## Numerical Methods for Shallow-Water FlowA wide variety of problems are associated with the flow of shallow water, such as atmospheric flows, tides, storm surges, river and coastal flows, lake flows, tsunamis. Numerical simulation is an effective tool in solving them and a great variety of numerical methods are available. The first part of the book summarizes the basic physics of shallow-water flow needed to use numerical methods under various conditions. The second part gives an overview of possible numerical methods, together with their stability and accuracy properties as well as with an assessment of their performance under various conditions. This enables the reader to select a method for particular applications. Correct treatment of boundary conditions (often neglected) is emphasized. The major part of the book is about two-dimensional shallow-water equations but a discussion of the 3-D form is included. The book is intended for researchers and users of shallow-water models in oceanographic and meteorological institutes, hydraulic engineering and consulting. It also provides a major source of information for applied and numerical mathematicians. |

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### Contents

Shallowwater flows | 1 |

12 Atmospheric flows | 2 |

13 Tidal flows | 3 |

16 Storm surges | 6 |

18 Flows around structures | 8 |

110 Coastal flows | 10 |

113 Internal flows | 12 |

114 Planetary flows | 13 |

Effects of space discretization on wave propagation | 117 |

72 Gravity waves | 120 |

73 Vorticity waves | 123 |

74 Flood waves | 126 |

75 Rossby waves | 129 |

76 Rotated grids | 132 |

77 Irregular grids | 133 |

78 Discrete conservation | 134 |

Equations | 15 |

22 Surface and bottom boundary conditions | 18 |

23 Scales | 19 |

24 Boundarylayer form | 21 |

25 Twodimensional shallowwater equations | 22 |

26 Driving forces | 26 |

27 Bottom stress | 29 |

28 Lateral momentum exchange | 36 |

29 Forms of the shallowwater equations | 38 |

210 Curvilinear coordinates | 41 |

Some properties | 47 |

32 Correspondence with incompressible viscous flow | 48 |

33 Conservation laws | 49 |

34 Discontinuities | 53 |

Behaviour of solutions | 56 |

42 Wave equation | 59 |

44 Harmonic wave propagation | 63 |

Boundary conditions | 73 |

52 Energy arguments | 75 |

53 Initial conditions | 81 |

54 Reflection | 82 |

55 Moving boundaries | 88 |

Discretization in space | 89 |

62 Staggered grids | 90 |

63 Curvilinear grids | 95 |

64 Finite elements | 100 |

65 Finite elements for wave equation | 105 |

66 Grid generation | 106 |

67 Spectral methods | 112 |

Time integration methods | 139 |

83 Implicit methods | 142 |

84 Semiimplicit methods | 143 |

85 ADI methods | 149 |

86 Fractionalstep methods | 152 |

87 Riemann solvers | 153 |

Effects of time discretization on wave propagation | 168 |

92 Gravity waves | 171 |

93 Vorticity waves | 175 |

94 Flood waves | 177 |

95 Rossby waves | 179 |

96 Amount of work | 184 |

97 Evaluation | 185 |

Numerical treatment of boundary conditions | 196 |

102 Examples of boundary schemes | 197 |

103 Stability analysis by the energy method | 199 |

104 Normal mode analysis | 203 |

105 Accuracy of boundary treatment | 210 |

Threedimensional shallowwater flow | 217 |

112 3d Model equations | 218 |

113 Discretization in space | 226 |

114 Discretization in time | 230 |

115 Advectiondiffusion | 233 |

116 Accuracy | 238 |

List of notation | 247 |

249 | |

259 | |

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### Common terms and phrases

accuracy ADI method advective terms amplitude analysis applied approximation assumed bottom friction bottom stress boundary conditions coefficient computed conservation constant continuity equation coordinate Coriolis parameter Coriolis terms corresponding Courant number Crank-Nicolson method density depth depth-averaged derivatives direction discretization discretization error discussed in section dt dx dy eddy viscosity eigenvalues eigenvectors energy enstrophy example finite-difference finite-element first-order flood waves fluxes friction terms functions gives gravity waves grid points group velocity horizontal integration layer leap-frog leap-frog method linear matrix momentum equations Navier-Stokes equations nonlinear normal numerical methods obtained phase potential vorticity pressure gradient problem region roots Rossby waves second-order semi-discrete semi-implicit shallow-water equations shallow-water flow shown in fig similar solution solved spatial specified spectral method spectral space stability step surface transformation turbulent values variables vector velocity components vorticity wave Vreugdenhil water level wave damping wave length wave number wave propagation wave speed wind stress

### Popular passages

Page 258 - ZIMMERMAN, JTF 1978 Topographic generation of residual circulation by oscillatory (tidal) currents. Geophys. Astrophys. Fluid Dyn. 11, 35-47. ZIMMERMAN, JTF 1979 On the Euler-Lagrange transformation and the Stokes drift in the presence of oscillatory and residual currents.

Page 249 - A High-Resolution Godunov-Type Scheme in Finite Volumes for the 2D Shallow-Water Equations...

Page 252 - Forward— backward scheme modified to prevent two-grid-interval noise and its application in sigma coordinate models. Contrib. Atmos. Phys. 52, 69-84. Janjic, ZI, 1984. Non-linear advection schemes and energy cascade on semi-staggered grids.

Page 257 - Vichnevetsky, R. 1987 - Wave propagation analysis of difference schemes for hyperbolic equations: a review, Int. J.

Page 253 - ... zur naherungsweisen Integration totaler Differentialgleichungen. Z. Math. Phys. 46, 435-453. 27. JD LAMBERT (1973). Computational Methods in Ordinary Differential Equations, John Wiley & Sons, London. 28. P. LANCASTER (1969). Theory of Matrices, Academic Press, New York and London. 29. JJ LEENDERTSE (1967). Aspects of a Computational Model for Longperiod Water-wave Propagation, Rand Corp., Mem. RM-5294, Santa Monica. 30. JJ LEENDERTSE (1970). A water-quality simulation model for wellmixed estuaries...

Page 255 - The effect of spatial discretization on the steady-state and transient solutions of a dispersive wave equation.

Page 258 - L. van Stijn, GS Stelling, and GA Fokkema, 1988: A fully implicit splitting method for accurate tidal computations.

Page 256 - Press. Taylor, GI, 1919. Tidal friction in the Irish Sea. Phil. Trans.

Page 255 - Reid, RO, 1957: Modification of the quadratic bottom-stress law for turbulent channel flow in the presence of surface wind-stress. US Army Corps of Engineers, Beach Erosion Board, Tech. Memo.