Implementing Spectral Methods for Partial Differential Equations

Transcription

1 Implementing Spectral Methods for Partial Differential Equations

2 Scientific Computation Editorial Board J.-J.Chattot,Davis,CA,USA P. Colella, Berkeley, CA, USA W. Eist, Princeton, NJ, USA R. Glowinski, Houston, TX, USA Y. Hussaini, Tallahassee, FL, USA P. Joly, Le Chesnay, France H.B. Keller, Pasadena, CA, USA J.E. Marsden, Pasadena, CA, USA D.I. Meiron, Pasadena, CA, USA O. Pironneau, Paris, France A. Quarteroni, Lausanne, Switzerland and Politecnico of Milan, Milan, Italy J.Rappaz,Lausanne,Switzerland R.Rosner,Chicago,IL,USA P. Sagaut, Paris, France J.H. Seinfeld, Pasadena, CA, USA A. Szepessy, Stockholm, Sweden M.F. Wheeler, Austin, TX, USA For other titles published in this series, go to

3 David A. Kopriva Implementing Spectral Methods for Partial Differential Equations Algorithms for Scientists and Engineers

4 Prof. Dr. David A. Kopriva Dept. of Mathematics Florida State University Tallahassee, FL USA ISBN e-isbn DOI / Library of Congress Control Number: Springer Science + Business Media B.V No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper springer.com

5 To my Wife, my Mother, and in memory of my Dad.

6 Preface This book is aimed to be both a textbook for graduate students and a starting point for applications scientists. It is designed to show how to implement spectral methods to approximate the solutions of partial differential equations. It presents a systematic development of the fundamental algorithms needed to write spectral methods codes to solve basic problems of mathematical physics, including steady potentials, transport, and wave propagation. As such, it is meant to supplement, not replace, more general monographs on spectral methods like the recently updated Spectral Methods: Fundamentals in Single Domains and Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics by Canuto, Hussaini, Quarteroni and Zang, which provide detailed surveys of the variety of methods, their performance and theory. I was motivated by comments that I have heard over the years that spectral methods are too hard to implement. I hope to dispel this view or at least to remove the too. Although it is true that a spectral code is harder to hack together than a simple finite difference code (at least a low order finite difference method on a square domain), I show that only a few fundamental algorithms for interpolation, differentiation, FFT and quadrature the subjects of basic numerical methods courses form the building blocks of any spectral code, even for problems in complex geometries. I present the algorithms not only to solve problems in 1D, but 2D as well, to show the flexibility of spectral methods and to make as straightforward as possible the transition from simple, exploratory programs that illustrate the behavior of the methods to application programs. I assume that the reader has a basic knowledge of numerical algorithms. The most important topics include interpolation, quadrature and the numerical integration of ordinary differential equations. An understanding of other methods for the solution of PDEs, such as finite difference or finite element methods is very helpful. Although I assume some background in numerical methods, I have tried to make the presentation and the collection of algorithms self contained. The idea is to make it as straightforward as possible to implement the methods without the need to search for auxiliary routines. Of course, some of the routines (like FFTs, for example) should be later replaced by well-tested and optimally designed versions in the programmer s programming language of choice. Also, since I emphasize the implementation of spectral methods, I recommend having one of the many more general books, such as the two mentioned above, available to consult on the theory of the methods. I have chosen to present the algorithms in a detailed pseudocode, rather than in a specific language such as C or Fortran or application such as Matlab, Maple or Mathematica. The idea is to present the algorithms unencumbered by a language syntax that the reader might not know, but that can nevertheless be quickly translated into the programmer s favorite computer language. I hope that this will make

