  This paper is devoted to establishing the convergence rate of the hypersonic
similarity for the inviscid steady irrotational Euler flow over a
two-dimensional Lipschitz slender wedge in $BV\cap L^1$ space. The rate we
established is the same as the one predicted by Newtonian-Busemann law (see
(3.29) in \cite[Page 67]{anderson} for more details)as the incoming Mach number
$\textrm{M}_{\infty}\rightarrow\infty$ for a fixed hypersonic similarity
parameter $K$. The hypersonic similarity, which is also called the Mach-number
independence principle, is equivalent to the following Van Dyke's similarity
theory: For a given hypersonic similarity parameter $K$, when the Mach number
of the flow is sufficiently large, the governing equations after the scaling
are approximated by a simpler equation, that is called the hypersonic
small-disturbance equation. To achieve the convergence rate, we approximate the
curved boundary by piecewisely straight lines and find a new Lipschitz
continuous map $\mathcal{P}_{h}$ such that the trajectory can be obtained by
piecing together the Riemann solutions near the approximated boundary. Next, we
derive the $L^1$ difference estimates between the approximate solutions
$U^{(\tau)}_{h,\nu}(x,\cdot)$ to the initial-boundary value problem for the
scaled equations and the trajectories $\mathcal{P}_{h}(x,0)(U^{\nu}_{0})$ by
piecing together all the Riemann solvers. Then, by the uniqueness and the
compactness of $\mathcal{P}_{h}$ and $U^{(\tau)}_{h,\nu}$, we can further
establish the $L^1$ estimates of order $\tau^2$ between the solutions to the
initial-boundary value problem for the scaled equations and the solutions to
the initial-boundary value problem for the hypersonic small-disturbance
equations, if the total variations of the initial data and the tangential
derivative of the boundary are sufficiently small.
