Skip to main content
main-content

Über dieses Buch

1. 1 AREAS OF APPLICATION FOR THE SHALLOW WATER EQUATIONS The shallow water equations describe conservation of mass and mo­ mentum in a fluid. They may be expressed in the primitive equation form Continuity Equation _ a, + V. (Hv) = 0 L(l;,v;h) at (1. 1) Non-Conservative Momentum Equations a M("vjt,f,g,h,A) = at(v) + (v. V)v + tv - fkxv + gV, - AIH = 0 (1. 2) 2 where is elevation above a datum (L) ~ h is bathymetry (L) H = h + C is total fluid depth (L) v is vertically averaged fluid velocity in eastward direction (x) and northward direction (y) (LIT) t is the non-linear friction coefficient (liT) f is the Coriolis parameter (liT) is acceleration due to gravity (L/T2) g A is atmospheric (wind) forcing in eastward direction (x) and northward direction (y) (L2/T2) v is the gradient operator (IlL) k is a unit vector in the vertical direction (1) x is positive eastward (L) is positive northward (L) Y t is time (T) These Non-Conservative Momentum Equations may be compared to the Conservative Momentum Equations (2. 4). The latter originate directly from a vertical integration of a momentum balance over a fluid ele­ ment. The former are obtained indirectly, through subtraction of the continuity equation from the latter. Equations (1. 1) and (1. 2) are valid under the following assumptions: 1. The fluid is well-mixed vertically with a hydrostatic pressure gradient. 2. The density of the fluid is constant.

Inhaltsverzeichnis

Frontmatter

Chapter I. Introduction

Abstract
The shallow water equations describe conversation of mass and momentum in a fluid.
Ingemar Kinnmark

Chapter II. Equation Formulation

Abstract
We are interested in the vertically averaged form of the mass conservation (or continuity) equation and the horizontal components of the vertically averaged non-conservative momentum equations.
Ingemar Kinnmark

Chapter III. Fourier Analysis Methods

Abstract
Fourier analysis is based on the idea that a function defined on a finite interval can be expanded in a series of cosine or sine functions. In the context of numerical simulation of the shallow water equations, Fourier analysis is used to study the ability of various numerical schemes to accurately simulate waves of different wave lengths. Fourier analysis is typically applied to the linearized form of the shallow water equations (2.37) to (2.40). Linearization uncouples the different wave lengths from one another and allows the study of only one wave length at a time. This method provides stability criteria and an accuracy measure for the amplitude and phase characteristics of each wave length. The technique to obtain the two latter is outlined in Section 3.2. These amplitude and phase portraits have also been used to predict the ability of a scheme to suppress node-to-node oscillations. This is more thoroughly discussed in Chapter 7. The method has mostly been used on uniform one-dimensional meshes, although some applications to uniform, orthogonal two-dimensional meshes exist (Sobey, 1970; Kinnmark and Gray, 1984a). The method is however more general. It can be applied to the case of meshes with variable node-spacing, especially useful for finite element approximations, and for the case of variable parameters.
Ingemar Kinnmark

Chapter IV. Stability

Abstract
Why are we interested in knowing that a numerical approximation of a partial differential equation is stable? It is clearly of interest to know whether the approximate solution is close to the exact solution. One way of formulating this is to require that the approximate solution converge to the exact solution in the limit of small time steps and node spacings. In this case there is however a theorem stating the equivalence between convergence and stability for consistent schemes (see i.e. Richtmyer and Morton, 1967).
Ingemar Kinnmark

