Central schemes for the modified Buckley-Leverett equation

In this paper, we extend the second and third order classical central schemes for the hyperbolic conservation laws to solve the modified Buckley-Leverett (MBL) equation which is of pseudo-parabolic type. The MBL equation describes two-phase flow in porous media, and it differs from the classical Buckley-Leverett (BL) equation by including a balanced diffusive-dispersive combination. 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 profiles for certain Riemann problems as suggested by the experimental observations. Numerical results in this paper confirm the existence of non-monotone water saturation profiles consisting of constant states separated by shocks.