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Proper Orthogonal Decomposition Methods for Partial Differential Equations [Pehme köide]

(Professor of Mathematics and Aerospace Engineering, Texas A & M University), (North China Electric Power University, Beijing, China)
  • Formaat: Paperback / softback, 278 pages, kõrgus x laius: 229x152 mm, kaal: 450 g
  • Sari: Mathematics in Science & Engineering
  • Ilmumisaeg: 03-Dec-2018
  • Kirjastus: Academic Press Inc
  • ISBN-10: 012816798X
  • ISBN-13: 9780128167984
Teised raamatud teemal:
Proper Orthogonal Decomposition Methods for Partial Differential EquationsSuurem pilt
  • Formaat: Paperback / softback, 278 pages, kõrgus x laius: 229x152 mm, kaal: 450 g
  • Sari: Mathematics in Science & Engineering
  • Ilmumisaeg: 03-Dec-2018
  • Kirjastus: Academic Press Inc
  • ISBN-10: 012816798X
  • ISBN-13: 9780128167984
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9780128167984.html
Märksõnad:

Proper Orthogonal Decomposition Methods for Partial Differential Equations evaluates the potential applications of POD reduced-order numerical methods in increasing computational efficiency, decreasing calculating load and alleviating the accumulation of truncation error in the computational process. Introduces the foundations of finite-differences, finite-elements and finite-volume-elements. Models of time-dependent PDEs are presented, with detailed numerical procedures, implementation and error analysis. Output numerical data are plotted in graphics and compared using standard traditional methods. These models contain parabolic, hyperbolic and nonlinear systems of PDEs, suitable for the user to learn and adapt methods to their own R&D problems.

  • Explains ways to reduce order for PDEs by means of the POD method so that reduced-order models have few unknowns
  • Helps readers speed up computation and reduce computation load and memory requirements while numerically capturing system characteristics
  • Enables readers to apply and adapt the methods to solve similar problems for PDEs of hyperbolic, parabolic and nonlinear types

Arvustused

"This book details the application of the Proper Orthogonal Decomposition (POD) to instationary problems whose spatial semidiscretization is done either by Finite Difference (FD), Finite Element (FE) or Finite Volume (FV) methods. These three discretization methods correspond to the 3 main chapters of the book." --zbMATH

