The solid curve is for the approximate sensitivity found by using a finite different quotient involving multiple flow solves. The dashed curve is for the approximate sensitivity found by using the sensitivity equation method. In both cases, the sensitivity is large near the point of intersection of the shock wave and the comparison plane. The spike in the sensitivity is an approximation to the the delta function that arises when one differentiates a flow variable across the shock wave.
The exact solution of the Reimann problem
Exact sensitivities for the Reimann problem
Approximate sensitivities for the Lax-Wendroff method
Approximate sensitivities for the Godunov method
Approximate sensitivities for the Roe method.
Approximate sensitivities for the Lax-Wendroff method on the medium grid
Approximate sensitivities for the Lax-Wendroff method on the coarse grid.