A relaxation method via the Born–Infeld system
Michaël Baudin, Frédéric Coquel and Quang Huy Tran
Abstract: We propose an original relaxation scheme for scalar conservation laws of the form ∂tu+∂x(u(1−u)g(u))=0. Those scalar conservation laws come from a drift-flux model for two-phase flows. Unlike Jin and Xin's approach, the new relaxation strategy does not involve any tuning parameter, but makes use of the Born-Infeld system. The advantage of this new method is that it enables us to achieve a maximum principle on the velocities w=(1−u)g and z=−ug, and to control the sign of the numerical flux. Stability is established for a wide class of eligible functions g. This method can be plugged into a scheme for two-phase fluids, for which numerical experiments are shown.