ebook img

Numerical Analysis and Optimization: An Introduction to Mathematical Modelling and Numerical Simulation (Numerical Mathematics and Scientific Computation) PDF

472 Pages·2007·6.77 MB·English
Save to my drive
Quick download
Download
Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.

Preview Numerical Analysis and Optimization: An Introduction to Mathematical Modelling and Numerical Simulation (Numerical Mathematics and Scientific Computation)

NUMERICAL MATHEMATICS AND SCIENTIFIC COMPUTATION Series Editors G.H. GOLUB, A. GREENBAUM A.M. STUART, E. SU¨LI NUMERICAL MATHEMATICS AND SCIENTIFIC COMPUTATION Books in the series Monographs marked with an asterix (*) appeared in the series ‘Monographs in Numerical Analysis’ which has been folded into, and is continued by, the current series. * P. Dierckx: Curve and surface fittings with splines * J.H. Wilkinson: The algebraic eigenvalue problem * I. Duff, A. Erisman, and J. Reid: Direct methods for sparse matrices * M.J. Baines: Moving finite elements * J.D. Pryce: Numerical solution of Sturm–Liouville problems K. Burrage: Parallel and sequential methods for ordinary differential equations Y. Censor and S.A. Zenios: Parallel optimization: theory, algorithms and applications M. Ainsworth, J. Levesley, W. Light, and M. Marletta: Wavelets, multilevel methods and elliptic PDEs W.Freeden,T.Gervens,andM.Schreiner: Constructiveapproximationonthesphere: theory and applications to geomathematics C.H. Schwab: p- and hp- finite element methods: theory and applications to solid and fluid mechanics J.W. Jerome: Modelling and computation for applications in mathematics, science, and engineering Alfio Quarteroni and Alberto Valli: Domain decomposition methods for partial differential equations G.E. Karniadakis and S.J. Sherwin: Spectral/hp element methods for CFD I. Babuˇska and T. Strouboulis: The finite element method and its reliability B. Mohammadi and O. Pironneau: Applied shape optimization for fluids S. Succi: The Lattice Boltzmann Equation for fluid dynamics and beyond P. Monk: Finite element methods for Maxwell’s equations A. Bellen and M. Zennaro: Numerical methods for delay differential equations J. Modersitzki: Numerical methods for image registration M. Feistauer, J. Felcman, and I. Straˇskraba: Mathematical and computational methods for compressible flow W. Gautschi: Orthogonal polynomials: computation and approximation M.K. Ng: Iterative methods for Toeplitz systems Michael Metcalf, John Reid, and Malcolm Cohen: Fortran 95/2003 explained George Em Karniadakis and Spencer Sherwin: Spectral/hp element methods for CFD, second edition DarioA.Bini,GuyLatouche,andBeatriceMeini: NumericalmethodsforstructuredMarkov chains Howard Elman, David Silvester, and Andy Wathen: Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics Moody Chu and Gene Golub: Inverse eigenvalue problems: theory and applications Jean-Fr´ed´eric Gerbeau, Claude Le Bris, and Tony Leli`evre: Mathematical methods for the magnetohydrodynamics of liquid metals Gr´egoire Allaire: Numerical analysis and optimization umerical nalysis and ptimization N A O An introduction to mathematical modelling and numerical simulation Gr´egoire Allaire ´ Ecole Polytechnique Translated by Dr Alan Craig University of Durham 1 3 GreatClarendonStreet,OxfordOX26DP OxfordUniversityPressisadepartmentoftheUniversityofOxford. ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship, andeducationbypublishingworldwidein Oxford NewYork Auckland CapeTown DaresSalaam HongKong Karachi KualaLumpur Madrid Melbourne MexicoCity Nairobi NewDelhi Shanghai Taipei Toronto Withofficesin Argentina Austria Brazil Chile CzechRepublic France Greece Guatemala Hungary Italy Japan Poland Portugal Singapore SouthKorea Switzerland Thailand Turkey Ukraine Vietnam OxfordisaregisteredtrademarkofOxfordUniversityPress intheUKandincertainothercountries PublishedintheUnitedStates byOxfordUniversityPressInc.,NewYork (cid:1)c Gr´egoireAllaire2007 Themoralrightsoftheauthorhavebeenasserted DatabaserightOxfordUniversityPress(maker) Firstpublished2007 Allrightsreserved. Nopartofthispublicationmaybereproduced, storedinaretrievalsystem,ortransmitted,inanyformorbyanymeans, withoutthepriorpermissioninwritingofOxfordUniversityPress, orasexpresslypermittedbylaw,orundertermsagreedwiththeappropriate reprographicsrightsorganization. Enquiriesconcerningreproduction outsidethescopeoftheaboveshouldbesenttotheRightsDepartment, OxfordUniversityPress,attheaddressabove Youmustnotcirculatethisbookinanyotherbindingorcover andyoumustimposethesameconditiononanyacquirer BritishLibraryCataloguinginPublicationData Dataavailable LibraryofCongressCataloginginPublicationData Dataavailable TypesetbyNewgenImagingSystems(P)Ltd.,Chennai,India PrintedinGreatBritain onacid-freepaperby BiddlesLtd.,King’sLynn,Norfolk ISBN978–0–19–920521–9(Hbk.) ISBN978–0–19–920522–6(Pbk.) 1 3 5 7 9 10 8 6 4 2 Contents 1 Introduction ix 1 Introduction to mathematical modelling and numerical simulation 1 1.1 General introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 An example of modelling. