x_{\rm max}^-}  q^{(1)}(x) > 0, \qquad
\lim_{x\to x_{\rm max}^+}  q^{(1)}(x) = 0$$ making the convergence more difficult. We recall that in deriving equations (\[SP2\]) and (\[SP3\]) differentiability of $q(x)$ was assumed. The use of lower order integration schemes, as second-order Adam-Bashfort or Euler schemes, does not give sensible improvements.

Larger values of $N_x$ requires larger needs of computer memory therefore to increase the precision we adopted a different approach. Since $\dot{q}(x)=0$ for $x>x_{\rm max}$ equations (\[SP2\]) and (\[SP3\]) are trivial in this range and we can reduce the upper bound of the $x$-integration from $x=1$ to $x=x_0=x_{\rm max}$. This obviously requires the knowledge of $x_{\rm max}$ for the given temperature. However if we assume [*no a priori*]{} knowledge of $x_{\rm max}$ we must proceeds for successive approximations: we start from $x_0=1$ an then reducing it until we ‘hit’ the value of $x_{\rm max}$. This procedure is simplified by the fact that if $x_0<x_{\rm max}$ the shape of $q(x)$ near $x_0$ changes dramatically with the concavity passing from negative values for $x_0>x_{\rm max}$ to positive values for $x_0<x_{\rm max}$. In Figure \[fig:qx-pdq.T0.6-x0\] panel (a) are shown $q(x)$ (panel a) and $P(q)$ (panel b) at $T=0.6$ for different values of $x_0$, the 