7 viii Preface the methods useful to the widest possible audience and for the widest range of applications. This is not a programming book, however, and the translation process of the pseudocode will depend on the particular computer language chosen. Fortran programmers will most likely replace loops with vector operations. C/C++ programmers will often have to adjust array indices. The object oriented concepts used for many of the algorithms will be natural to C++ programmers, but a number of tutorials on the web for Fortran programmers show how to implement object oriented ideas using modules, at least until F2003 compilers become available. The book consists of two parts. The first is a quick introduction to spectral approximation to provide a reference point and to define the notation. It is followed by the development of the basic algorithms that form the building blocks of spectral codes. These algorithms include how to use the complex FFT to compute real transforms, how to compute Chebyshev and Legendre polynomials and then how to use them to approximate integrals and derivatives. The second part presents algorithms to approximate the solutions of PDEs. It includes a short survey of spectral approximations that shows how to use the building blocks in one space dimension. Our main interest, however, is how to solve problems in complex geometries in two space dimensions, so we put the emphasis on collocation and nodal versions of Galerkin spectral methods. The development of the algorithms starts from basic approximation on the square, then moves to more complex geometries through the introduction of boundary-fitted mappings, and ends with spectral multidomain methods. I am grateful for the efforts of my students and colleagues who did so much to help me prepare this work. I d like to thank my students Wuming Zhu, James De- Marco, Matt Willyard and Cesar Acosta who gave suggestions on what they wanted to see in a book on spectral methods and implemented algorithms. Tom Zang deserves many thanks for taking the time to read through an entire draft of the manuscript and provide detailed comments. Last, but certainly not least, I want to thank Yousuff Hussaini, who started my research in spectral methods. He has mentored me and encouraged my career for two and a half decades. Many of the topics covered here came out of projects that we have worked on together. Tallahassee, FL David A. Kopriva October 2008

8 Contents Preface... vii Part I Approximating Functions, Derivatives and Integrals 1 Spectral Approximation Preamble:SeriesSolutionofPDEs The Fourier Basis Functions and Fourier Series Series Truncation Modal vs. Nodal Approximation Discrete Orthogonality and Quadrature FourierInterpolation DirectComputationoftheFourierInterpolation ErroroftheFourierInterpolation TheDerivativeoftheFourierInterpolant Polynomial Basis Functions The Legendre Polynomials The Chebyshev Polynomials Polynomial Series Polynomial Series Truncation Derivatives of Truncated Series Polynomial Quadrature Orthogonal Polynomial Interpolation Algorithms for Periodic Functions HowtoComputetheDiscreteFourierTransform Fourier Transforms of Complex Sequences Fourier Transforms of Real Sequences The Fourier Transform in Two Space Variables TheRealFourierTransform How to Evaluate the Fourier Interpolation Derivative by FFT How to Compute Derivatives by Matrix Multiplication Algorithms for Non-Periodic Functions How to Compute the Legendre and Chebyshev Polynomials How to Compute the Gauss Quadrature Nodes and Weights Legendre Gauss Quadrature Legendre Gauss-Lobatto Quadrature Chebyshev Gauss Quadratures How to Evaluate Chebyshev Interpolants via the FFT The Fast Chebyshev Transform How to Evaluate Polynomial Interpolants in Lagrange Form ix

9 x Contents 3.5 How to Evaluate Polynomial Derivatives DirectEvaluationoftheDerivative Evaluation of Derivatives by Matrix Multiplication Even-Odd Decomposition Evaluation by Transform Methods Performance of Various Polynomial Derivative Algorithms 84 Part II Approximating Solutions of PDEs 4 Survey of Spectral Approximations TheFourierCollocationMethod HowtoImplementtheFourierCollocationMethod Benchmark Solution TheFourierGalerkinMethod HowtoImplementtheFourierGalerkinMethod Benchmark Solution Nonlinear and Product Terms TheGalerkinApproximation HowtoComputetheConvolutionSum TheCollocationApproximation Polynomial Collocation Methods ApproximationoftheDiffusionEquation How to Implement the Methods Benchmark Solution Approximation of Scalar Advection The Legendre Galerkin Method HowtoImplementtheMethod The Nodal Continuous Galerkin Method HowtoImplementtheMethod Benchmark Solution The Nodal Discontinuous Galerkin Method HowtoImplementtheMethod Benchmark Solution Summary and Some Broad Generalizations Spectral Approximation on the Square Approximation of Functions in Multiple Space Dimensions Potential Problems on the Square TheCollocationApproximation The Nodal Galerkin Approximation Approximation of Time Dependent Advection-Diffusion TheCollocationApproximation The Nodal Galerkin Approximation TimeIntegration HowtoImplementtheApproximations Benchmark Solution: Advection and Diffusion of a Spot inauniformflow...200

