifurcation in the dense limit depends on the choice of the contact radial distribution function and the constitutive relations. The scalings of the first Landau coefficient, the equilibrium amplitude and the phase diagram, in terms of mode number and inelasticity, have been demonstrated. The granular plane Couette flow serves as a paradigm that supports all three possible types of pitchfork bifurcations, with the mean density (φ0) being the single control parameter that dictates the nature of the bifurcation. The predicted bifurcation scenario for the shear-band formation is in qualitative agreement with particle dynamics simulations and the experiment in the rapid shear regime of the granular plane Couette flow.
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TL;DR: In this article, the authors analyzed the non-Newtonian stress tensor, collisional dissipation rate and heat flux in the plane shear flow of smooth inelastic disks using the anisotropic Gaussian as a reference.
Abstract: The non-Newtonian stress tensor, collisional dissipation rate and heat flux in the plane shear flow of smooth inelastic disks are analysed from the Grad-level moment equations using the anisotropic Gaussian as a reference. For steady uniform shear flow, the balance equation for the second moment of velocity fluctuations is solved semi-analytically, yielding closed-form expressions for the shear viscosity , pressure , first normal stress difference and dissipation rate as functions of (i) density or area fraction , (ii) restitution coefficient , (iii) dimensionless shear rate , (iv) temperature anisotropy (the difference between the principal eigenvalues of the second-moment tensor) and (v) angle between the principal directions of the shear tensor and the second-moment tensor. The last two parameters are zero at the Navier–Stokes order, recovering the known exact transport coefficients from the present analysis in the limit , and are therefore measures of the non-Newtonian rheology of the medium. An exact analytical solution for leading-order moment equations is given, which helped to determine the scaling relations of , and with inelasticity. We show that the terms at super-Burnett order must be retained for a quantitative prediction of transport coefficients, especially at moderate to large densities for small values of the restitution coefficient ( ). Particle simulation data for a sheared inelastic hard-disk system are compared with theoretical results, with good agreement for , and over a range of densities spanning from the dilute to close to the freezing point. In contrast, the predictions from a constitutive model at Navier–Stokes order are found to deviate significantly from both the simulation and the moment theory even at moderate values of the restitution coefficient ( ). Lastly, a generalized Fourier law for the granular heat flux, which vanishes identically in the uniform shear state, is derived for a dilute granular gas by analysing the non-uniform shear flow via an expansion around the anisotropic Gaussian state. We show that the gradient of the deviatoric part of the kinetic stress drives a heat current and the thermal conductivity is characterized by an anisotropic second-rank tensor, for which explicit analytical expressions are given.
26 citations
Cites background from "Weakly nonlinear theory of shear-ba..."
...…constitutive relations for the stress tensor (§ 4.2) and the heat flux (§ 6) along with extended hydrodynamic equations (2.11)–(2.13) can also be tested in dynamic simulations of granular flows, including the stability analyses of shear flows (Gayen & Alam 2006; Shukla & Alam 2009, 2011a,b)....
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TL;DR: In this paper, a cubic Landau equation is derived to study the limiting value of growth of instabilities under nonlinear effects, and the influence of nonlinear interaction of different harmonics on the heat transfer rate, friction coefficient, nonlinear kinetic energy spectrum and disturbance flow is also studied in both supercritical as well as subcritical regimes.
Abstract: This paper addresses the finite amplitude instability of stably stratified non-isothermal parallel flow in a vertical channel filled with a highly permeable porous medium. A cubic Landau equation is derived to study the limiting value of growth of instabilities under nonlinear effects. The non-Darcy model is considered to describe the flow instabilities in a porous medium. The nonlinear results are presented for air as well as water. The analysis is carried out in the vicinity of as well as away from the critical point (bifurcation point). It is found that when the medium is saturated by water then supercritical bifurcation is the only type of bifurcation at and beyond the bifurcation point. However, for air, depending on the strength of the flow and permeability of the medium, both supercritical and subcritical bifurcations are observed. The influence of nonlinear interaction of different harmonics on the heat transfer rate, friction coefficient, nonlinear kinetic energy spectrum and disturbance flow is also studied in both supercritical as well as subcritical regimes. The variation of neutral stability curves of parallel mixed convection flow of air with wavenumber reveals that a bifurcation that is supercritical for some wavenumber may be subcritical or vice versa at other nearby wavenumbers. The analysis of the nonlinear kinetic energy spectrum of the fundamental disturbance also supports the existence of supercritical/subcritical bifurcation at and away from the critical point. The effect of different harmonics on the pattern of secondary flow, based on linear stability theory, is also studied and a significant influence is found, especially in the subcritical regime.
