  The effective and efficient numerical solution of Riemann-Hilbert problems
has been demonstrated in recent work. With the aid of ideas from the method of
nonlinear steepest descent for Riemann-Hilbert problems, the resulting
numerical methods have been shown numerically to retain accuracy as values of
certain parameters become arbitrarily large. Remarkably, this numerical
approach does not require knowledge of local parametrices; rather, the deformed
contour is scaled near stationary points at a specific rate. The primary aim of
this paper is to prove that this observed asymptotic accuracy is indeed
achieved. To do so, we first construct a general theoretical framework for the
numerical solution of Riemann-Hilbert problems. Second, we demonstrate the
precise link between nonlinear steepest descent and the success of numerics in
asymptotic regimes. In particular, we prove sufficient conditions for numerical
methods to retain accuracy. Finally, we compute solutions to the homogeneous
Painlev\'e II equation and the modified Korteweg-de Vries equations to
explicitly demonstrate the practical validity of the theory.
