Shock-capturing methods for free-surface shallow flows

The first of its kind in the field, this title examines the use of modern, shock-capturing finite volume numerical methods, in the solution of partial differential equations associated with free-surface flows, which satisfy the shallow-water type assumption (including shallow water flows, dense gases and mixtures of materials as special samples).
Starting with a general presentation of the governing equations for free-surface shallow flows and a discussion of their physical applicability, the book goes on to analyse the mathematical properties of the equations, in preparation for the presentation of the exact solution of the Riemann problem for wet and dry beds. After a general introduction to the finite volume approach, several chapters are then devoted to describing a variety of modern shock-capturing finite volume numerical methods, including Godunov methods of the upwind and centred type. Approximate Riemann solvers following various approaches are studied in detail as is their use in the Godunov approach for constructing low and high-order upwind TVD methods. Centred TVD schemes are also presented. Two chapters are then devoted to practical applications. The book finishes with an overview of potential practical applications of the methods studied, along with appropriate reference to sources of further information.
Features include:
* Algorithmic and practical presentation of the methods
* Practical applications such as dam-break modelling and the study of bore reflection patterns in two space dimensions
* Sample computer programs and accompanying numerical software (details available at www.numeritek.com)
The book is suitable for teaching postgraduate students of civil, mechanical, hydraulic and environmental engineering, meteorology, oceanography, fluid mechanics and applied mathematics. Selected portions of the material may also be useful in teaching final year undergraduate students in the above disciplines. The contents will also be of interest to research scientists and engineers in academia and research and consultancy laboratories.

Author Notes

Eleuterio F. Toro is the author of Shock-Capturing Methods for Free-Surface Shallow Flows, published by Wiley.

