Question on Sediment Mass Conservation Scheme

Wen Long

Dear CFD friends:

I got question on solving a very simple sea bottom sediment mass conservation equation using Finite Difference, can you give me hint on which scheme to use?

The problem is to simulation the elevation change of sea bottom near shore due to waves and currents.

The equation for sea bottom elevation look like:

d(zb)/dt= - d(qb)/dx

for 1D case, where zb is sea bottom elevation, and qb is volume transport rate of sands/sediments. Usually qb is calculated using near bottom fluid velocities. Say qb=A*ub^3 with A a coefficient.

The problem been that I'm using wave phase resolving method to predict near bottom velocities, so that ub=ub(x,t) is instantaneous, and have both positive value and negative value in a wave period. So does transport rate qb.

It's very hard to relate qb to zb in functional form, so it's not easy to use Lax-Wendroff scheme directly. Maybe some modified version of Lax-Wendroff scheme will be okay. Any suggestions?

FTCS scheme will be definitely unstable. I tend to find some upwind scheme. Question is that, which direction is upwind? Can I use sign of qb to judge that? Also, can I use dx more than a wavelength? I mean can we discretize with step larger than the length scale of the qb(x,t) oscillation? ("wind" changes direction in a wave length)

I'm looking for bedlevel change over a month. Then, suppose that we can use time averaged(hourly average) equation, right?

Then equation becomes

d<zb>/dt= -d<qb>/dx

The time average <qb> should not have fluctuation in x and time less then the scale of wavelength and wave period, right? Then in this case, we should be able to use dx lareger than wavelength.

But I really don't see difference in the two equations as far as time integration is concerned.

Thank you in advance, and I'm waiting for your input.

Wen