19 citations
Cites background from "Weakly nonlinear theory of shear-ba..."
...As has been discussed in Drazin & Reid (2004) and Shukla & Alam (2011), when αci and (a1)r are of the opposite sign, the existence of a finite amplitude solution is guaranteed....
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TL;DR: In this article, the first evidence of a variety of nonlinear equilibrium states of travelling and stationary waves is provided in a two-dimensional granular plane Couette flow via nonlinear stability analysis.
Abstract: The first evidence of a variety of nonlinear equilibrium states of travelling and stationary waves is provided in a two-dimensional granular plane Couette flow via nonlinear stability analysis. The relevant order-parameter equation, the Landau equation, has been derived for the most unstable two-dimensional perturbation of finite size. Along with the linear eigenvalue problem, the mean-flow distortion, the second harmonic, the distortion to the fundamental mode and the first Landau coefficient are calculated using a spectral-based numerical method. Two types of bifurcations, Hopf and pitchfork, that result from travelling and stationary instabilities, respectively, are analysed using the first Landau coefficient. The present bifurcation theory shows that the flow is subcritically unstable to stationary finite-amplitude perturbations of long wavelengths (k x ∼0, where k x is the streamwise wavenumber) in the dilute limit that evolve from subcritical shear-banding modes (k x = 0), but at large enough Couette gaps there are stationary instabilities with k x = O(1) that lead to supercritical pitchfork bifurcations. At moderate-to-large densities, in addition to supercritical shear-banding modes, there are long-wave travelling instabilities that lead to Hopf bifurcations. It is shown that both supercritical and subcritical nonlinear states exist at moderate-to-large densities that originate from the dominant stationary and travelling instabilities for which k x = O(1). Nonlinear patterns of density, velocity and granular temperature for all types of instabilities are contrasted with their linear eigenfunctions. While the supercritical solutions appear to be modulated forms of the fundamental mode, the structural features of unstable subcritical solutions are found to be significantly different from their linear counterparts. It is shown that the granular plane Couette flow is prone to nonlinear resonances in both stable and unstable regimes, the signature of which is implicated as a discontinuity in the first Landau coefficient. Our analysis identified two types of modal resonances that appear at the quadratic order in perturbation amplitude: (i) a 'mean-flow resonance' which occurs due to the interaction between a streamwise-independent shear-banding mode (k x = 0) and a linear/fundamental mode k x ≠ 0, and (ii) an exact '1:2 resonance' that results from the interaction between two waves with their wavenumber ratio being 1 :2.
18 citations
Cites background or methods or result from "Weakly nonlinear theory of shear-ba..."
...They employed both the direct method of centre manifold reduction (Shukla & Alam 2009) and the indirect method of amplitude expansion (Shukla & Alam 2011), and showed the equivalence between these two methods that result in the same expression for the first Landau coefficient, which is the…...
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...In addition to the well-known shear-banding instability (kx = 0) whose nonlinear saturation has been studied recently by us (Shukla & Alam 2009, 2011), there are long-wave (kx ∼ 0) stationary and travelling instabilities in a granular plane Couette flow....
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...Starting with granular hydrodynamic equations, Shukla & Alam (2009, 2011) derived the Landau–Stuart equation for the shear-banding instabilities (for the first time in granular flows) via weakly nonlinear analyses....
