Nonlinear steepest descent and the numerical solution of Riemann–Hilbert problems

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. The primary aim of this paper is to prove that this observation is valid. 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 Painlevé II equation and the modified Korteweg–de Vries equations to explicitly demonstrate the practical validity of the theory.