  A multi-scale method for the hyperbolic systems governing sediment transport
in subcritical case is developed. The scale separation of this problem is due
to the fact that the sediment transport is much slower than flow velocity. We
first derive a zeroth order homogenized model, and then propose a first order
correction. It is revealed that the first order correction for hyperbolic
systems has to be applied on the characteristic speed of slow variables in one
dimensional case. In two dimensional case, besides the characteristic speed,
the source term is also corrected. We develop a second order numerical scheme
following the framework of heterogeneous multi-scale method. The numerical
results in both one and two dimensional cases demonstrate the effectiveness and
efficiency of our method.