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...In our previous papers (Shukla & Alam 2009, 2011), we considered one-dimensional streamwise-independent equations which can be obtained by putting ∂/∂x(·) = 0 in (2.11)–(2.14) since our focus was on the shear-banding instability, for which the associated patterns have no variations along the…...
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...For the special case of shear-banding modes (kx = 0), it has been shown (Shukla & Alam 2011) that the second harmonic X[2;2] is real and X[2;2] = X̃[2;2] = X[0;2]....
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TL;DR: Chen et al. as mentioned in this paper developed a weakly nonlinear stability theory in terms of Landau equation to analyze the nonlinear saturation of stably stratified non-isothermal Poiseuille flow in a vertical channel.
Abstract: A weakly nonlinear stability theory in terms of Landau equation is developed to analyze the nonlinear saturation of stably stratified non-isothermal Poiseuille flow in a vertical channel. The results are presented with respect to fluids: mercury, gases, liquids, and heavy oils. The weakly nonlinear stability results predict only the supercritical instability, in agreement with the published result [Y. C. Chen and J. N. Chung, “A direct numerical simulation of K and H-type flow transition in heated vertical channel,” Comput. Fluids 32, 795–822 (2003)] based on direct numerical simulation. Apart from this, the influence of nonlinear interaction among different superimposed waves on the heat transfer rate, real part of wavespeed, and friction coefficient on the wall is also investigated. A substantial enhancement (reduction) in heat transfer rate (friction coefficient) is found for liquids and heavy oils from the basic state beyond the critical Rayleigh number. The amplitude analysis indicates that the equilibrium amplitude decreases on increasing the value of Reynolds number. However, in the case of mercury, influence of nonlinear interaction on the variation of equilibrium amplitude, heat transfer rate, wavespeed, as well as friction coefficient is complex and subtle. The analysis of the nonlinear energy spectra for the disturbance also supports the supercritical instability at and beyond the critical point. Finally, the effect of superimposed waves on the pattern of secondary flow, based on linear stability theory, is also studied. It has been found that the impact of nonlinear interaction of waves on the pattern of secondary flow for mercury is weak compared to gases, which is the consequence of negligible modification in the buoyant production of disturbance kinetic energy of the mercury.
18 citations
TL;DR: In this paper, a weakly nonlinear analysis of a two dimensional sheared granular flow is carried out under the Lees-Edwards boundary condition and the time dependent Ginzburg-Landau equation of a disturbance amplitude starting from a set of granular hydrodynamic equations is derived.
Abstract: Weakly nonlinear analysis of a two dimensional sheared granular flow is carried out under the Lees-Edwards boundary condition. We derive the time dependent Ginzburg–Landau equation of a disturbance amplitude starting from a set of granular hydrodynamic equations and discuss the bifurcation of the steady amplitude in the hydrodynamic limit.
12 citations
Cites methods from "Weakly nonlinear theory of shear-ba..."
...a physical boundary condition starting from a set of granular hydrodynamic equations [54–56] by the method of Reynolds and Potter [57]....
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17,845 citations
"Weakly nonlinear theory of shear-ba..." refers methods in this paper
...We employed the method of singular value decomposition (Press et al. 1992) for solving AX = b system in each case of (4.25), (4.26) and (4.31)....
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...We employed the method of singular value decomposition (Press et al. 1992) for solving AX = b system in each case of (4....
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TL;DR: A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented in this article, with emphasis on comparisons between theory and quantitative experiments, and a classification of patterns in terms of the characteristic wave vector q 0 and frequency ω 0 of the instability.