Preface	p. XIII
1. Introduction	p. 1
2. The Shallow Water Equations	p. 15
2.1 Introduction	p. 15
2.2 Conservation Principles	p. 16
2.3 Water Flow with a Free Surface	p. 18
2.4 The Shallow Water Equations	p. 21
2.5 The de St Venant Equations for River Flows	p. 24
2.6 Conclusions	p. 26
3. Properties of the Equations	p. 29
3.1 Introduction	p. 29
3.2 Recalling the Equations	p. 30
3.3 Eigenstructure in Terms of Conserved Variables	p. 31
3.4 Eigenstructure in Terms of Primitive Variables	p. 35
3.5 Hyperbolic Character of the Equations	p. 38
3.6 Nature of Characteristic Fields	p. 40
3.7 The Dam-Break Problem	p. 41
3.8 Elementary Wave Solutions of the Riemann Problem	p. 44
3.8.1 Wave Relations	p. 46
3.8.2 Rarefaction Waves	p. 47
3.8.3 A Right Rarefaction	p. 48
3.8.4 A Left Rarefaction	p. 50
3.8.5 Shear Waves	p. 51
3.8.6 Shock Waves	p. 52
3.8.7 Left Shock	p. 53
3.8.8 Right Shock	p. 56
3.9 Non-Conservative Formulations and Shock Waves	p. 61
3.10 Integral Forms and Rotational Invariance	p. 64
3.11 Steady Supercritical Flow	p. 66
3.12 Problems	p. 69
3.12.1 Flux Properties	p. 69
3.12.2 Analysis of the de St Venant Equations	p. 70
3.12.3 The Two-Dimensional Steady Supercritical Equations	p. 70
3.12.4 Equations with Variable Bed Elevation	p. 71
4. Linearised Shallow Water	p. 73
4.1 Introduction	p. 73
4.2 Linearised Models	p. 73
4.3 Eigenstructure and Characteristic Variables	p. 76
4.4 The General Initial Value Problem	p. 78
4.4.1 The General IVP for the Scalar Case	p. 78
4.4.2 The General IVP for the System Case	p. 80
4.5 The Riemann Problem	p. 83
4.5.1 The Scalar Case	p. 83
4.5.2 The System Case	p. 85
4.5.3 Example	p. 86
4.6 A Linear Model with Source Terms	p. 88
4.7 Problems	p. 90
4.7.1 Solution of the Riemann Problem	p. 90
4.7.2 The Linearisation	p. 90
4.7.3 Riemann Invariants	p. 90
4.7.4 Rankine-Hugoniot Conditions	p. 91
5. Exact Riemann Solver: Wet Bed	p. 93
5.1 Introduction	p. 93
5.2 The Riemann Problem and a Solution Strategy	p. 94
5.3 Solution for h* and u*	p. 96
5.4 Behaviour of the Depth Function f(h)	p. 98
5.5 Iterative Solution for h*	p. 101
5.6 The Complete Solution	p. 102
5.7 Transport of Pollutants and Passive Scalars	p. 104
5.8 Sampling the Solution	p. 106
5.8.1 Sampling Point is to the Left of the Shear	p. 106
5.8.2 Sampling Point is to the Right of the Shear	p. 107
5.9 Conclusions	p. 107
6. Exact Riemann Solver: Dry Bed	p. 109
6.1 Introduction	p. 109
6.2 Admissible Wet/Dry Interface Waves	p. 110
6.3 Three Possible Cases	p. 111
6.3.1 The Dry Bed is on the Right Side	p. 111
6.3.2 The Dry Bed is on the Left Side	p. 114
6.3.3 Generation of Dry Bed in the Middle	p. 115
6.4 Pollutant Transport and Passive Scalars	p. 115
7. Tests with Exact Solution	p. 119
7.1 Introduction	p. 119
7.2 Homogeneous Problems	p. 119
7.2.1 Test 1: Left Critical Rarefaction and Right Shock	p. 120
7.2.2 Test 2: Two Rarefactions and Nearly Dry Bed	p. 121
7.2.3 Test 3: Right Dry Bed Riemann Problem	p. 122
7.2.4 Test 4: Left Dry Bed Riemann Problem	p. 122
7.2.5 Test 5: Generation of a Dry Bed	p. 123
7.3 Test Problems with Constant Slope	p. 125
7.4 Fortran Program for the Exact Riemann Solver	p. 126
8. Basics on Numerical Methods	p. 141
8.1 Introduction	p. 141
8.2 Conservative Methods	p. 142
8.2.1 The One-Dimensional Case	p. 142
8.2.2 The Two-Dimensional Case	p. 144
8.3 Non-Conservative Methods	p. 147
8.4 Theoretical Issues	p. 149
9. First-Order Methods	p. 151
9.1 Introduction	p. 151
9.2 The Godunov Upwind Method	p. 152
9.2.1 Introduction	p. 152
9.2.2 The Scheme	p. 153
9.2.3 Godunov's Scheme for Linear Advection	p. 156
9.2.4 Stability Condition for Godunov's Method	p. 158
9.2.5 Boundary Conditions	p. 158
9.3 The Random Choice Method	p. 159
9.4 The Flux Vector Splitting Approach	p. 162
9.5 Centred Methods	p. 163
9.5.1 The Lax-Friedrichs Scheme	p. 163
9.5.2 The Force Scheme	p. 163
9.5.3 The Godunov Centred Scheme	p. 164
9.6 Numerical Results	p. 165
10. Approximate Riemann Solvers	p. 173
10.1 Introduction	p. 173
10.2 The Riemann Problem and the Godunov Flux	p. 174
10.3 Approximate-State Riemann Solvers	p. 175
10.3.1 A Primitive Variable Riemann Solver	p. 177
10.3.2 A Riemann Solver Based on Exact Depth Positivity	p. 178
10.3.3 A Two-Rarefaction Riemann Solver	p. 178
10.3.4 A Two-Shock Riemann Solver	p. 179
10.4 The HLL and HLLC Riemann Solvers	p. 179
10.4.1 The HLL Riemann Solver	p. 180
10.4.2 The HLLC Solver	p. 181
10.5 Relating Centred to Upwind Methods	p. 183
10.5.1 Rusanov-Type Schemes	p. 183
10.5.2 Upwinding in Centred Methods	p. 185
10.6 The Approximate Riemann Solver of Roe	p. 187
10.6.1 The Basic Scheme	p. 187
10.6.2 Entropy Fix for the Roe Solver	p. 189
10.7 The Riemann Solver of Osher and Solomon	p. 191
10.8 Wet/Dry Fronts	p. 194
10.8.1 Sources of Error	p. 194
10.8.2 Dry-Bed Approximate Riemann Solvers	p. 197
11. TVD Methods	p. 199
11.1 Introduction	p. 199
11.2 The Weighted Average Flux (WAF) Method	p. 200
11.2.1 The Basic WAF Scheme	p. 200
11.2.2 TVD Version of the WAF Scheme	p. 202
11.2.3 Critical Flow	p. 204
11.3 The MUSCL-Hancock Scheme	p. 205
11.3.1 Step I: Data reconstruction	p. 206
11.3.2 Step II: Evolution of extrapolated values	p. 207
11.3.3 Step III: The Riemann problem	p. 208
11.3.4 TVD Version of the MUSCL-Hancock Scheme	p. 208
11.3.5 Non-Conservative Upwind Methods	p. 210
11.4 Centred TVD Schemes: the SLIC Method	p. 211
11.5 Other Methods	p. 213
11.6 Numerical Results	p. 213
11.7 Conclusions	p. 216
12. Sources and Multi-Dimensions	p. 227
12.1 Introduction	p. 227
12.2 Treatment of Source Terms	p. 228
12.2.1 Model Problems with Source Terms	p. 228
12.2.2 Splitting for Non-linear Systems with Sources	p. 232
12.2.3 Relation between Upwinding and Splitting	p. 234
12.3 Two-Dimensional Problems	p. 236
12.3.1 Dimensional-Splitting Schemes	p. 236
12.3.2 Unsplit Finite Volume Schemes	p. 238
13. Dam-Break Modelling	p. 243
13.1 Introduction	p. 243
13.2 Idealised Circular Dam: Reference Solutions	p. 245
13.3 Physical Models: Experiments and Numerics	p. 249
13.3.1 Introduction	p. 249
13.3.2 Dam with Channel with 45[degree] Bend	p. 250
13.3.3 Comparison of Numerical and Experimental Results	p. 251
13.4 Conclusions	p. 253
14. Mach Reflection of Bores	p. 269
14.1 Introduction	p. 269
14.2 The Problem	p. 271
14.3 Analytical Study	p. 273
14.3.1 Oblique Bore Relations	p. 274
14.3.2 Regular Reflection	p. 276
14.3.3 Transition from Regular to Mach Reflection	p. 277
14.4 Numerical Computations	p. 278
14.5 Closing Remarks	p. 280
15. Concluding Remarks	p. 287
References	p. 291
Index	p. 305

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Table of Contents