  We study compactness properties of time-discrete and continuous time BGK-type
schemes for scalar conservation laws, in which microscopic interactions occur
only when the state of a system deviates significantly from an equilibrium
distribution. The threshold deviation, $\epsilon,$ is a parameter of the
problem. In the vanishing relaxation time limit we obtain solutions of a
conservation law in which flux is pointwisely close (of order $\epsilon$) to
the flux of the original equation and derive several other properties of such
solutions, including an example of approximate solution to a shock for Burger's
equation.
