Initial Boundary-Value Problems for a Pair of Conservation Laws
We describe a multiple-scale technique for solving the initial boundary-value problem over the positive x-axis for a one-dimensional pair of hyperbolic conservation laws. This technique involves decomposing the solution into waves and incorporating slow temporal and stretched spatial scales in different parts of the solution domain. We apply these ideas to a wavemaker problem for shallow water flow and show why the presence of source terms in the conservation laws makes the analytic solution more complicated.
Yong, DH, Kevorkian, J. Initial boundary-value problems for a pair of conservation laws. Stud Appl Math. 2002;108(4): 351–367. .