Abstract: A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented, with emphasis on comparisons between theory and quantitative experiments. Examples include patterns in hydrodynamic systems such as thermal convection in pure fluids and binary mixtures, Taylor-Couette flow, parametric-wave instabilities, as well as patterns in solidification fronts, nonlinear optics, oscillatory chemical reactions and excitable biological media. The theoretical starting point is usually a set of deterministic equations of motion, typically in the form of nonlinear partial differential equations. These are sometimes supplemented by stochastic terms representing thermal or instrumental noise, but for macroscopic systems and carefully designed experiments the stochastic forces are often negligible. An aim of theory is to describe solutions of the deterministic equations that are likely to be reached starting from typical initial conditions and to persist at long times. A unified description is developed, based on the linear instabilities of a homogeneous state, which leads naturally to a classification of patterns in terms of the characteristic wave vector q0 and frequency ω0 of the instability. Type Is systems (ω0=0, q0≠0) are stationary in time and periodic in space; type IIIo systems (ω0≠0, q0=0) are periodic in time and uniform in space; and type Io systems (ω0≠0, q0≠0) are periodic in both space and time. Near a continuous (or supercritical) instability, the dynamics may be accurately described via "amplitude equations," whose form is universal for each type of instability. The specifics of each system enter only through the nonuniversal coefficients. Far from the instability threshold a different universal description known as the "phase equation" may be derived, but it is restricted to slow distortions of an ideal pattern. For many systems appropriate starting equations are either not known or too complicated to analyze conveniently. It is thus useful to introduce phenomenological order-parameter models, which lead to the correct amplitude equations near threshold, and which may be solved analytically or numerically in the nonlinear regime away from the instability. The above theoretical methods are useful in analyzing "real pattern effects" such as the influence of external boundaries, or the formation and dynamics of defects in ideal structures. An important element in nonequilibrium systems is the appearance of deterministic chaos. A greal deal is known about systems with a small number of degrees of freedom displaying "temporal chaos," where the structure of the phase space can be analyzed in detail. For spatially extended systems with many degrees of freedom, on the other hand, one is dealing with spatiotemporal chaos and appropriate methods of analysis need to be developed. In addition to the general features of nonequilibrium pattern formation discussed above, detailed reviews of theoretical and experimental work on many specific systems are presented. These include Rayleigh-Benard convection in a pure fluid, convection in binary-fluid mixtures, electrohydrodynamic convection in nematic liquid crystals, Taylor-Couette flow between rotating cylinders, parametric surface waves, patterns in certain open flow systems, oscillatory chemical reactions, static and dynamic patterns in biological media, crystallization fronts, and patterns in nonlinear optics. A concluding section summarizes what has and has not been accomplished, and attempts to assess the prospects for the future.
6,145 citations
"Weakly nonlinear theory of shear-ba..." refers methods in this paper
...The Ginzburg–Landau-type order-parameter equations have been widely used to study the bifurcation scenario, nonlinear waves and patterns in fluid mechanics (Newell, Passot & Lega 1993; Manneville 1990; Cross & Hohenberg 1993)....
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...…complex Ginzburg–Landau equation has been used to obtain a qualitative (and sometimes quantitative) understanding of a host of phenomena, namely phase transitions, superconductivity, superfluidity, liquid crystals, vortex glass and defect turbulence (Cross & Hohenberg 1993; Aranson & Kramer 2002)....
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Book•
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01 Jan 1973
TL;DR: This website becomes a very available place to look for countless perturbation methods sources and sources about the books from countries in the world are provided.
Abstract: Following your need to always fulfil the inspiration to obtain everybody is now simple. Connecting to the internet is one of the short cuts to do. There are so many sources that offer and connect us to other world condition. As one of the products to see in internet, this website becomes a very available place to look for countless perturbation methods sources. Yeah, sources about the books from countries in the world are provided.
5,427 citations
"Weakly nonlinear theory of shear-ba..." refers background or methods in this paper
...1993) by calculating the first 10 or more Landau coefficients, and then finding the nearest singularity from Domb–Sykes plots (Hinch 1991) to estimate the radius of convergence....
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...Such studies have been carried out for incompressible shear flows of Newtonian fluids (Herbert 1980; Newell et al. 1993) by calculating the first 10 or more Landau coefficients, and then finding the nearest singularity from Domb–Sykes plots (Hinch 1991) to estimate the radius of convergence....