10 Contents xi 5.4 Approximation of Wave Propagation Problems The Nodal Discontinuous Galerkin Approximation How to Implement the Nodal Discontinuous Galerkin Approximation Benchmark Solution: Plane Wave Propagation Benchmark Solution: Propagation of a Circular Sound Wave Transformation Methods from Square to Non-Square Geometries Mappings and Coordinate Transformations Mapping a Straight Sided Quadrilateral How to Approximate Curved Boundaries How to Map the Reference Square to a Curved-Sided Quadrilateral Transformation of Equations under Mappings Two-Dimensional Forms HowtoApproximatetheMetricTerms HowtoComputetheMetricTerms Spectral Methods in Non-Square Geometries Steady Potentials in a Quadrilateral Domain TheCollocationApproximation The Nodal Galerkin Approximation Solution of the Linear Systems Benchmark Solution: Potential in Non-Square Domains Benchmark Solution: Incompressible Flow over a Circular Obstacle Steady Potentials in an Annulus Benchmark Solution: Potential in an Annulus with a Source Advection and Diffusion in Quadrilateral Domains TransformationoftheAdvection-DiffusionEquation TheCollocationApproximation The Nodal Galerkin Approximation HowtoImplementtheApproximations Benchmark Solution: Advection and Diffusion in a Non-Square Geometry Benchmark Solution: Advection and Diffusion of a Pollutant in a Curved Channel Conservation Laws in Quadrilateral Domains The Nodal Discontinuous Galerkin Approximation How to Implement the Nodal Discontinuous Galerkin Approximation Benchmark Solution: Acoustic Scattering off a Cylinder Spectral Element Methods Spectral Element Methods in One Space Dimension The Continuous Galerkin Spectral Element Method...297

11 xii Contents How to Implement the Continuous Galerkin Spectral ElementMethod Benchmark Solution: Cooling of a Temperature Spot The Discontinuous Galerkin Spectral Element Method How to Implement the Discontinuous Galerkin Spectral ElementMethod Benchmark Solution: Wave Propagation and Reflection The Two-Dimensional Mesh and Its Specification How to Construct a Two-Dimensional Mesh Benchmark Solution: A Spectral Element Mesh for a Disk The Spectral Element Method in Two Space Dimensions How to Implement the Spectral Element Method Benchmark Solution: Steady Temperatures in a Long CylindricalRod The Discontinuous Galerkin Spectral Element Method How to Implement the Discontinuous Galerkin Spectral ElementMethod Benchmark Solution: Propagation of a Circular Wave inacirculardomain Benchmark Solution: Transmission and Reflection from amaterialinterface A Pseudocode Conventions B Floating Point Arithmetic C Basic Linear Algebra Subroutines (BLAS) D Linear Solvers D.1 DirectSolvers D.1.1 Tri-Diagonal Solver D.1.2 LUFactorization D.2 IterativeSolvers E Data Structures E.1 LinkedLists E.1.1 Example: Elements that Share a Node E.2 HashTables E.2.1 Example: Avoiding Duplicate Edges in a Mesh References Index of Algorithms Subject Index...389

