On the Riemann-Hilbert approach to the numerical solution of boundary-value problems for evolution partial differential equations

Abstract

Integrable systems play an important role in many research areas in Mathematics and Physics. For such systems, the Inverse Scattering Transform provides an alternate way to solve the initial-value problem in terms of Riemann-Hilbert problems. The Riemann-Hilbert approach allows not only a new way for the analysis but also a new way for numerical methods. On the other hand, as an extension of the Inverse Scattering Transform, the Unified Transform Method provides an alternate way to solve initial-boundary-value problems for integrable systems in terms of Riemann-Hilbert problems. In this dissertation, I develop the Numerical Unified Transform Method as a generalization of the Numerical Inverse Scattering Transform. Compared with traditional numerical methods for evolution partial differential equations, methods based on the Riemann-Hilbert approach can give the solution at a given point in the physical domain without time-stepping and can compute the solution with fixed computational costs to reach a given accuracy.