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Book•
01 Jan 1987TL;DR: Spectral methods have been widely used in simulation of stability, transition, and turbulence as discussed by the authors, and their applications to both compressible and incompressible flows, to viscous as well as inviscid flows, and also to chemically reacting flows are surveyed.
Abstract: Fundamental aspects of spectral methods are introduced. Recent developments in spectral methods are reviewed with an emphasis on collocation techniques. Their applications to both compressible and incompressible flows, to viscous as well as inviscid flows, and also to chemically reacting flows are surveyed. The key role that these methods play in the simulation of stability, transition, and turbulence is brought out. A perspective is provided on some of the obstacles that prohibit a wider use of these methods, and how these obstacles are being overcome.
4,632 citations
TL;DR: In this article, the authors present a set of methods for the estimation of two-dimensional fluid flow, including a Fourier Galerkin method and a Chebyshev Collocation method.
Abstract: 1. Introduction.- 1.1. Historical Background.- 1.2. Some Examples of Spectral Methods.- 1.2.1. A Fourier Galerkin Method for the Wave Equation.- 1.2.2. A Chebyshev Collocation Method for the Heat Equation.- 1.2.3. A Legendre Tau Method for the Poisson Equation.- 1.2.4. Basic Aspects of Galerkin, Tau and Collocation Methods.- 1.3. The Equations of Fluid Dynamics.- 1.3.1. Compressible Navier-Stokes.- 1.3.2. Compressible Euler.- 1.3.3. Compressible Potential.- 1.3.4. Incompressible Flow.- 1.3.5. Boundary Layer.- 1.4. Spectral Accuracy for a Two-Dimensional Fluid Calculation.- 1.5. Three-Dimensional Applications in Fluids.- 2. Spectral Approximation.- 2.1. The Fourier System.- 2.1.1. The Continuous Fourier Expansion.- 2.1.2. The Discrete Fourier Expansion.- 2.1.3. Differentiation.- 2.1.4. The Gibbs Phenomenon.- 2.2. Orthogonal Polynomials in ( - 1, 1).- 2.2.1. Sturm-Liouville Problems.- 2.2.2. Orthogonal Systems of Polynomials.- 2.2.3. Gauss-Type Quadratures and Discrete Polynomial Transforms.- 2.3. Legendre Polynomials.- 2.3.1. Basic Formulas.- 2.3.2. Differentiation.- 2.4. Chebyshev Polynomials.- 2.4.1. Basic Formulas.- 2.4.2. Differentiation.- 2.5. Generalizations.- 2.5.1. Jacobi Polynomials.- 2.5.2. Mapping.- 2.5.3. Semi-Infinite Intervals.- 2.5.4. Infinite Intervals.- 3. Fundamentals of Spectral Methods for PDEs.- 3.1. Spectral Projection of the Burgers Equation.- 3.1.1. Fourier Galerkin.- 3.1.2. Fourier Collocation.- 3.1.3. Chebyshev Tau.- 3.1.4. Chebyshev Collocation.- 3.2. Convolution Sums.- 3.2.1. Pseudospectral Transform Methods.- 3 2 2 Aliasing Removal by Padding or Truncation.- 3.2.3. Aliasing Removal by Phase Shifts.- 3.2.4. Convolution Sums in Chebyshev Methods.- 3.2.5. Relation Between Collocation and Pseudospectral Methods.- 3.3. Boundary Conditions.- 3.4. Coordinate Singularities.- 3.4.1. Polar Coordinates.- 3.4.2. Spherical Polar Coordinates.- 3.5. Two-Dimensional Mapping.- 4. Temporal Discretization.- 4.1. Introduction.- 4.2. The Eigenvalues of Basic Spectral Operators.