A RELAXATION METHOD VIA THE BORN-INFELD SYSTEM

The semilinear relaxation was introduced by Jin and Xin [Comm. Pure Appl. Math. 48, 235, (1995)] in order to approximate the conservation law partial derivative(t)u + partial derivative(x)f(u) = 0 for any flux function f is an element of l(1) (R; R). In this paper, we propose an alternative relaxation technique for scalar conservation laws of the form partial derivative(t)u + partial derivative(x)u(1 - u)g(u) = 0, where g is an element of l(1) ([0, 1]; R) and 0 is not an element of g(]0, 1[). We extend this new philosophy to an arbitrary flux function f whenever possible. Unlike the semilinear approach, the new relaxation strategy does not involve any tuning parameter, but makes use of the Born-Infeld system. Another advantage of this method is that it enables us to achieve a maximum principle on the velocities omega = (1 - u)g and z = -ug, which turns out to be a physically interesting and helpful feature in the context of some two-phase flow problems.