12 List of Algorithms 1 DiscreteFourierCoefficients: Direct Evaluation of the Discrete Fourier Coefficients FourierInterpolantFromModes: Direct Evaluation of the Fourier Interpolant from Its Modes FourierInterpolantFromNodes: Direct Evaluation of the Fourier Interpolant from Its Nodes LegendreDerivativeCoefficients: Evaluate the Legendre Coefficients of the Derivative of a Polynomial ChebyshevDerivativeCoefficients: Evaluate the Chebyshev Coefficients of the Derivative of a Polynomial DFT: Direct (and Slow) Evaluation of the Discrete Fourier Transform InitializeFFT: Initialization Routine for FFT Radix2FFT: Temperton sradix2selfsortingcomplexfft FFFTOfTwoRealVectors: Simultaneous Computation of the DFT of Two Real Sequences. The Forward Transform BFFTForTwoRealVectors: Simultaneous Computation of the DFT of Two Real Sequences. The Backward Transform FFFTEO: The Forward DFT by Even-Odd Decomposition BFFTEO: The Backward DFT by Even-Odd Decomposition Forward2DFFT: A Two-Dimensional Forward FFT of a Real Array withanevennumberofpointsineachdirection Backward2DFFT: Two-Dimensional Backward FFT of a Real Array withanevennumberofpointsineachdirection ForwardRealFFT: TheForwardRealTransform BackwardRealFFT: The Backward Real Transform FourierDerivativeByFFT: Fast Evaluation of the Fourier Polynomial Derivative FourierDerivativeMatrix: Computation of the Fourier Derivative MatrixusingtheNegativeSumTrick MxVDerivative: A Matrix-Vector Multiplication Procedure LegendrePolynomial: Evaluate the Legendre Polynomial of Degree k using Three Term Recursion ChebyshevPolynomial: The Chebyshev Polynomial of Degree k using Three Term Recursion and Trigonometric Forms LegendrePolynomialAndDerivative: The Legendre Polynomial of Degree k and Its Derivative using the Three Term Recursion LegendreGaussNodesAndWeights: qandlevaluation: Combined Algorithm to Compute L N (x), q(x) = L N+1 L N 1, and q (x) LegendreGaussLobattoNodesAndWeights: xiii

13 xiv List of Algorithms 26 ChebyshevGaussNodesAndWeights: ChebyshevGaussLobattoNodesAndWeights: FastCosineTransform: The Cosine Transform Computed with the Real FFT FastChebyshevTransform: The Fast Chebyshev Transform using thefastcosinetransform BarycentricWeights: Weights for Lagrange Interpolation LagrangeInterpolation: Lagrange Interpolant from Barycentric Form PolynomialInterpolationMatrix: Matrix for Interpolation Between Two SetsofPoints InterpolateToNewPoints: Interpolation Between Two Sets of Points by Matrix Multiplication LagrangeInterpolatingPolynomials: l j (x) DCoarseToFineInterpolation: Interpolation from a Coarse to a Fine Gridin2D LagrangeInterpolantDerivative: Direct Computation of the Polynomial Derivative in Barycentric Form PolynomialDerivativeMatrix: First Derivative Approximation Matrix mthorderpolynomialderivativematrix: Derivative Matrix for mth OrderDerivatives EOMatrixDerivative: Computation of First Derivative by Even-Odd Decomposition FastChebyshevDerivative: Computation of the Derivative by the Fast Chebyshev Transform FourierCollocationTimeDerivative: The Fourier Collocation Time DerivativefortheAdvection-DiffusionEquation CollocationStepByRK3: Low-Storage Runge-Kutta Integration ofthefouriercollocationapproximation FourierCollocationDriver: A Driver for the Fourier Collocation Approximation AdvectionDiffusionTimeDerivative: Advection-Diffusion Time DerivativeforFourierGalerkin FourierGalerkinStep: Take One Time Step of the Fourier Galerkin Method EvaluateFourierGalerkinSolution: Direct Synthesis of the Fourier GalerkinSolution FourierGalerkinDriver: A Driver for the Fourier Galerkin Approximation DirectConvolutionSum: Direct (Slow) Computation of the Convolution Sum FastConvolutionSum: Computation of the Convolution Sum withthefft CollocationStepByRK3: Low Storage Runge-Kutta Integration of a Polynomial Collocation Approximation LegendreCollocation: Drivers for Legendre Collocation Approximation 118