Chapter V. Explicit Methods using Various Spatial Discretizations

Abstract
Stability and accuracy properties of finite element schemes for the shallow water equations have been investigated earlier (Gray and Lynch, 1977) for different time marching schemes. This chapter concentrates on two time marching procedures, the primitive equation leapfrog approximation and the wave equation formulation, in conjunction with various spatial discretizations. Consistent and lumped, linear and quadratic isoparametric Lagrangian finite elements as well as second and fourth order finite differences are studied on a uniform mesh with constant bathymetry in Section 5.2. It is shown that quadratic Lagrangian finite elements require the use of a smaller time step than linear elements to remain stable (Kinnmark and Gray, 1984c). This effect is more severe for the leapfrog method than for the wave equation formulation. For the two-dimensional cases considered, in Section 5.5, with equal node spacing, constant bathymetry and the Coriolis force neglected, the stability constraint is identical in form to the one-dimensional case with the square of the Courant number replaced by the sums of squares of the Courant number in x- and y-directions.
Ingemar Kinnmark

Chapter VI. Implicit Methods

Abstract
In the previous chapter we studied explicit time marching procedures, i.e. methods where the unknown quantity at the new time level is computed using only quantities at the known preceeding time level. These methods offer efficient computation of each time step but usually stability constraints, rather than accuracy considerations, limit the size of the time step and therefore lead to costly simulations. In the present chapter we will investigate implicit time marching procedures. This class of methods allows the value of an unknown quantity at the new time level to depend also on values of other unknown quantities at the new time level. This eliminates or strongly reduces the restriction on the time step by the stability constraint. The computational effort per time step is however in general increased due to the following factors:
  • --- A matrix equation is generated which requires the simultaneous solution of elevation and both horizontal velocity components in Equations (2.19), (2.2) and (2.3).
  • --- The matrix of the equation system has to be reformulated at every time step.
  • --- The matrix has to be decomposed into a lower and upper triangular matrix at every time step.
  • --- The two resulting triangular equation systems have to be solved using forward solution and back substitution at every time step.
Ingemar Kinnmark

Chapter VII. Spatial Oscillations

Abstract
There is an upper limit to the frequency of a wave which can be represented on a discrete grid using a numerical approximation of a partial differential equation. If a higher frequency wave is utilized it will be aliased into (i.e. be indistinguishable from) a wave of a lower frequency on the given mesh. It is important to realize that this limiting maximum frequency, or equivalently limiting minimum wave length, stems from two inherently different effects.
Ingemar Kinnmark

Chapter VIII. Temporal Oscillations

Abstract
The previous chapter shows that the wave continuity equation suppresses node-to-node oscillations in space. For a centered three time level approximation of the momentum equations there remain, however, node-to-node oscillations in time for the velocity solution (Kinnmark and Gray, 1982). It is shown in Section 8.2 that three time level momentum equations introduce an additional non-physical root, a numerical artifact. By using different three time level approximations of the momentum equations, the magnitude of this numerical artifact can however be made smaller, as shown in Section 8.3. Except for the nonlinear convective term we do however preserve second order accuracy in time. Finally it is shown, in Section 8.4, that a two time level, Crank-Nicolson type approximation of the momentum equations completely eliminates the numerical artifact. We still maintain second order accuracy in time, except for the non-linear convective terms. If lumping, through integration, is applied to the momentum equations, velocities are computed from a simple block diagonal matrix equation.
Ingemar Kinnmark

Chapter IX. Applications

Abstract
In this chapter we will apply the shallow water models developed in earlier chapters. In Section 9.2 we will investigate the generalized wave continuity equation developed in Section 6.5. The test case, Quarter Circle Harbor, has a known analytic solution. It has been used successfully in comparing different numerical solution procedures (Lynch and Gray, 1979; Walters, 1983a). We will assess the effect of varying the numerical parameter G in the generalized wave continuity equation.
Ingemar Kinnmark

Chapter X. Conclusions

Abstract
The steps involved in obtaining the wave continuity equation from the primitive continuity and momentum equations were brought out more clearly using operator notation. A potential formulation (Platzman, 1978), cast in operator form, and the wave continuity equation were both shown to contain second order spatial derivatives originating from a spatial derivative of the conservative momentum equation. It was shown that the operator form naturally leads to the formulation of a more general wave continuity equation.
Ingemar Kinnmark

Backmatter

Weitere Informationen