Graphics Reference
In-Depth Information
Of course, for a moving solid wall, this should instead be (
u
solid
,w
solid
)
n
.
To maintain that velocity in the normal direction, following the velocity
equations (12.2), we also need
∂h
·
∂x
,
∂h
·
n
=0
.
∂z
This also applies at an inflow/outflow boundary, where we pump water
in or out of the simulation. The utility of such a boundary may be en-
hanced by adding a source term to the height equation, directly adding
(or subtracting) water in some regions; such source terms are also perfect
for modeling vertical sinks or sources of water (such as a drop falling from
above, perhaps in a particle system, or a drainage hole).
It's much more dicult dealing with the edge of the simulation domain,
if it's assumed that the water continues on past the edge. If you expect
all the waves in the system to travel parallel to the edge, it's perfectly
reasonable to put an invisible solid wall boundary there. If you determine
waves should be entering along one edge, perhaps from a simple sinusoid
model (see the next section for how to choose such a wave), you can further
specify normal velocity and height. However, if you also expect waves
to leave through the edge, things are much, much trickier: solid walls,
even if invisible or specifying fancy normal velocities and heights, reflect
incoming waves. Determining a
non-reflecting
(or
absorbing
) boundary
condition is not at all simple and continues as a subject of research in
numerical methods. The usual approach taken is to gradually blend away
the simulated velocities and heights with a background field (such as a basic
sinusoid wave, or flat water at rest), over the course of many grid cells: if
the blend is smooth and gradual enough, reflections should be minimal.
Finally one boundary condition of prime importance for many shallow
water simulations is at the
moving contact line
: where the depth of the
water drops to zero, such as where the water ends on a beach. In fact,
no boundary conditions need to be applied in this case: if desired for a
numerical method, the velocity can be extrapolated to the dry land as
usual, and the depth is zero (
h
=
b
).
12.2 The Wave Equation
Before jumping to numerical methods for solving the shallow water equa-
tions, it's worth taking a quick look at a further simplification. For very
