Claudio Canuto

Bio: Claudio Canuto is an academic researcher from Polytechnic University of Turin. The author has contributed to research in topics: Spectral method & Galerkin method. The author has an hindex of 24, co-authored 112 publications receiving 8372 citations.

Spectral Methods in Fluid Dynamics.

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.

Spectral Methods: Fundamentals in Single Domains

Abstract: Since the publication of ''Spectral Methods in Fluid Dynamics'' 1988, spectral methods have become firmly established as a mainstream tool for scientific and engineering computation. The authors of that book have incorporated into this new edition the many improvements in the algorithms and the theory of spectral methods that have been made since then. This latest book retains the tight integration between the theoretical and practical aspects of spectral methods, and the chapters are enhanced with material on the Galerkin with numerical integration version of spectral methods. The discussion of direct and iterative solution methods is also greatly expanded. (orig.)

Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics

Abstract: Spectral methods, particularly in their multidomain version, have become firmly established as a mainstream tool for scientific and engineering computation. This book provides an extensive and critical overview of the essential algorithmic and theoretical aspects of spectral methods for complex geometries, in addition to detailed discussions of spectral algorithms for fluid dynamics in simple and complex geometries. Modern strategies for constructing spectral approximations in complex domains, such as spectral elements, mortar elements, and discontinuous Galerkin methods, as well as patching collocation, are introduced, analyzed in their mathematical aspects, and demonstrated by means of numerous numerical examples. Efficient domain decomposition preconditioners (of both Schwarz and Schur type) that are amenable to parallel implementation are deeply investigated. Representative simulations from continuum mechanics are also shown

