  This paper investigates the convergence of density approximations for
stochastic heat equation in both uniform convergence topology and total
variation distance. The convergence order of the densities in uniform
convergence topology is shown to be exactly $1/2$ in the nonlinear case and
nearly $1$ in the linear case. This result implies that the distributions of
the approximations always converge to the distribution of the origin equation
in total variation distance. As far as we know, this is the first result on the
convergence of density approximations to the stochastic partial differential
equation.
