Harten solution for one-dimensional unsteady equation (Q684854)

In order to use the second-order 5-point difference scheme mentioned [\textit{Ami Harten}, J. Comput. Phys. 49, 357-393 (1983; Zbl 0565.65050)] to compute the solution of one-dimensional unsteady equations of the direct reflection of the strong plane detonation wave meeting a solid wall barrier, we technically construct the difference schems on the boundary and sub-boundary of the problem, and deduce the auto-similar analytic solutions of the initial value problem. At the same time, we present a method taking into account the singular property of the initial value problem, from which we can get a satisfactory computation result.