  We study the numerical approximation of the invariant measure of a viscous
scalar conservation law, one-dimensional and periodic in the space variable,
and stochastically forced with a white-in-time but spatially correlated noise.
The flux function is assumed to be locally Lipschitz continuous and to have at
most polynomial growth. The numerical scheme we employ discretises the SPDE
according to a finite-volume method in space, and a split-step backward Euler
method in time. As a first result, we prove the well-posedness as well as the
existence and uniqueness of an invariant measure for both the semi-discrete and
the split-step scheme. Our main result is then the convergence of the invariant
measures of the discrete approximations, as the space and time steps go to
zero, towards the invariant measure of the SPDE, with respect to the
second-order Wasserstein distance. We investigate rates of convergence
theoretically, in the case where the flux function is globally Lipschitz
continuous with a small Lipschitz constant, and numerically for the Burgers
equation.