Foreword and Introduction xi
1 Reduced-Order Extrapolation Finite Difference Schemes Based on Proper Orthogonal Decomposition
1.1 Review of Classical Basic Finite Difference Theory
1(9)
1.1.1 Approximation of Derivative
1(2)
1.1.2 Difference Operators
3(1)
1.1.3 The Formation of Difference Equations
4(1)
1.1.4 The Effectiveness of Finite Difference Schemes
4(6)
1.2 A POD-Based Reduced-Order Extrapolation Finite Difference Scheme for the 2D Parabolic Equation
10(10)
1.2.1 A Classical Finite Difference Scheme for the 2D Parabolic Equation
11(1)
1.2.2 Formulation of the POD Basis
12(2)
1.2.3 Establishment of the POD-Based Reduced-Order Finite Difference Scheme for the 2D Parabolic Equation
14(3)
1.2.4 Error Estimates of the Reduced-Order Finite Difference Solutions for the 2D Parabolic Equation
17(1)
1.2.5 The Implementation of the Algorithm of the POD-Based Reduced-Order Finite Difference Scheme for the 2D Parabolic Equation
18(1)
1.2.6 A Numerical Example for the 2D Parabolic Equation
19(1)
1.3 A POD-Based Reduced-Order Extrapolation Finite Difference Scheme for the 2D Nonstationary Stokes Equation
20(12)
1.3.1 Background for the 2D Nonstationary Stokes Equation
21(1)
1.3.2 A Classical Finite Difference Scheme for the 2D Nonstationary Stokes Equation and the Generation of Snapshots
22(1)
1.3.3 Formulations of the POD Basis and the POD-Based Reduced-Order Extrapolating Finite Difference Scheme
23(2)
1.3.4 Error Estimates and a Criterion for Renewing the POD Basis
25(2)
1.3.5 Implementation for the POD-Based Reduced-Order Extrapolating Finite Difference Scheme
27(2)
1.3.6 Some Numerical Experiments for the 2D Nonstationary Stokes Equation
29(3)
1.4 POD-Based Reduced-Order Extrapolating Finite Difference Scheme for 2D Shallow Water Equation
32(24)
1.4.1 Model Background and Survey for the 2D Shallow Water Equation
32(2)
1.4.2 The Governing Equations and the Classical FD Scheme for the 2D Shallow Water Equation Including Sediment Concentration
34(6)
1.4.3 Establishment of the POD-Based Reduced-Order Extrapolating Finite Difference Scheme
40(2)
1.4.4 Error Estimates for the POD-Based Reduced-Order Extrapolating Finite Difference Solutions
42(2)
1.4.5 Algorithm Implementation for the POD-Based Reduced-Order Extrapolating Finite Difference Scheme
44(1)
1.4.6 Some Numerical Experiments for the 2D Shallow Water Equation With Sediment Concentration
45(11)
1.5 Conclusions and Discussion About POD-Based Reduced-Order Extrapolation Finite Difference Schemes
56(2)
2 Reduced-Order Extrapolation Finite Element Methods Based on Proper Orthogonal Decomposition
2.1 Basic Theory of the Finite Element Method and Mixed Finite Element Method
58(19)
2.1.1 Sobolev Spaces
58(4)
2.1.2 Imbedding and Trace Theorems of Sobolev Spaces
62(2)
2.1.3 Finite Element Spaces
64(4)
2.1.4 Interpolation Error Estimates in Sobolev Spaces
68(2)
2.1.5 Variational Problems and Their Finite Element Approximations
70(2)
2.1.6 Mixed Variational Problems and Their Mixed Finite Element Approximations
72(3)
2.1.7 L2 Projection, the Ritz Projection, and Their Properties
75(1)
2.1.8 Green's Formulas, the Cauchy--Schwarz Inequality, and the Holder Inequality
76(1)
2.2 POD-Based Reduced-Order Extrapolation Finite Element Algorithm for 2D Viscoelastic Wave Equation
77(23)
2.2.1 Generalized Solution for the 2D Viscoelastic Wave Equation
77(2)
2.2.2 Semidiscretized Formulation About Time for the 2D Viscoelastic Wave Equation
79(2)
2.2.3 Classical Fully Discretized Finite Element Method for the 2D Viscoelastic Wave Equation
81(4)
2.2.4 The POD Basis and the Reduced-Order Finite Element Algorithm for the 2D Viscoelastic Wave Equation
85(7)
2.2.5 Error Estimates of the Reduced-Order Solutions for the 2D Viscoelastic Wave Equation
92(4)
2.2.6 The Implementation of the Reduced-Order Algorithm for the 2D Viscoelastic Wave Equation
96(1)
2.2.7 A Numerical Example for the 2D Viscoelastic Wave Equation
97(3)
2.3 POD-Based Reduced-Order Extrapolation Finite Element Method for the Two-Dimensional Nonstationary Burgers Equation
100(25)
2.3.1 Generalized Solution for the 2D Nonstationary Burgers Equation
101(4)
2.3.2 Semidiscretized Formulation With Respect to Time for the 2D Nonstationary Burgers Equation
105(2)
2.3.3 Classical Fully Discretized Finite Element Method for the 2D Nonstationary Burgers Equation
107(3)
2.3.4 Formulating the POD Basis and Establishing a Reduced-Order Method for the 2D Nonstationary Burgers Equation
110(7)
2.3.5 Error Estimates of Reduced-Order Solutions for the 2D Nonstationary Burgers Equation
