Computational Algorithms for Shallow Water Equations

Preface
	This Book
	Genesis of This Book
	Contents of This Book
	Who Will Benefit From This Book
	Acknowledgements
	Reference
Contents
1 The Shallow Water Equations
	1.1 Introduction
	1.2 Conservation Principles
	1.3 Water Flow with a Free Surface
	1.4 The Shallow Water Equations
	1.5 The Saint Venant Equations for River Flows
	1.6 Conclusions
	References
2 Notions on Hyperbolic Equations
	2.1 The Linear Advection Equation and Basic Concepts
		2.1.1 The Initial-Value Problem
		2.1.2 The Riemann Problem
	2.2 Linear Hyperbolic Systems
		2.2.1 Eigenstructure and Hyperbolicity
		2.2.2 Diagonalization and Characteristic Variables
		2.2.3 The Riemann Problem
	2.3 Non-linear Scalar Equations
		2.3.1 Definitions and Examples
		2.3.2 Solution Along Characteristics
		2.3.3 Integral Forms of the Equation
		2.3.4 Generalised Solutions
		2.3.5 Non-uniqueness
		2.3.6 The Riemann Problem for Burgers\'s Equation
	2.4 First-Order Non-linear Systems
	References
3 Linear Shallow Water Equations
	3.1 Linearised Models
	3.2 Eigenstructure and Characteristic Variables
	3.3 The General Initial-Value Problem
		3.3.1 Recalling the General IVP for the Scalar Case
		3.3.2 The General IVP for the System Case
	3.4 The Riemann Problem
		3.4.1 Recalling the Scalar Case
		3.4.2 The System Case
		3.4.3 Example
	3.5 A Linear Model with Source Terms
	3.6 Case Study: Alternative Linearisation
		3.6.1 The Linear Equations
		3.6.2 The Eigenstructure
		3.6.3 Equations in Characteristic Variables
		3.6.4 The General Initial-Value Problem
		3.6.5 The Riemann Problem
	References
4 Properties of the Nonlinear Equations
	4.1 Recalling the Equations
	4.2 Eigenstructure in Terms of Conserved Variables
	4.3 Eigenstructure in Terms of Primitive Variables
	4.4 Hyperbolic Character of the 2D Equations
	4.5 Nature of Characteristic Fields
	4.6 Integral Forms and Rotational Invariance
	4.7 Steady Supercritical Flow
	4.8 Concluding Remarks
	4.9 Suggested Exercises
	References
5 Elementary Waves in Shallow Water
	5.1 Elementary Waves in the Riemann Problem
		5.1.1 Motivation: The Dam-Break Problem
		5.1.2 The Riemann Problem and Wave Patterns
		5.1.3 Relations Across the Wave Structure
	5.2 Single Rarefaction Wave
		5.2.1 Left Rarefaction Wave
		5.2.2 Right Rarefaction Wave
	5.3 Single Shock Wave
		5.3.1 Right-Facing Shock Wave
		5.3.2 Left-Facing Shock Wave
	5.4 Contact Discontinuity and Shear Wave
	5.5 The Full Wave System
	5.6 Useful Shock Relations
		5.6.1 Left Shock
		5.6.2 Right Shock
	5.7 Non-conservative Formulation and Shocks
		5.7.1 Conservative System in Non-conserved Variables
		5.7.2 Shock Waves
	5.8 Concluding Remarks
	5.9 Suggested Exercises
	References
6 Exact Riemann Solver: Wet Bed
	6.1 Introduction
	6.2 The Riemann Problem and Solution Strategy
	6.3 Solution in the Star Region
		6.3.1 Non-linear Equation for Water Depth h *
		6.3.2 Analysis of the Depth Function f (h)
		6.3.3 Iterative Solution for h Subscript asteriskh*
	6.4 Sampling the Complete Solution
		6.4.1 Passive Scalars
		6.4.2 Left of Contact/shear: S = x/t∗ ≤ u∗
		6.4.3 Right of Contact/shear: S = x/t∗ ≥ u∗
	6.5 Conclusions
	6.6 Exercises
	References
