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Next: 2.10.3 C grid staggering
Up: 2.10 Spatial discretization of
Previous: 2.10.1 Notation
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The finite volume method is used to discretize the equations in
space. The expression ``finite volume'' actually has two meanings; one
is the method of embedded or intersecting boundaries (shaved or lopped
cells in our terminology) and the other is non-linear interpolation
methods that can deal with non-smooth solutions such as shocks
(i.e. flux limiters for advection). Both make use of the integral form
of the conservation laws to which the weak solution is a
solution on each finite volume of (sub-domain). The weak solution can
be constructed out of piece-wise constant elements or be
differentiable. The differentiable equations can not be satisfied by
piece-wise constant functions.
As an example, the 1-D constant coefficient advection-diffusion
equation:
can be discretized by integrating over finite sub-domains, i.e.
the lengths
:
is exact if is piece-wise constant over the interval
or more generally if is defined as the average
over the interval
.
The flux, , must be approximated:
and this is where truncation errors can enter the solution. The
method for obtaining
is unspecified and a wide
range of possibilities exist including centered and upwind
interpolation, polynomial fits based on the the volume average
definitions of quantities and non-linear interpolation such as
flux-limiters.
Choosing simple centered second-order interpolation and differencing
recovers the same ODE's resulting from finite differencing for the
interior of a fluid. Differences arise at boundaries where a boundary
is not positioned on a regular or smoothly varying grid. This method
is used to represent the topography using lopped cell, see
[5]. Subtle difference also appear in more than one
dimension away from boundaries. This happens because the each
direction is discretized independently in the finite difference method
while the integrating over finite volume implicitly treats all
directions simultaneously. Illustration of this is given in
[2].
Next: 2.10.3 C grid staggering
Up: 2.10 Spatial discretization of
Previous: 2.10.1 Notation
Contents
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