14 List of Algorithms xv 52 ModifiedLegendreBasis: The Legendre Basis Modified to Vanish at Endpoints EvaluateLegendreGalerkinSolution: Synthesis of the Legendre GalerkinSolution InitTMatrix: Legendre Galerkin Tridiagonal Matrix ModifiedCoefsFromLegendreCoefs: Computing the Modified Legendre Coefficients from Legendre Coefficients LegendreGalerkinStep: Take One Time Step by Trapezoidal Rule CGDerivativeMatrix: Matrix for Legendre Galerkin Approximation NodalDiscontinuousGalerkin: A Discontinuous Galerkin Class Definition NodalDiscontinuousGalerkin:Construct: Constructor for the Discontinuous Galerkin Class NodalDiscontinuousGalerkin:DGDerivative: First Spatial Derivative viathegalerkinapproximation NodalDiscontinuousGalerkin:DGTimeDerivative: Time Derivative via the Discontinuous Galerkin Approximation DGStepByRK3: Low Storage Runge-Kutta Integration of a Nodal Discontinuous Galerkin Approximation Nodal2DStorage: Storage for a Nodal Spectral Method NodalPotentialClass: A Class for the Potential Problem on the Square NodalPotentialClass:Construct: Constructor for the Chebyshev CollocationApproximationofthePotentialProblem NodalPotentialClass:LaplacianOnTheSquare: Collocation ApproximationtotheLaplaceOperator MaskSides: Set Boundary Values to Zero According to a Mask Function NodalPotentialClass:MatrixAction: Collocation Approximation tothelaplaceoperator CollocationRHSComputation: Right Hand Side Construction for Direct SolutionoftheCollocationEquations LaplaceCollocationMatrix: Matrix Construction for Direct Solution ofthecollocationapproximationforthepoissonproblem Residual: Residual for a Polynomial Collocation Approximation to the Potential Equation on the Square FDPreconditioner: A Class for a Finite Difference Preconditioner FDPreconditioner:Construct: Constructor for the Finite Difference Preconditioner on the Square FDPreconditioner:Solve: Solver for the ILU Preconditioner H ILU u = R BiCGSSTABSolve: BiCGStab Iterative Solver for Nodal Spectral Methods CollocationPotentialDriver: Driver for a Polynomial Collocation Approximation to the Potential on the Square LaplacianOnTheSquare: Nodal Galerkin Approximation to the Laplace Operator...178

15 xvi List of Algorithms 78 ApproximateFEMStencil: Computing the Approximate Finite Element Stencil on the Square SSORSweep: SSOR Sweep for the Finite Element Preconditioner PreconditionedConjugateGradientSolve: Conjugate Gradient Iterative Solver for Nodal Spectral Methods NodalAdvDiffClass: A Class for the Advection-Diffusion Problem on the Square NodalAdvDiffClass:Construct: Constructor for the Chebyshev CollocationApproximationoftheAdvection-DiffusionProblem NodalAdvDiffClass:Transport: Approximation to q Φ NodalAdvDiffClass:ExplicitRHS: Explicit Part of the BDF ApproximationoftheAdvection-DiffusionEquation NodalAdvDiffClass:MatrixAction: Matrix Action for the BDF ApproximationoftheAdvection-DiffusionEquation NodalAdvDiffClass:Residual: Iteration Residual for the BDF ApproximationoftheAdvection-DiffusionEquation MultistepIntegration: One Step of the Linear Multistep Integration oftheadvection-diffusionequation RiemannSolver: TheNumericalFluxfortheWaveEquation NodalDG2DStorage: Data Storage for a Nodal Spectral Method NodalDG2DClass: A Discontinuous Galerkin Class Definition NodalDG2D:Construct: Constructor for the Discontinuous Galerkin Class SystemDGDerivative: Compute the First Derivative via the Discontinuous Galerkin Approximation NodalDG2D:DG2DTimeDerivative: Time Derivative in 2D for the Discontinuous Galerkin Approximation WaveEquationFluxes: Flux Vectors for the Two Dimensional Wave Equation QuadMap: Mapping of the Reference Square to a Straight Sided Quadrilateral CurveInterpolant: A Curve Interpolant Class Definition CurveInterpolantProcedures: TransfiniteQuadMap: Mapping of the Reference Square to a Curve-Bounded Quadrilateral TransfiniteQuadMetrics: Computation of the Metric Terms on a Curve-Bounded Quadrilateral QuadMapMetrics: Computation of the Metric Terms on a Straight Sided Quadrilateral MappedGeometryClass: Manage Geometry and Metric Terms for Quadrilateral Domains MappedGeometry:Construct: Constructor for Geometry and Metric Terms for Quadrilateral Domains MappedNodalPotentialClass: A Class for the Potential Problem in a Mapped Domain...250

