Shallow Water Hydrodynamics (eBook)

Front Cover 1
Shallow Water Hydrodynamics: Mathematical Theory and Numerical Solution for a Two-dimensional System of Shallow Water Equations 4
Copyright Page 5
Contents 14
Foreword 6
Preface 8
List of Symbols 10
Abbreviations 11
CHAPTER 1. BACKGROW IN MECHANICS 16
1.1 Physical objects 16
1.2 System of fluid-dynamics equations for homogeneous isotropic incompresaible flows 18
1.3 Two-dimensional system of shallow water equations (2-D SSWE) 32
1.4 Physical meanings of various terms in 2-D SSWE 41
1.5 Various forms of 2-D SSWE 53
CHAPTER 2. PROPERTIES OF 2-D SYSTEM OF SHALLOW-WATER EQUATIONS (2-D) 75
2.1 Conceptual Mechanical Behavior 75
2.2 Dimensional Analysis of 2-D SSWE 81
2.2 Basic Mathematics for Systems of First-Order Quasilinear Hyperbolic Equations 87
2.4 Geometric Theory of Characteristics 99
2.5 Riemann Invariants 108
2.6 Theory of Nonlinear Wave Propagation 115
CHAPTER 3. PROPERTIES OF THE SOLUTIONS OF 2-D SSWE 122
3.1 Initial And Boundary Conditions for Well-Posed Problems 122
3.2 Behavior of Solutions 136
CHAPTER 4. DISCONTINOUS SOLUTIONS OF SSWE 145
4.1 Isentropic-Flow Simulation of SSWE And Its Limitations 145
4.2 Discontinuous Solutions of I-D First-Order Hyperbolic Systems 150
4.3 Introductions to 2-D Discontinuous Solutions 165
4.4 Mathematical Conditions of Shock Waves for 2-D SSWE 170
CHAPTER 5. PRELIMINARY REVIEW OF FINITE DIFFERENCE METHODS 176
5.1 General Description 176
5.2 Basic Performance of A Difference Scheme 183
5.3 Basic Difference Schemes for First-Order Hyperbolic Systems In One Space Dimension 202
5.4 Fdms for the Computation of I-D Unsteady Open Flows 208
CHAPTER 6. DIFFERENCE SCHEMES FOR 2-D SSWE 221
6.1 Fdms for the Solutions of 2-D SSWE In Nonconservative Form 221
6.2 Fdms for the Solutions of 2-D SSWE In Conservative Form 230
6.3 Fractional-step methods and splitting-up algorithms 234
6.4 Fractional-step difference schemes for 2-D unsteady flow computations 238
6.5 FDMs for curvilinear meshes 249
6.6 Finite Volume Method (FVM) 256
CHAPTER 7. NUMERICAL SOLUTIONS USING FINITE ELEMENT METHODS 260
7.1 Related principles in variational calculus 260
7.2 Piecewise approximation of plane problems and convergence of FEM solutions 266
7.3 FEM for 2-D unsteady open flows 279
7.4 Several Classes of special FEWs 287
CHAPTER 8. TECHNIQUES FOR THE IMPLEMENTATION OF ALGORITHMS 298
8.1 Computational Mesh 298
8.2 Classical techniques for improving computational stability and accuracy 323
CHAPTER 9. NEW DEVELOPMENTS OF DIFFERENCE SCHEMES FOR 2-D FIRST-ORDER HYPERBOLIC SYSTEMS OF EQUATIONS 349
9.1 General description 349
9.2 Two.dimensioal methods of characteristics 357
9.3 Characteristic-baled splitting 362
9.4 Riemann Approach 375
9.5 Approximate Factorizatio of Implicit Schemes 381
9.6 Fct Algorithms and Tvd Schemes 384
9.7 Square Conservation Schemes 397
CHAPTER 10. STABILITY ANALYSIS AND BOUNDARY PROCEDURES 403
10.1 Mathematical Definitions of Stability for Difference Schemes 403
10.2 Von Neumann Linear Stability Analysis 405
10.3 Nonlinear Instability 410
10.4 Boundary Procedures And Their Influence on Numerical Solutions 415
10.5 Stability Theory for Mixed Problems 428
CHAPTER 11. CONCLUDING REMARKS 433
11.1 Requirements for An Ideal Finite-Difference Scheme 433
11.2 Comparison of Performance,Merits and Drawbacks Between FDM and FEM 434
11.3 Brief Introduction to Other Algorithms 435
11.4 Towards A Truly 2-D Algorithm 439
Index 446