Nonlinear Steepest Descent and 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, in practice, 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 Painlevé II equation and the modified Korteweg–de Vries equation to explicitly demonstrate the practical validity of the theory. © 2014 Wiley Periodicals, Inc.