16 List of Algorithms xvii 104 MappedNodalPotentialClass:Construct: Constructor for Collocation Potential Solution on a Mapped Domain MappedNodalPotentialClass:MappedLaplacian: The Collocation Approximation to the Laplace Operator on a Mapped Domain TransposeMatrixMultiply: Matrix Transpose-Vector Multiplication Algorithm MappedNodalPotentialClass:MappedLaplacian: Nodal Galerkin Approximation to the Laplace Operator on a Mapped Domain MappedCollocationDriver: Driver for the Collocation Approximation to Steady Potential in a Non-Square Geometry PotentialOnAnnulus: Use of the FFT to Compute Potentials with One PeriodicDirection DGSolutionStorage: Storage of Interior and Boundary Solutions MappedNodalDG2DClass: A Discontinuous Galerkin Class Definition DG2DProlongToFaces: Interpolate the Solution from Gauss Points to the Boundaries MappedDG2DBoundaryFluxes: Boundary Fluxes in 2D for the Discontinuous Galerkin Approximation MappedDG2DTimeDerivative: Time Derivative in 2D for the Discontinuous Galerkin Approximation GlobalTimeDerivative: Full Time Derivative in 2D for the Discontinuous Galerkin Approximation SEM1DClass: Data Storage for the One-Dimensional Spectral Element Method SEMGlobalProcedures1D: Global Operations for the One-Dimensional Spectral Element Method SEM1DProcedures: Spatial Approximations for the One-Dimensional Spectral Element Method TrapezoidalRuleIntegration: Integration of the One-Dimensional Spectral Element Method in Time DGSEM1DClasses: Element and Mesh Definitions for the One-Dimensional Discontinuous Galerkin Spectral Element Method LocalDSEMProcedures: Local Procedures for the Discontinuous Galerkin Spectral Element Method GlobalMeshProcedures: Mesh Global Procedures for the Discontinuous Galerkin Spectral Element Approximation CornerNodeClass: Corner Node for Two-Dimensional Spectral Element Methods QuadElementClass: Quadrilateral Element Definition for Two-Dimensional Spectral Element Methods EdgeClass: Edge Definition for Two-Dimensional Spectral Element Methods QuadMesh: Mesh Definition for Two-Dimensional Spectral Element Methods...325

17 xviii List of Algorithms 127 QuadMesh:Construct: Constructor for a Two Dimensional Spectral ElementMesh SEMPotentialClass: A Class Definition for the Spectral Element ApproximationofthePotentialProblem SEMPotentialClass:Construct: Constructor for the Spectral Element ApproximationofthePotentialProblem SEMMask: Mask Edges and Corners for the Spectral Element Method SEMUnMask: UnMask for the Spectral Element Method SEMGlobalSum: Sum Edge Contributions for the Two-Dimensional Spectral Element Method SEMGlobalSum: Sum Edge Contributions for the Two-Dimensional Spectral Element Method (continued) SEMPotentialClass:MatrixAction: Matrix Action for the Spectral ElementApproximationtothePotentialEquation Residual: Residual Computation for the Spectral Element ApproximationtothePotentialEquation SetBoundaryValues: Set Dirichlet Boundary Conditions for the Two-Dimensional Spectral Element Method DGSEMClass: A Discontinuous Galerkin Class Definition EdgeFluxes: Compute the Riemann Problem Along Mesh Edges DGSEMClass:TimeDerivative: Compute the Time Derivative for the Discontinuous Galerkin Approximation AlmostEqual: Testing Equality of Two Floating Point Numbers BLAS_Level1: A Selection of Basic Linear Algebra Subroutines TriDiagonalSolve: LUFactorization: Factorization and Solve Procedures to Solve Ax = y Record: An Example Linked List Record Definition LinkedList: A Linked List Class Definition LinkedList:Procedures: SparseMatrix: A Sparse Matrix Class Definition SparseMatrix:Procedures: ConstructMeshEdges: Construct Edge Information for a Spectral ElementMesh...383