  We establish the optimal convergence rate to the hypersonic similarity law,
which is also called the Mach number independence principle, for steady
compressible full Euler flows over two-dimensional slender Lipschitz wedges.
The problem can be formulated as the comparison of the entropy solutions in
$BV\cap L^{1}$ between the two initial-boundary value problems for the
compressible full Euler equations with parameter $\tau>0$ and the hypersonic
small-disturbance equations with curved characteristic boundaries. We establish
the $L^1$--convergence estimate of these two solutions with the optimal
convergence rate, which justifies Van Dyke's similarity theory rigorously for
the compressible full Euler flows. This is the first mathematical result on the
comparison of two solutions of the compressible Euler equations with
characteristic boundary conditions. To achieve this, we first employ the
special structures of the two systems and establish the global existence and
the $L^1$--stability of the entropy solutions under the smallness assumptions
on the total variation of both the initial data and the tangential slope
function of the wedge boundary. Based on the $L^1$--stability properties of the
approximate solutions to the scaled equations with parameter $\tau$, a uniform
Lipschtiz continuous map with respect to the initial data and the wedge
boundary is obtained. Next, we compare the solutions given by the Riemann
solvers of the two systems by taking the boundary perturbations into account
case by case. Then, for a given fixed hypersonic similarity parameter, as the
Mach number tends to infinity, we establish the desired $L^1$--convergence
estimate with the optimal convergence rate. Finally, we show the optimality of
the convergence rate by investigating a special solution.
