The method of successive approximations for solving the problem of the decay of a discontinuity and its application to the equations of two-layer shallow water theory

The method of successive approximations for solving the problem of the decay of a small amplitude discontinuity is proposed for hyperbolic systems of conservation laws. In the linear approximation, a Cauchy problem for a linear hyperbolic system is obtained. Its solution represents lines of discontinuity separating the regions in which the solution is constant. Most attention is paid to the first and second approximations, within the limits of which the discontinuities obtained in the first approximation decay into stable shock waves and rarefaction waves. An analysis of the qualitatively different flow conditions that arise when solving the problem of the failure of a dam for a two-layer shallow water model with a free boundary is presented as an example.