- 4.2.1. The First-Derivative Operator.- 4.2.2. The Second-Derivative Operator.- 4.3. Some Standard Schemes.- 4.3.1. Multistep Schemes.- 4.3.2. Runge-Kutta Methods.- 4.4. Special Purpose Schemes.- 4.4.1. High Resolution Temporal Schemes.- 4.4.2. Special Integration Techniques.- 4.4.3. Lerat Schemes.- 4.5. Conservation Forms.- 4.6. Aliasing.- 5. Solution Techniques for Implicit Spectral Equations.- 5.1. Direct Methods.- 5.1.1. Fourier Approximations.- 5.1.2. Chebyshev Tau Approximations.- 5.1.3. Schur-Decomposition and Matrix-Diagonalization.- 5.2. Fundamentals of Iterative Methods.- 5.2.1. Richardson Iteration.- 5.2.2. Preconditioning.- 5.2.3. Non-Periodic Problems.- 5.2.4. Finite-Element Preconditioning.- 5.3. Conventional Iterative Methods.- 5.3.1. Descent Methods for Symmetric, Positive-Definite Systems.- 5.3.2. Descent Methods for Non-Symmetric Problems.- 5.3.3. Chebyshev Acceleration.- 5.4. Multidimensional Preconditioning.- 5.4.1. Finite-Difference Solvers.- 5.4.2. Modified Finite-Difference Preconditioners.- 5.5. Spectral Multigrid Methods.- 5.5.1. Model Problem Discussion.- 5.5.2. Two-Dimensional Problems.- 5.5.3. Interpolation Operators.- 5.5.4. Coarse-Grid Operators.- 5.5.5. Relaxation Schemes.- 5.6. A Semi-Implicit Method for the Navier-Stokes Equations.- 6. Simple Incompressible Flows.- 6.1. Burgers Equation.- 6.2. Shear Flow Past a Circle.- 6.3. Boundary-Layer Flows.- 6.4. Linear Stability.- 7. Some Algorithms for Unsteady Navier-Stokes Equations.- 7.1. Introduction.- 7.2. Homogeneous Flows.- 7.2.1. A Spectral Galerkin Solution Technique.- 7.2.2. Treatment of the Nonlinear Terms.- 7.2.3. Refinements.- 7.2.4. Pseudospectral and Collocation Methods.- 7.3. Inhomogeneous Flows.- 7.3.1. Coupled Methods.- 7.3.2. Splitting Methods.- 7.3.3. Galerkin Methods.- 7.3.4. Other Confined Flows.- 7.3.5. Unbounded Flows.- 7.3.6. Aliasing in Transition Calculations.- 7.4. Flows with Multiple Inhomogeneous Directions.- 7.4.1. Choice of Mesh.- 7.4.2. Coupled Methods.- 7.4.3. Splitting Methods.- 7.4.4. Other Methods.- 7.5. Mixed Spectral/Finite-Difference Methods.- 8. Compressible Flow.- 8.1. Introduction.- 8.2. Boundary Conditions for Hyperbolic Problems.- 8.3. Basic Results for Scalar Nonsmooth Problems.- 8.4. Homogeneous Turbulence.- 8.5. Shock-Capturing.- 8.5.1. Potential Flow.- 8.5.2. Ringleb Flow.- 8.5.3. Astrophysical Nozzle.- 8.6. Shock-Fitting.- 8.7. Reacting Flows.- 9. Global Approximation Results.- 9.1. Fourier Approximation.- 9.1.1. Inverse Inequalities for Trigonometric Polynomials.- 9.1.2. Estimates for the Truncation and Best Approximation Errors.- 9.1.3. Estimates for the Interpolation Error.- 9.2. Sturm-Liouville Expansions.- 9.2.1. Regular Sturm-Liouville Problems.- 9.2.2. Singular Sturm-Liouville Problems.- 9.3. Discrete Norms.- 9.4. Legendre Approximations.- 9.4.1. Inverse Inequalities for Algebraic Polynomials.- 9.4.2. Estimates for the Truncation and Best Approximation Errors.- 9.4.3. Estimates for the Interpolation Error.- 9.5. Chebyshev Approximations.- 9.5.1. Inverse Inequalities for Polynomials.- 9.5.2. Estimates for the Truncation and Best Approximation Errors.