  The focus of the present study is the modified Buckley-Leverett (MBL)
equation describing two-phase flow in porous media. The MBL equation differs
from the classical Buckley-Leverett (BL) equation by including a balanced
diffusive-dispersive combination. The dispersive term is a third order mixed
derivatives term, which models the dynamic effects in the pressure difference
between the two phases. The classical BL equation gives a monotone water
saturation profile for any Riemann problem; on the contrast, when the
dispersive parameter is large enough, the MBL equation delivers non-monotone
water saturation profile for certain Riemann problems as suggested by the
experimental observations. In this paper, we first show that the solution of
the finite interval [0,L] boundary value problem converges to that of the
half-line [0,+\infty) boundary value problem for the MBL equation as L->
+\infty. This result provides a justification for the use of the finite
interval boundary value problem in numerical studies for the half line problem.
Furthermore, we extend the classical central schemes for the hyperbolic
conservation laws to solve the MBL equation which is of pseudo-parabolic type.
Numerical results confirm the existence of non-monotone water saturation
profiles consisting of constant states separated by shocks.
