High-Order Accurate Flux-Splitting Scheme for Conservation Laws with Discontinuous Flux Function in Space

A new modified Engquist–Osher-type flux-splitting scheme is proposed to approximate the scalar conservation laws with discontinuous flux function in space. The fact that the discontinuity of the fluxes in space results in the jump of the unknown function may be the reason why it is difficult to design a high-order scheme to solve this hyperbolic conservation law. In order to implement the WENO flux reconstruction, we apply the new modified Engquist–Osher-type flux to compensate for the discontinuity of fluxes in space. Together the third-order TVD Runge–Kutta time discretization, we can obtain the high-order accurate scheme, which keeps equilibrium state across the discontinuity in space, to solve the scalar conservation laws with discontinuous flux function. Some examples are given to demonstrate the good performance of the new high-order accurate scheme.

Numerical Study for Two-Phase Flow with Gravity in Homogeneous and Piecewise-Homogeneous Porous Media

In this work we study two-phase flow with gravity either in 1-rock homogeneous media or 2-rocks composed media, this phenomenon can be modeled by a non-linear scalar conservation law with continuous flux function or discontinuous flux function, respectively. Our study is essentially from a numerical point of view, we apply the new Lagrangian-Eulerian finite difference method developed by Abreu and Pérez and the Lax-Friedrichs classic method to obtain numerical entropic solutions. Comparisons between numerical and analytical solutions show the efficiency of the methods even for discontinuous flux function.