Approximation results for orthogonal polynomials in Sobolev spaces

Abstract: We analyze the approximation properties of some interpolation operators and some L2-orthogonal projection operators related to systems of polynomials which are orthonormal with respect to a weight function o(x1, . . ., Xd), d > 1. The error estimates for the Legendre system and the Chebyshev system of the first kind are given in the norms of the Sobolev spaces H'. These results are useful in the numerical analysis of the approximation of partial differential equations by spectral methods. 0. Introduction. Spectral methods are a classical and largely used technique to solve differential equations, both theoretically and numerically. During the years they have gained new popularity in automatic computations for a wide class of physical problems (for instance in the fields of fluid and gas dynamics), due to the use of the Fast Fourier Transform algorithm. These methods appear to be competitive with finite difference and finite element methods and they must be decisively preferred to the last ones whenever the solution is highly regular and the geometric dimension of the domain becomes large. Moreover, by these methods it is possible to control easily the solution (filtering) of those numerical problems affected by oscillation and instability phenomena. The use of spectral and pseudo-spectral methods in computations in many fields of engineering has been matched by deeper theoretical studies; let us recall here the pioneering works by Orszag [25], [26], Kreiss and Oliger [14] and the monograph by Gottlieb and Orszag [13]. The theoretical results of such works are mainly concerned with the study of the stability of approximation of parabolic and hyperbolic equations; the solution is assumed to be infinitely differentiable, so that by an analysis of the Fourier coefficients an infinite order of convergence can be achieved. More recently (see Pasciak [27], Canuto and Quarteroni [10], [11], Maday and Quarteroni [20], [211, [22], Mercier [23]), the spectral methods have been studied by the variational techniques typical of functional analysis, to point out the dependence of the approximation error (for instance in the L2-norm, or in the energy norm) on the regularity of the solution of continuous problems and on the discretization parameter (the dimension of the space in which the approximate solution is sought). Indeed, often the solution is not infinitely differentiable; on the other hand, sometimes even if the solution is smooth, its derivatives may have very Received August 9, 1980; revised June 12, 1981. 1980 Mathematics Subject Classification. Primary 41A25; Secondary 41A 10, 41A05. ? 1982 American Mathematical Society 0025-571 8/82/0000-0470/$06.00 (67 This content downloaded from 207.46.13.111 on Tue, 09 Aug 2016 06:29:39 UTC All use subject to http://about.jstor.org/terms 68 C. CANUTO AND A. QUARTERONI large norms which affect negatively the rate of convergence (for instance in problems with boundary layers). Both spectral and pseudo-spectral methods are essentially Ritz-Galerkin methods (combined with some integration formulae in the pseudo-spectral case). It is well known that when Galerkin methods are used the distance between the exact and the discrete solution (approximation error) is bounded by the distance between the exact solution and its orthogonal projection upon the subspace (projection error), or by the distance between the exact solution and its interpolated polynomial at some suitable points (interpolation error). This upper bound is often realistic, in the sense that the asymptotic behavior of the approximation error is not better than the one of the projection (or even the interpolation) error. Even more, in some cases the approximate solution coincides with the projection of the true solution upon the subspace (for instance when linear problems with constant coefficients are approximated by spectral methods). This motivates the interest in evaluating the projection and the interpolation errors in differently weighted Sobolev norms. So we must face a situation different from the one of the classical approximation theory where the properties of approximation of orthogonal function systems, polynomial and trigonometric, are studied in the LP-norms, and mostly in the maximum norm (see, e.g., Butzer and Berens [6], Butzer and Nessel [7], Nikol'skiT [24], Sansone [291, Szego [30], Triebel [31], Zygmund [32]; see also Bube [5]). Approximation results in Sobolev norms for the trigonometric system have been obtained by Kreiss and Oliger [15]. In this paper we consider the systems of Legendre orthogonal polynomials, and of Chebyshev orthogonal polynomials of the first kind in dimension d > 1. The reason for this interest must be sought in the applications to spectral approximations of boundary value problems. Indeed, if the boundary conditions are not periodic, Legendre approximation seems to be the easiest to be investigated (the weight w is equal to 1). On the other hand, the Chebyshev approximation is the most effective for practical computations since it allows the use of the Fast Fourier Transform algorithm. The techniques used to obtain our results are based on the representation of a function in the terms of a series of orthogonal polynomials, on the use of the so-called inverse inequality, and finally on the operator interpolation theory in Banach spaces. For the theory of interpolation we refer for instance to Calderon [8], Lions [17], Lions and Peetre [19], Peetre [28]; a recent survey is given, e.g., by Bergh and Lofstrom [4]. An outline of the paper is as follows. In Section 1 some approximation results for the trigonometric system are recalled; the presentation of the results to the interpolation is made in the spirit of what will be its application to Chebyshev polynomials. In Section 2 we consider the La-projection operator upon the space of polynomials of degree at most N in any variable (w denotes the Chebyshev or Legendre weight). In Section 3 a general interpolation operator, built up starting by integration formulas which are not necessarily the same in different spatial dimensions, is considered, and its approximation properties are studied. In [22] Maday and Quarteroni use the results of Section 2 to study the approximation properties of some projection operators in higher order Sobolev norms. Recently, an interesting method which lies inbetween finite elements and This content downloaded from 207.46.13.111 on Tue, 09 Aug 2016 06:29:39 UTC All use subject to http://about.jstor.org/terms ORTHOGONAL POLYNOMIALS IN SOBOLEV SPACES 69 spectral methods has been investigated from the theoretical point of view by Babuska, Szabo and Katz [3]. In particular they obtain approximation properties of polynomials in the norms of the usual Sobolev spaces. Acknowledgements. Some of the results of this paper were announced in [9]; we thank Professor J. L. Lions for the presentation to the C. R. Acad. Sci. of Paris. We also wish to express our gratitude to Professors F. Brezzi and P. A. Raviart for helpful suggestions and continuous encouragement. Notations. Throughout this paper we shall use the following notations: I will be an open bounded interval c R, whose variable is denoted by x; Q the product Id C Rd (d integer > 1) whose variable is denoted by x = (x(.')I_ d; for a multi-integer k E Zd, we set ikV = jd X I'12 and IkloK = m x 1, Dj = a/ax@). The symbol X'J=p (q eventually + oo) will denote the summation over all integral k such that p 0 in U. Set L2(Q) = ({: Q -C I 0 is measurable and ( 0, set Hs ( () = C E L(Q) I 1111ksI, < +?}, where /d 2 11I412I= kENd f DI L/)4 D w dx.

