Conservation laws with discontinuous coefficients have been applied in different science and engineering fields, such as fluid dynamics, traffic flow, etc. Due to the complexity of analytical solutions, numerical methods for conservation laws with discontinuous considerations have been proposed and widely applied in in science and engineering fields. However, one of the particular applications of the conservation laws with discontinuous considerations is the LWR for interrupted flow conditions. \\\\ In this research, a novel numerical scheme is proposed to solve the problem of conservation laws with discontinuous interface and the proposed scheme is then applied to solve the LWR model under interrupted flows, especially signals. The proposed scheme is constructed based on the Godunov's scheme with special treatment at interfaces. \\\\ The proposed scheme is then illustrated and evaluated through different numerical experiments. Some numerical results from basic and simply experiments are compared with those obtained from analytical models. Then, the proposed scheme is applied to solve the problem of the LWR model with signals under the assumption of the Greenshield function. \\\\ The numerical results show the promising applications of the proposed numerical scheme in different science and engineering fields.