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Some classical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.1 The heat flow equation . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.2 The wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.3 The Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.4 Schr¨odinger’s equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3.5 The Lam´e system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3.6 The Stokes system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3.7 The plate equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 Numerical calculation by finite differences . . . . . . . . . . . . . . . . . . . . 14 1.4.1 Principles of the method . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4.2 Numerical results for the heat flow equation . . . . . . . . . . . . . . . 17 1.4.3 Numerical results for the advection equation . . . . . . . . . . . . . . 21 1.5 Remarks on mathematical models. . . . . . . . . . . . . . . . . . . . . . . . . 25 1.5.1 The idea of a well-posed problem . . . . . . . . . . . . . . . . . . . . . 25 1.5.2 Classification of PDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2 Finite difference method 31 2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2 Finite differences for the heat equation . . . . . . . . . . . . . . . . . . . . . . 32 2.2.1 Various examples of schemes . . . . . . . . . . . . . . . . . . . . . . . 32 2.2.2 Consistency and accuracy . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.2.3 Stability and Fourier analysis . . . . . . . . . . . . . . . . . . . . . . . 36 2.2.4 Convergence of the schemes . . . . . . . . . . . . . . . . . . . . . . . . 42 2.2.5 Multilevel schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.2.6 The multidimensional case. . . . . . . . . . . . . . . . . . . . . . . . . 46 2.3 Other models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.3.1 Advection equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.3.2 Wave equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 v vi CONTENTS 3 Variational formulation of elliptic problems 65 3.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.1.2 Classical formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.1.3 The case of a space of one dimension . . . . . . . . . . . . . . . . . . . 67 3.2 Variational approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.2.1 Green’s formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.2.2 Variational formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.3 Lax–Milgram theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.3.1 Abstract framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.3.2 Application to the Laplacian . . . . . . . . . . . . . . . . . . . . . . . 76 4 Sobolev spaces 79 4.1 Introduction and warning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.2 Square integrable functions and weak differentiation . . . . . . . . . . . . . . 80 4.2.1 Some results from integration . . . . . . . . . . . . . . . . . . . . . . . 80 4.2.2 Weak differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.3 Definition and principal properties . . . . . . . . . . . . . . . . . . . . . . . . 84 4.3.1 The space H1(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.3.2 The space H1(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 0 4.3.3 Traces and Green’s formulas . . . . . . . . . . . . . . . . . . . . . . . 89 4.3.4 A compactness result. . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.3.5 The spaces Hm(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 4.4 Some useful extra results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.4.1 Proof of the density theorem 4.3.5 . . . . . . . . . . . . . . . . . . . . 98 4.4.2 The space H(div) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.4.3 The spaces Wm,p(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.4.4 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.5 Link with distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 5 Mathematical study of elliptic problems 109 5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.2 Study of the Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 5.2.1 Dirichlet boundary conditions . . . . . . . . . . . . . . . . . . . . . . . 109 5.2.2 Neumann boundary conditions . . . . . . . . . . . . . . . . . . . . . . 116 5.2.3 Variable coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.2.4 Qualitative properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.3 Solution of other models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.3.1 System of linear elasticity . