A RELAXATION METHOD VIA THE BORN–INFELD SYSTEM
Abstract
The semilinear relaxation was introduced by Jin and Xin [Comm. Pure Appl. Math.48, 235, (1995)] in order to approximate the conservation law ∂tu + ∂xf(u) = 0 for any flux function f ∈ 𝒞1 (ℝ;ℝ). In this paper, we propose an alternative relaxation technique for scalar conservation laws of the form ∂tu + ∂xu(1 - u)g(u) = 0, where g ∈ 𝒞1 ([0, 1]; ℝ) and 0 ∉ 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 w = (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.
