Hugoniot–Maslov Chain for Shock Waves in Buckley–Leverett Equations

In this paper, we apply the asymptotic method developed by V. P. Maslov [1] to obtain the approximated shock-type solutions of the generalized Riemann problem (GRP) to the Buckley–Leverett equation. We calculate the the Hugoniot–Maslov chain (an infinite ODE system) whose fulfillment is a necessary condition that must be satisfied by the coefficients of the asymptotic expansion of the shock-type solution. Numerical simulations based on the truncated Hugoniot–Maslov chain show the efficiency of this method which captures the shock wave unlike some classical finite differences schemes. Finally, we compare the results obtained in this paper with the results obtained via the same asymptotic method, but based in a previous polynomial approximation of the Buckley–Leverett flux as explained in [2]. It was observed that the application of the asymptotic method preceded by a polynomial approximation of the flux function, does not work well for long time simulation values.