7 Exact Riemann Solver: Dry Bed
	7.1 Introduction
	7.2 Admissible Wet/Dry Interface Waves
	7.3 Dry Bed: Three Possible Cases
		7.3.1 The Dry Bed Is on the Right Side
		7.3.2 The Dry Bed Is on the Left Side
		7.3.3 Generation of Vacuum from Wet-Bed States
	7.4 Passive Scalars
	7.5 Conclusions
	References
8 Tests with Exact Solution
	8.1 Introduction
	8.2 Test 1: Left Critical Rarefaction and Right Shock
	8.3 Test 2: Two Rarefactions and Nearly Dry Bed
	8.4 Test 3: Right Dry-Bed Riemann Problem
	8.5 Test 4: Left Dry-Bed Riemann Problem
	8.6 Test 5: Generation of a Dry-Bed Region
	8.7 Test Problems with Constant Slope
	8.8 Closing Remarks
	8.9 Computer Program for the Exact Riemann Solver
	References
9 Notions on Numerical Methods
	9.1 Numerical Approximation of Hyperbolic Equations
		9.1.1 Finite Difference Approximation to PDEs
		9.1.2 Well-Known Finite Difference Methods
	9.2 Basic Properties of Numerical Methods
		9.2.1 Forms of Expressing a Numerical Scheme
		9.2.2 Monotonicity, Accuracy and Godunov\'s Theorem
		9.2.3 Viscous Form of a Scheme
		9.2.4 Conservative Form of a Scheme
	9.3 Computational Results
		9.3.1 Test Problems, Methods and Parameters
		9.3.2 Results and Discussion
	9.4 Conclusions and Further Reading
	References
10 First-Order Methods for Systems
	10.1 The Finite Volume Framework
		10.1.1 Balance Laws in Integral Form
		10.1.2 The Finite Volume Formula
	10.2 The Godunov Upwind Method
		10.2.1 The Numerical Flux from the Riemann Problem
		10.2.2 Godunov\'s Method for the Linear Advection Equation
		10.2.3 Godunov\'s Method and the Source Term
		10.2.4 Godunov\'s Method for Shallow Water
	10.3 Initial, Boundary and Stability Conditions
		10.3.1 Initial Conditions
		10.3.2 Boundary Conditions
		10.3.3 Stability Condition
	10.4 The Random Choice Method
	10.5 Alternative Conservative Schemes
		10.5.1 The Flux Vector Splitting Approach
		10.5.2 Centred Methods
	10.6 Finite Volume Schemes in Multidimensions
		10.6.1 Unstructured Meshes
		10.6.2 The Numerical Flux
		10.6.3 The Cartesian Case
		10.6.4 The Telescopic Property
	10.7 Numerical Results
	10.8 Conclusions
	References
11 Approximate Riemann Solvers
	11.1 Recalling the Godunov Upwind Method
	11.2 Approximate-State Riemann Solvers
		11.2.1 The Framework
		11.2.2 A Primitive Variable Riemann Solver
		11.2.3 Riemann Solver Based on Exact Depth Positivity
		11.2.4 A Two-Rarefaction Riemann Solver
		11.2.5 A Two-Shock Riemann Solver
	11.3 HLL Riemann Solvers
	11.4 HLLC Riemann Solvers
	11.5 The Rusanov and Lax-Friedrichs Schemes
	11.6 Roe\'s Approximate Riemann Solver
		11.6.1 The Basic Scheme
		11.6.2 Entropy Fix for the Roe Solver
	11.7 The Riemann Solver of Osher and Solomon
	11.8 The Dumbser-Osher-Toro Riemann Solver: DOT
		11.8.1 Notation
		11.8.2 The DOT Numerical Flux
	11.9 Path-Conservative Methods
		11.9.1 Non-conservative Methods
		11.9.2 The Framework
		11.9.3 DOT Path-Conservative Scheme
		11.9.4 FORCE-alphaα Path-Conservative Scheme
		11.9.5 Choosing alphaα: Accuracy Versus Size of Time Step
		11.9.6 FORCE-alphaα in DG Finite Element Methods
	11.10 Computation of Wet/Dry Fronts
		11.10.1 Artificial Bed Wetting
		11.10.2 Conservative-Form Induced Errors
