(These plots are for the nondimensional time t= 0.148.) The solid line is for the exact sensitivities. Although they are difficult to distinguish, there are three other curves in the figures. There is a dotted curve for the ADIFOR approximate sensitivity, a dashed curve for the sensitivity equation method approximate sensitivity, and a dotted-dashed curve for the finite difference quotient approximate sensitivity.
The results of using all three methods for defining approximate sensitivities are very similar and are very poor approximationsto the exact sensitivities. Since we have ignored the presence of the flow discontinuities,
all three methods attempt to approximate the delta functions that result from differentiating the flow across surfaces of discontinuity.This is, of course, a difficult task to do well and is the source of most of the inaccuracies in the approximate sensitivities. Note also that
the jump in the sensitivites at the edges of the rarefaction wave are very poorly resolved.Finally, note that the inacurracies due to
the attempt to approximate delta functions and other discontinuities severly polute the approximations even in the continuous portions of the flow.
A flow sensitivity along the comparison plane
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 coarse grid.