Abstract <p>When numerically solving quasi-linear hyperbolic equations, so-called acoustic points can arise; these occur when the slope coefficient of the characteristic changes sign in neighboring computational cells. This requires special treatment within the framework of the balance–characteristic approach. The problem can be circumvented by approximating the original equation with a system of two hyperbolic relaxation-type equations that exclude the formation of acoustic points (the Jin–Xin model). A solution to the problem of acoustic points in the CABARET scheme for scalar hyperbolic conservation laws is proposed.</p>