Numerical Methods for Shallow-Water Flow
A 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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12 Atmospheric flows
13 Tidal flows
16 Storm surges
18 Flows around structures
110 Coastal flows
113 Internal flows
114 Planetary flows
Effects of space discretization on wave propagation
72 Gravity waves
73 Vorticity waves
74 Flood waves
75 Rossby waves
76 Rotated grids
77 Irregular grids
78 Discrete conservation
22 Surface and bottom boundary conditions
24 Boundarylayer form
25 Twodimensional shallowwater equations
26 Driving forces
27 Bottom stress
28 Lateral momentum exchange
29 Forms of the shallowwater equations
210 Curvilinear coordinates
32 Correspondence with incompressible viscous flow
33 Conservation laws
Behaviour of solutions
42 Wave equation
44 Harmonic wave propagation
52 Energy arguments
53 Initial conditions
55 Moving boundaries
Discretization in space
62 Staggered grids
63 Curvilinear grids
64 Finite elements
65 Finite elements for wave equation
66 Grid generation
67 Spectral methods
Time integration methods
83 Implicit methods
84 Semiimplicit methods
85 ADI methods
86 Fractionalstep methods
87 Riemann solvers
Effects of time discretization on wave propagation
92 Gravity waves
93 Vorticity waves
94 Flood waves
95 Rossby waves
96 Amount of work
Numerical treatment of boundary conditions
102 Examples of boundary schemes
103 Stability analysis by the energy method
104 Normal mode analysis
105 Accuracy of boundary treatment
Threedimensional shallowwater flow
112 3d Model equations
113 Discretization in space
114 Discretization in time
List of notation
Other editions - View all
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
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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.
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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.