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.3.2 Stokes equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 6 Finite element method 149 6.1 Variational approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 6.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 6.1.2 General internal approximation . . . . . . . . . . . . . . . . . . . . . . 150 6.1.3 Galerkin method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6.1.4 Finite element method (general principles). . . . . . . . . . . . . . . . 153 6.2 Finite elements in N =1 dimension . . . . . . . . . . . . . . . . . . . . . . . 154 CONTENTS vii 6.2.1 P1 finite elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 6.2.2 Convergence and error estimation. . . . . . . . . . . . . . . . . . . . . 159 6.2.3 P2 finite elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 6.2.4 Qualitative properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 6.2.5 Hermite finite elements . . . . . . . . . . . . . . . . . . . . . . . . . . 168 6.3 Finite elements in N ≥2 dimensions . . . . . . . . . . . . . . . . . . . . . . . 171 6.3.1 Triangular finite elements . . . . . . . . . . . . . . . . . . . . . . . . . 171 6.3.2 Convergence and error estimation. . . . . . . . . . . . . . . . . . . . . 184 6.3.3 Rectangular finite elements . . . . . . . . . . . . . . . . . . . . . . . . 191 6.3.4 Finite elements for the Stokes problem . . . . . . . . . . . . . . . . . . 195 6.3.5 Visualization of the numerical results. . . . . . . . . . . . . . . . . . . 201 7 Eigenvalue problems 205 7.1 Motivation and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 7.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 7.1.2 Solution of nonstationary problems . . . . . . . . . . . . . . . . . . . . 206 7.2 Spectral theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 7.2.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 7.2.2 Spectral decomposition of a compact operator. . . . . . . . . . . . . . 210 7.3 Eigenvalues of an elliptic problem . . . . . . . . . . . . . . . . . . . . . . . . . 213 7.3.1 Variational problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 7.3.2 Eigenvalues of the Laplacian . . . . . . . . . . . . . . . . . . . . . . . 218 7.3.3 Other models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 7.4 Numerical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 7.4.1 Discretization by finite elements . . . . . . . . . . . . . . . . . . . . . 224 7.4.2 Convergence and error estimates . . . . . . . . . . . . . . . . . . . . . 227 8 Evolution problems 231 8.1 Motivation and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 8.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 8.1.2 Modelling and examples of parabolic equations . . . . . . . . . . . . . 232 8.1.3 Modelling and examples of hyperbolic equations . . . . . . . . . . . . 233 8.2 Existence and uniqueness in the parabolic case . . . . . . . . . . . . . . . . . 234 8.2.1 Variational formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 234 8.2.2 A general result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236 8.2.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 8.3 Existence and uniqueness in the hyperbolic case. . . . . . . . . . . . . . . . . 246 8.3.1 Variational formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 246 8.3.2 A general result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 8.3.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 8.4 Qualitative properties in the parabolic case . . . . . . . . . . . . . . . . . . . 253 8.4.1 Asymptotic behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 8.4.2 The maximum principle . . . . . . . . . . . . . . . . . . . . . . . . . . 255 8.4.3 Propagation at infinite velocity . . . . . . . . . . . . . . . . . . . . . . 256 8.4.4 Regularity and regularizing effect . . . . . . . . . . . . . . . . . . . . . 257 8.4.5 Heat equation in the entire space . . . . . . . . . . . . . . . . . . . . . 259 8.5 Qualitative properties in the hyperbolic case. . . . . . . . . . . . . . . . . . . 261 8.5.1 Reversibility in time . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 viii CONTENTS 8.5.2 Asymptotic behaviour and equipartition of energy . . . . . . . . . . . 262 8.5.3 Finite velocity of propagation . . . . . . . . . . . . . . . . . . . . . . . 263 8.6 Numerical methods in the parabolic case . . . . . . . . . . . . . . . . . . . . . 264 8.6.1 Semidiscretization in space . . . . . . . . . . . . . . . . . . . . . . . . 264 8.6.2 Total discretization in space-time . . . . . . . . . . . . . . . . . . . . . 266 8.7 Numerical methods in the hyperbolic case . . . . . . . . . . . . . . . . . . . . 269 8.7.1 Semidiscretization in space . . . . . . . . . . . . . . . . . . . . . . . . 270 8.7.2 Total discretization in space-time . . . . . . . . . . . . . . . . . . . . . 271 9 Introduction to optimization 277 9.1 Motivation and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 9.