  This article devises a new numerical method for first-order transport
problems by using the primal-dual weak Galerkin (PD-WG) finite element method
recently developed in scientific computing. The PD-WG method is based on a
variational formulation of the modeling equation for which the differential
operator is applied to the test function so that low regularity for the exact
solution of the original equation is sufficient for computation. The PD-WG
finite element method indeed yields a symmetric system involving both the
original equation for the primal variable and its dual for the dual variable
(also known as Lagrangian multiplier). For the linear transport problem, it is
shown that the PD-WG method offers numerical solutions that conserve mass
locally on each element. Optimal order error estimates in various norms are
derived for the numerical solutions arising from the PD-WG method with weak
regularity assumptions on the modelling equations. A variety of numerical
results are presented to demonstrate the accuracy and stability of the new
method.
