Abstract

We consider the numerical solution of Riemann—Hilbert problems that are asymptotically posed on a union of disjoint intervals.   The singularities present in the solution of the problem are captured accurately using singular weight functions. A collocation method using orthogonal polynomials with respect to these weights is developed. The primary applications we discuss are to (1) the numerical solution of the computation of the finite-genus solutions of the KdV equation and to (2) the computation of the three-term recurrence coefficients for polynomials orthogonal on multiple intervals.  With regard to (1), we give a new numerical method to solve the periodic initial-value problem for the KdV equation and compute large-genus solutions.  For (2), an outcome of the work is a new implementation of inner-product free iterative solvers for indefinite linear systems.