Numerical methods for conservation laws

Abstract: I Mathematical Theory- 1 Introduction- 11 Conservation laws- 12 Applications- 13 Mathematical difficulties- 14 Numerical difficulties- 15 Some references- 2 The Derivation of Conservation Laws- 21 Integral and differential forms- 22 Scalar equations- 23 Diffusion- 3 Scalar Conservation Laws- 31 The linear advection equation- 311 Domain of dependence- 312 Nonsmooth data- 32 Burgers' equation- 33 Shock formation- 34 Weak solutions- 35 The Riemann Problem- 36 Shock speed- 37 Manipulating conservation laws- 38 Entropy conditions- 381 Entropy functions- 4 Some Scalar Examples- 41 Traffic flow- 411 Characteristics and "sound speed"- 42 Two phase flow- 5 Some Nonlinear Systems- 51 The Euler equations- 511 Ideal gas- 512 Entropy- 52 Isentropic flow- 53 Isothermal flow- 54 The shallow water equations- 6 Linear Hyperbolic Systems 58- 61 Characteristic variables- 62 Simple waves- 63 The wave equation- 64 Linearization of nonlinear systems- 641 Sound waves- 65 The Riemann Problem- 651 The phase plane- 7 Shocks and the Hugoniot Locus- 71 The Hugoniot locus- 72 Solution of the Riemann problem- 721 Riemann problems with no solution- 73 Genuine nonlinearity- 74 The Lax entropy condition- 75 Linear degeneracy- 76 The Riemann problem- 8 Rarefaction Waves and Integral Curves- 81 Integral curves- 82 Rarefaction waves- 83 General solution of the Riemann problem- 84 Shock collisions- 9 The Riemann problem for the Euler equations- 91 Contact discontinuities- 92 Solution to the Riemann problem- II Numerical Methods- 10 Numerical Methods for Linear Equations- 101 The global error and convergence- 102 Norms- 103 Local truncation error- 104 Stability- 105 The Lax Equivalence Theorem- 106 The CFL condition- 107 Upwind methods- 11 Computing Discontinuous Solutions- 111 Modified equations- 1111 First order methods and diffusion- 1112 Second order methods and dispersion- 112 Accuracy- 12 Conservative Methods for Nonlinear Problems- 121 Conservative methods- 122 Consistency- 123 Discrete conservation- 124 The Lax-Wendroff Theorem- 125 The entropy condition- 13 Godunov's Method- 131 The Courant-Isaacson-Rees method- 132 Godunov's method- 133 Linear systems- 134 The entropy condition- 135 Scalar conservation laws- 14 Approximate Riemann Solvers- 141 General theory- 1411 The entropy condition- 1412 Modified conservation laws- 142 Roe's approximate Riemann solver- 1421 The numerical flux function for Roe's solver- 1422 A sonic entropy fix- 1423 The scalar case- 1424 A Roe matrix for isothermal flow- 15 Nonlinear Stability- 151 Convergence notions- 152 Compactness- 153 Total variation stability- 154 Total variation diminishing methods- 155 Monotonicity preserving methods- 156 l1-contracting numerical methods- 157 Monotone methods- 16 High Resolution Methods- 161 Artificial Viscosity- 162 Flux-limiter methods- 1621 Linear systems- 163 Slope-limiter methods- 1631 Linear Systems- 1632 Nonlinear scalar equations- 1633 Nonlinear Systems- 17 Semi-discrete Methods- 171 Evolution equations for the cell averages- 172 Spatial accuracy- 173 Reconstruction by primitive functions- 174 ENO schemes- 18 Multidimensional Problems- 181 Semi-discrete methods- 182 Splitting methods- 183 TVD Methods- 184 Multidimensional approaches

Turbulence, Coherent Structures, Dynamical Systems and Symmetry

Abstract: Preface Part I. Turbulence: 1. Introduction 2. Coherent structures 3. Proper orthogonal decomposition 4. Galerkin projection Part II. Dynamical Systems: 5. Qualitative theory 6. Symmetry 7. One-dimensional 'turbulence' 8. Randomly perturbed systems Part III. 9. Low-dimensional Models: 10. Behaviour of the models Part IV. Other Applications and Related Work: 11. Some other fluid problems 12. Review: prospects for rigor Bibliography.

Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for Hyperbolic Conservation Laws

Abstract: In these lecture notes we describe the construction, analysis, and application of ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) schemes for hyperbolic conservation laws and related Hamilton-Jacobi equations. ENO and WENO schemes are high order accurate finite difference schemes designed for problems with piecewise smooth solutions containing discontinuities. The key idea lies at the approximation level, where a nonlinear adaptive procedure is used to automatically choose the locally smoothest stencil, hence avoiding crossing discontinuities in the interpolation procedure as much as possible. ENO and WENO schemes have been quite successful in applications, especially for problems containing both shocks and complicated smooth solution structures, such as compressible turbulence simulations and aeroacoustics.