  We present a robust and accurate numerical method to solve the modified
Buckley-Leverett equation in two-phase porous media flow with dynamic capillary
pressure effect. A symmetric interior penalty discontinuous Galerkin method is
used to discretize the equation in the space direction. For accuracy and
stability issues, the third-order strong stability preserving implicit-explicit
Runge-Kutta method is adopted to solve the nonlinear semi-discrete system: the
linear diffusion term is discretized implicitly while the nonlinear flux term
is discretized explicitly. The spatial accuracy of the discontinuous Galerkin
method depends on the limiters applied to the solution: we test a minmod-TVB
limiter, a simple WENO limiter and a high-order shock-capturing moment limiter
to demonstrate that a suitable shock capturing moment limiter leads to more
accurate approximation of solution. A set of representative numerical
experiments are presented to show the accuracy and efficiency of the proposed
approach. The results indicate that the moment limiter proposed by Moe et al.
[Arxiv:1507.03024, 2015] is the most suitable one to be used in solving the
modified Buckley- Leverett equation, and high order schemes perform much better
than lower order schemes. Our simulation results are consistent with the
previous results in Kao et al.[J. Sci. Comput., 64(3) (2015), 837-857], Zhang
and Zegeling [J. Comput. Phys., 345 (2017), 510-527 and Commun. Comput. Phys.,
22(4) (2017), 935-964].