- 9.5.3. Estimates for the Interpolation Error.- 9.5.4. Proofs of Some Approximation Results.- 9.6. Other Polynomial Approximations.- 9.6.1. Jacobi Polynomials.- 9.6.2. Laguerre and Hermite Polynomials.- 9.7. Approximation Results in Several Dimensions.- 9.7.1. Fourier Approximations.- 9.7.2. Legendre Approximations.- 9.7.3. Chebyshev Approximations.- 9.7.4. Blended Fourier and Chebyshev Approximations.- 10. Theory of Stability and Convergence for Spectral Methods.- 10.1. The Three Examples Revisited.- 10.1.1. A Fourier Galerkin Method for the Wave Equation.- 10.1.2. A Chebyshev Collocation Method for the Heat Equation.- 10.1.3. A Legendre Tau Method for the Poisson Equation.- 10.2. Towards a General Theory.- 10.3. General Formulation of Spectral Approximations to Linear Steady Problems.- 10.4. Galerkin, Collocation and Tau Methods.- 10.4.1. Galerkin Methods.- 10.4.2. Tau Methods.- 10.4.3. Collocation Methods.- 10.5. General Formulation of Spectral Approximations to Linear Evolution Equations.- 10.5.1. Conditions for Stability and Convergence: The Parabolic Case.- 10.5.2. Conditions for Stability and Convergence: The Hyperbolic Case.- 10.6. The Error Equation.- 11. Steady, Smooth Problems.- 11.1. The Poisson Equation.- 11.1.1. Legendre Methods.- 11.1.2. Chebyshev Methods.- 11.1.3. Other Boundary Value Problems.- 11.2. Advection-Diffusion Equation.- 11.2.1. Linear Advection-Diffusion Equation.- 11.2.2. Steady Burgers Equation.- 11.3. Navier-Stokes Equations.- 11.3.1. Compatibility Conditions Between Velocity and Pressure.- 11.3.2. Direct Discretization of the Continuity Equation: The \"inf-sup\" Condition.- 11.3.3. Discretizations of the Continuity Equation by an Influence-Matrix Technique: The Kleiser-Schumann Method.- 11.3.4. Navier-Stokes Equations in Streamfunction Formulation.- 11.4. The Eigenvalues of Some Spectral Operators.- 11.4.1. The Discrete Eigenvalues for Lu = ? uxx.- 11.4.2. The Discrete Eigenvalues for Lu = ? vuxx + bux.- 11.4.3. The Discrete Eigenvalues for Lu = ux.- 12. Transient, Smooth Problems.- 12.1. Linear Hyperbolic Equations.- 12.1.1. Periodic Boundary Conditions.- 12.1.2. Non-Periodic Boundary Conditions.- 12.1.3. Hyperbolic Systems.- 12.1.4. Spectral Accuracy for Non-Smooth Solutions.- 12.2. Heat Equation.- 12.2.1. Semi-Discrete Approximation.- 12.2.2. Fully Discrete Approximation.- 12.3. Advection-Diffusion Equation.- 12.3.1. Semi-Discrete Approximation.- 12.3.2. Fully Discrete Approximation.- 13. Domain Decomposition Methods.- 13.1. Introduction.- 13.2. Patching Methods.- 13.2.1. Notation.- 13.2.2. Discretization.- 13.2.3. Solution Techniques.- 13.2.4. Examples.- 13.3. Variational Methods.- 13.3.1. Formulation.- 13.3.2. The Spectral-Element Method.- 13.4. The Alternating Schwarz Method.- 13.5. Mathematical Aspects of Domain Decomposition Methods.- 13.5.1. Patching Methods.- 13.5.2. Equivalence Between Patching and Variational Methods.- 13.6. Some Stability and Convergence Results.- 13.6.1. Patching Methods.- 13.6.2. Variational Methods.- Appendices.- A. Basic Mathematical Concepts.- B. Fast Fourier Transforms.- C. Jacobi-Gauss-Lobatto Roots.- References.
3,753 citations