Two most basic steps are required in ensuring that a numerical model
works for predicting evolution and inundation. While the first step
is ensuring that the model conserves mass, the second basic step is
checking convergence of the numerical code to a certain asymptotic
limit.
The conservation of mass equation is part of the equations of motion
that are solved in any numerical procedure, but cumulative numerical
approximations can sometimes produce results that violate mass
conservation. This is particularly the case when friction factors
are used, or smoothing to stabilize inundation computations for
breaking waves.
Conservation of mass can be checked by calculating the water volume
at the beginning and at the end of the computation. This should be
done by integrating disturbed water depth over the
entire flow domain, i.e., if the flow domain extends from the
maximum penetration during inundation to the outer
location of the source region , and to ,
then total displaced volume is

(1) 
The integral of should be used instead of the
integral of the entire flow depth
where is the undisturbed water
depthbecause the latter is likely to conceal errors in the
calculation. Typically, at offshore integrating will simply produce the entire volume of the flow domain and will
mask errors. Note that testing of the conservation of mass as above
involves placing a closed domain within reflective boundaries.
Numerical models use absorbing boundaries to allow the wave to
radiate outwards. While some loss of mass may in theory occur due to
the differencing at the boundary, placing the computational
boundaries far enough from the source ensures this loss is minimal.
Nonetheless, testing conservation of mass with open boundaries and
monitoring the volume fluxes over the corresponding boundaries at and is computationally possible. The net flux
needs to be added or subtracted to the total displaced volume.
The calculation of the entire volume of the wave motion of the flow
domain at the beginning and end of the computationto ensure that
mass is conservedis very important in code validation. Once the
associated mass fluxes at the boundaries are considered, numerical
errors can be highly additive, and mass invariably decreases in
numerical computations. It is essential that codes selfmonitor the
entire volume; if the difference between start and finish is greater
than , the grid needs to be readjusted. Usually with few
changes in , and , the conservation
of mass can be improved.
Calculations of conservation of mass should be such that the total
initial displaced volume should be within of the
total displaced volume at the end of the computation where represents the computation end time. It is assumed that the end
of the computation is when the initial wave is entirely reflected
and offshore.
