Stability of a Relaxation Model with a Nonconvex Flux∗

In this paper, we study the nonlinear stability of travelling wave solutions with shock profile for a relaxation model with a nonconvex flux, which is proposed by Jin and Xin [Comm. Pure Appl. Math., 48 (1995), pp. 555–563] to approximate an original hyperbolic system numerically under the subcharacteristic condition introduced by T. P. Liu [Comm. Math. Phys., 108 (1987), pp. 153–175]. The travelling wave solutions with strong shock profile are shown to be asymptotically stable under small disturbances with integral zero using an elementary but technical energy method. Proofs involve detailed study of the error equation for disturbances using the same weight function introduced in [Comm. Math. Phys., 165 (1994), pp. 83–96].