117(4)
2.3.6 Implementation of the Reduced-Order Method for the 2D Nonstationary Burgers Equation
121(1)
2.3.7 Numerical Examples for the 2D Nonstationary Burgers Equation
122(3)
2.4 POD-Based Reduced-Order Stabilized Crank--Nicolson Extrapolation Mixed Finite Element Formulation for Two-Dimensional Nonstationary Parabolized Navier--Stokes Equation
125(29)
2.4.1 Physical Background for the 2D Nonstationary Parabolized Navier--Stokes Equation
125(2)
2.4.2 Generalized Solution for the 2D Nonstationary Parabolized Navier--Stokes Equation
127(2)
2.4.3 Semidiscretized Formulation About Time for the 2D Nonstationary Parabolized Navier--Stokes Equation
129(4)
2.4.4 Classical Fully Discretized Stabilized Crank--Nicolson Mixed Finite Element Method for the 2D Nonstationary Parabolized Navier--Stokes Equation
133(5)
2.4.5 Formulating the POD Basis and Establishing the Reduced-Order Algorithm for the 2D Nonstationary Parabolized Navier--Stokes Equation
138(3)
2.4.6 Existence, Stability, and Error Estimates of the Reduced-Order Solutions for the 2D Nonstationary Parabolized Navier--Stokes Equation
141(5)
2.4.7 Implementation of the Reduced-Order Algorithm for the 2D Nonstationary Parabolized Navier--Stokes Equation
146(1)
2.4.8 Numerical Examples for the 2D Nonstationary Parabolized Navier--Stokes Equation
147(7)
2.5 Concluding Remarks on POD-Based Reduced-Order Extrapolation Finite Element Methods
154(4)
3 Reduced-Order Extrapolation Finite Volume Element Methods Based on Proper Orthogonal Decomposition
3.1 POD-Based Reduced-Order Extrapolating Finite Volume Element Algorithm for Two-Dimensional Hyperbolic Equation
158(27)
3.1.1 Classical Finite Volume Element Method for the 2D Hyperbolic Equation and Generation of Snapshots
158(13)
3.1.2 Formulating the POD Basis and Establishing the Reduced-Order Algorithm for the 2D Hyperbolic Equation
171(3)
3.1.3 Error Estimates of the Reduced-Order Solutions for the 2D Hyperbolic Equation
174(6)
3.1.4 Implementation of the Reduced-Order Algorithm for the 2D Hyperbolic Equation
180(2)
3.1.5 Numerical Experiments for the 2D Hyperbolic Equation
182(3)
3.2 POD-Based Reduced-Order Extrapolation Finite Volume Element Algorithm for the Two-Dimensional Sobolev Equation
185(20)
3.2.1 The Classical Finite Volume Method for the 2D Sobolev Equation
185(6)
3.2.2 Formulation of the POD Basis and the Reduced-Order Algorithm for the 2D Sobolev Equation
191(3)
3.2.3 Error Estimations of the Reduced-Order Solutions for the 2D Sobolev Equation
194(5)
3.2.4 The Implementation of the Reduced-Order Algorithm for the 2D Sobolev Equation
199(1)
3.2.5 Numerical Experiments
200(5)
3.3 POD-Based Reduced-Order Stabilized Crank--Nicolson Extrapolation Mixed Finite Volume Element Model for the Two-Dimensional Nonstationary Incompressible Boussinesq Equation
205(39)
3.3.1 Model Background and Survey for the 2D Nonstationary Incompressible Boussinesq Equation
205(3)
3.3.2 Semidiscretized Crank--Nicolson Formulation About Time for the 2D Nonstationary Incompressible Boussinesq Equation
208(8)
3.3.3 Fully Discretized Stabilized Crank--Nicolson Mixed Finite Volume Element Formulation for the 2D Nonstationary Incompressible Boussinesq Equation
216(2)
3.3.4 Existence, Stability, and Convergence of the Stabilized Crank--Nicolson Mixed Finite Volume Element Solutions for the 2D Nonstationary Incompressible Boussinesq Equation
218(9)
3.3.5 Formulations of POD Bases and the Reduced-Order Model for the 2D Nonstationary Incompressible Boussinesq Equation
227(4)
3.3.6 Existence, Uniqueness, Stability, and Convergence of the Reduced-Order Solutions for the 2D Nonstationary Incompressible Boussinesq Equation
231(7)
3.3.7 Algorithm Implementation of the Reduced-Order Model for the 2D Nonstationary Incompressible Boussinesq Equation
238(1)
3.3.8 Numerical Experiments for the 2D Nonstationary Incompressible Boussinesq Equation
239(5)
3.4 Concluding Remarks
244(3)
4 Epilogue and Outlook
Bibliography 247(10)
Index 257
Zhendong Luo is Professor of Mathematics at North China Electric Power University, Beijing, China. Luo is heavily involved in the areas of Optimizing Numerical Methods of PDEs; Finite Element Methods; Finite Difference Scheme; Finite Volume Element Methods; Spectral-Finite Methods; and Computational Fluid Dynamics. For the last 12 years, Luo has worked mainly on Reduced Order Numerical Methods based on Proper Orthogonal Decomposition Technique for Time Dependent Partial Differential Equations.