We carry out an error analysis for the heterogeneous multi-scale method for the case
when the macroscale process is that of gas dynamics or more generally nonlinear conservation laws
and the microscale process is an atomistic model such as kinetic Monte Carlo methods or molecular
dynamics (MD). We will consider problems of type B as defined in [4], i.e. the macroscale constitutive
relations are unknown and are extracted from the microscopic model. In addition to the standard
error in the macroscale solver, a new error term occurs in estimating the data, here the fluxes. This
new error term consists of three parts: the relaxation error, the sampling error and the error due to
the finite size of the atomistic simulation. Our results serve as guidelines for designing multiscale
methods, as was done in [13, 16].