  In this paper, we present and analyze a weak Galerkin finite element (WG)
method for solving the symmetric hyperbolic systems. This method is highly
flexible by allowing the use of discontinuous finite elements on element and
its boundary independently of each other. By introducing special weak
derivative, we construct a stable weak Galerkin scheme and derive the optimal
$L_2$-error estimate of $O(h^{k+\frac{1}{2}})$-order for the discrete solution
when the $k$-order polynomials are used for $k\geq 0$. As application, we
discuss this WG method for solving the singularly perturbed
convection-diffusion-reaction equation and derive an $\varepsilon$-uniform
error estimate of order $k+1/2$. Numerical examples are provided to show the
effectiveness of the proposed WG method.
