We further investigate the high order positivity-preserving
discontinuous Galerkin (DG) methods for linear hyperbolic
and radiative transfer equations developed in \cite{yuan}.
The DG methods in \cite{yuan} can maintain positivity and high
order accuracy, but they rely both on the scaling limiter
in \cite{zhang1} and a rotational limiter, the latter
may alter cell averages of the unmodulated DG scheme, thereby
affecting conservation. Even though a Lax-Wendroff type theorem
is proved in \cite{yuan}, guaranteeing convergence to weak
solutions with correct shock speed when such rotational limiter is
applied, it would still be desirable if a conservative DG method
without changing the cell averages can be obtained which has
both high order accuracy and positivity-preserving capability.
In this paper, we develop and analyze such a DG method for
both linear hyperbolic equations and radiative transfer equations.
In the one-dimensional case, the method uses traditional DG space
$P^k$ of piecewise polynomials of degree at most $k$. A key
result is proved that the unmodulated DG solver in this case
can maintain positivity of the cell average if the inflow
boundary value and the source term are both positive, therefore
the positivity-preserving framework
in \cite{zhang1} can be used to obtain a high order conservative
positivity-preserving DG scheme. Unfortunately, in two-dimensions
this is no longer the case. We show that the unmodulated DG solver
based either on $P^k$ or $Q^k$ spaces (piecewise $k$-th degree polynomials
or piecewise tensor-product $k$-th degree polynomials) could generate
negative cell averages. We augment the DG space with additional
functions so that the positivity of cell averages from the
unmodulated DG solver can be restored, thereby leading to
high order conservative
positivity-preserving DG scheme based on these augmented DG spaces
following the framework
in \cite{zhang1}. Computational results are provided to
demonstrate the good performance of our DG schemes.