Environmental Engineering Reference
In-Depth Information
to a flux out of a neighbouring one through the
same boundary). In 1D Finite Volume methods
are equivalent to Finite Difference methods.
Finite Volume methods are increasingly popular
and have become the most widely used method in
the area of shallow water flow modelling (see,
e.g., Sleigh et al. 1998; LeVeque, 2002; Caleffi
et al. 2003; Alcrudo and Mulet 2005; Danish
Hydraulic Institute 2007b; Kramer and Stelling
2008). This is explained by their advantages in
terms of conservativeness, geometric flexibility
and conceptual simplicity (Alcrudo 2004).
their popularity is currently in decay in the aca-
demic community (Alcrudo 2004), as unstruc-
tured grids
lend themselves better
to the
modelling of environmental flows.
Finite elementmethods In Finite Elementmeth-
ods, the solution space in divided into a number of
2D elements. In each element, the unknown vari-
ables are approximated by a linear combination of
piecewise linear functions called trial functions.
There are as many such functions as there are
vertices defining the element, and each takes the
value 1 at one vertex and the value 0 at all other
vertices. A global function based on this approx-
imation is substituted into the governing partial
differential equations. This equation is then inte-
grated with weighting functions and the resulting
error is minimized to give coefficients for the trial
functions that represent an approximate solution
(Wright 2005). A number of methods to do this
exist, including the Galerkin method (see, e.g.,
Ottosen and Petersson 1992). Finite Element
methods benefit from a rigorous mathematical
foundation (Alcrudo 2004), which allows a better
understanding of their accuracy (Hervouet 2007);
however, the technique has not beenused asmuch
as other approaches in commercial software, per-
haps because it is less accessible conceptually and
produces models that result in large run times.
Also, generating meshes can be time consuming
when a suitable mesh generation tool is not avail-
able (Sauvaget et al. 2000).
Properties of numerical schemes When applying
a numerical model from any of the classes above,
the local accuracy of the approximation made is
controlled by the grid resolution and by the mag-
nitude of local gradients in the flow process.
Grid refinement is usually the most obvious way
to improve the accuracy of a numerical model. A
convergent solution is defined as a solution that
becomes independent of grid resolution as the grid
resolution is increased (Wright 2005).
An important consideration in numerical
modelling is the approach used to proceed through
the calculation in time. The solution is normally
obtained in time-step increments. However, nu-
merical schemes can be divided in two major
categories depending on the approach used to dis-
cretize the shallowwater equations through time.
In implicit schemes the discretization approach
applied to the space gradients involves values at
both the previous time step (n) and the new time
step (n
Finite volume methods In the Finite Volume
method, space is divided into so-called finite
volumes, which are 2D regions of any geometric
shape. The shallow water equations (in conser-
vative form) are integrated over each control vol-
ume to yield equations in terms of fluxes through
the control volume boundaries. Flux values
across a given boundary (calculated using inter-
polated variables) are used for both control
volumes separated by the boundary, resulting in
the theoretically perfect mass and momentum
conservativeness of the approach (a flux into a
finite volume through a boundary is always equal
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1). In explicit schemes it involves values
from the previous time step only. Implicit
schemes are of greater theoretical accuracy. How-
ever, the approach also implies that at each new
step the solution cannot proceed through the
computational grid one node (or finite volume) at
a time, and that a system of algebraic equations
covering the entire computational domain must
be solved. Explicit schemes (which represent the
vast majority of newly developed schemes) are
simpler to implement. However, they are subject
to some form of time step limitation (for stability)
analogous
to
the
Courant-Friedrichs-Lewy