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 9.1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 9.1.3 Definitions and notation . . . . . . . . . . . . . . . . . . . . . . . . . . 284 9.1.4 Optimization in finite dimensions . . . . . . . . . . . . . . . . . . . . . 285 9.2 Existence of a minimum in infinite dimensions. . . . . . . . . . . . . . . . . . 287 9.2.1 Examples of nonexistence . . . . . . . . . . . . . . . . . . . . . . . . . 287 9.2.2 Convex analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 9.2.3 Existence results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292 10 Optimality conditions and algorithms 297 10.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 10.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297 10.1.2 Differentiability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 10.2 Optimality conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 10.2.1 Euler inequalities and convex constraints . . . . . . . . . . . . . . . . 303 10.2.2 Lagrange multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 306 10.3 Saddle point, Kuhn–Tucker theorem, duality . . . . . . . . . . . . . . . . . . 317 10.3.1 Saddle point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317 10.3.2 The Kuhn–Tucker theorem . . . . . . . . . . . . . . . . . . . . . . . . 318 10.3.3 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 10.4 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 10.4.1 Dual or complementary energy . . . . . . . . . . . . . . . . . . . . . . 323 10.4.2 Optimal command . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326 10.4.3 Optimization of distributed systems . . . . . . . . . . . . . . . . . . . 330 10.5 Numerical algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 10.5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332 10.5.2 Gradient algorithms (case without constraints) . . . . . . . . . . . . . 333 10.5.3 Gradient algorithms (case with constraints) . . . . . . . . . . . . . . . 336 10.5.4 Newton’s method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342 11 Methods of operational research (Written in collaboration with St´ephane Gaubert) 347 11.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 11.2 Linear programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 11.2.1 Definitions and properties . . . . . . . . . . . . . . . . . . . . . . . . . 348 11.2.2 The simplex algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 11.2.3 Interior point algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . 357 CONTENTS ix 11.2.4 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 11.3 Integer polyhedra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 11.3.1 Extreme points of compact convex sets. . . . . . . . . . . . . . . . . . 362 11.3.2 Totally unimodular matrices . . . . . . . . . . . . . . . . . . . . . . . 364 11.3.3 Flow problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 11.4 Dynamic programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 11.4.1 Bellman’s optimality principle. . . . . . . . . . . . . . . . . . . . . . . 372 11.4.2 Finite horizon problem. . . . . . . . . . . . . . . . . . . . . . . . . . . 372 11.4.3 Minimum cost path, or optimal stopping, problem . . . . . . . . . . . 375 11.5 Greedy algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380 11.5.1 General points about greedy methods . . . . . . . . . . . . . . . . . . 380 11.5.2 Kruskal’s algorithm for the minimum spanning tree problem . . . . . . . . . . . . . . . . . . . . . . . . . . . 381 11.6 Separation and relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383 11.6.1 Separation and evaluation (branch and bound) . . . . . . . . . . . . . 383 11.6.2 Relaxation of combinatorial problems . . . . . . . . . . . . . . . . . . 388 12 Appendix Review of hilbert spaces 399 13 Appendix Matrix Numerical Analysis 405 13.1 Solution of linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405 13.1.1 Review of matrix norms . . . . . . . . . . . . . . . . . . . . . . . . . . 406 13.1.2 Conditioning and stability . . . . . . . . . . . . . . . . . . . . . . . . . 409 13.1.3 Direct methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 13.1.4 Iterative methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424 13.1.5 The conjugate gradient method . . . . . . . . . . . . . . . . . . . . . . 428 13.2 Calculation of eigenvalues and eigenvectors . . . . . . . . . . . . . . . . . . . 435 13.2.1 The power method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 13.2.2 The Givens–Householder method . . . . . . . . . . . . . . . . . . . . . 438 13.2.3 The Lanczos method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442 Index 451 Index notations 455

See more

The list of books you might like

Most books are stored in the elastic cloud where traffic is expensive. For this reason, we have a limit on daily download.