		11.10.3 Dry-Bed Approximate Riemann Solvers
	11.11 Concluding Remarks
	References
12 Second-Order Non-linear Methods
	12.1 Introduction
	12.2 The Weighted Average Flux (WAF) Method
		12.2.1 The Basic WAF Scheme
		12.2.2 TVD Version of the WAF Scheme
		12.2.3 Handling Critical Flow
	12.3 The MUSCL-Hancock Scheme
		12.3.1 The Basic Linear Scheme
		12.3.2 TVD Version of the MUSCL-Hancock Scheme
		12.3.3 ENO Version of the MUSCL-Hancock Scheme
	12.4 FORCE-Based TVD Schemes: The SLIC Method
	12.5 Numerical Results
	12.6 Conclusions
	References
13 Sources and Multidimensions
	13.1 Introductory Remarks
	13.2 Treatment of Source Terms by Splitting
		13.2.1 Preliminary Notions
		13.2.2 Splitting for Systems with Source Terms
		13.2.3 Upwinding and Advection-Reaction Splitting
	13.3 Solvers for Ordinary Differential Equations
		13.3.1 First-Order Systems of ODEs
		13.3.2 Conventional Numerical Methods for ODEs
		13.3.3 TVD Runge–Kutta Schemes for ODEs
		13.3.4 A Note on Stability
	13.4 Multidimensional Systems of PDEs
		13.4.1 Dimensional-Split Schemes
		13.4.2 Unsplit Finite Volume Schemes
	13.5 Stability of Multi-dimensional Schemes
		13.5.1 Von Neumann Linear Stability Analysis
		13.5.2 Examples of Stable Schemes in 2D/3D
	13.6 Unsplit 2D/3D Second-Order WAF Schemes
		13.6.1 Construction of Schemes in 2D/3D
		13.6.2 Stability of the Schemes in 2D/3D
		13.6.3 Extensions of WAF Framework
	13.7 Concluding Remarks
	References
14 ADER High-Order Methods
	14.1 Introduction
	14.2 ADER in the Finite Volume Framework
		14.2.1 Preliminaries
		14.2.2 The ADER Approach to High Order
	14.3 The Toro-Titarev Solver for GRPm
		14.3.1 Flux Leading Term
		14.3.2 Flux Higher-Order Terms
		14.3.3 The Numerical Source
	14.4 The HEOC Solver for GRPm
		14.4.1 Time-Evolution Step for the Flux
		14.4.2 Data Interaction Step for the Flux
		14.4.3 The Numerical Source
		14.4.4 Variations of the HEOC Solver
	14.5 The Montecinos-Toro Solver for GRPm
	14.6 Supplementary Topics
		14.6.1 Other Solvers forGRPm
		14.6.2 A Note on Spatial Reconstruction
		14.6.3 ADER in the DG Framework
	14.7 Examples: Second-Order ADER Schemes
		14.7.1 ADER2 with TT Solver for GRP1
		14.7.2 ADER2 with HEOC Solver for upper G R P 1GRP1
	14.8 ADER Applied to Shallow Water Flows
	14.9 High Accuracy is Efficiency
		14.9.1 Efficiency for the Linear Advection Equation
		14.9.2 Efficiency for the Euler Equations
	14.10 Concluding Remarks and Further Reading
	References
15 DAM-BREAK Modelling
	15.1 Introduction
	15.2 Circular Dam: Computation of Wave Phenomena
		15.2.1 Geometry, Equations and Methods
		15.2.2 Computational Results
	15.3 Physical Models: Experiments and Numerics
		15.3.1 Introduction
		15.3.2 The Problem: A Dam with Channel Bend at 45°
		15.3.3 Numerical Methods and Computational Geometry
		15.3.4 Comparison of Numerical Results with Experiments
	15.4 Conclusions
	References
16 Mach Reflection in Tsunamis
	16.1 Introduction
	16.2 The Problem
	16.3 Analytical Study
		16.3.1 Oblique Bore Relations
		16.3.2 Regular Reflection
		16.3.3 Transition from Regular to Mach Reflection
	16.4 Numerical Study
	16.5 Closing Remarks
	References
17 Concluding Remarks
	References
Index