On the numerical solution of Riemann--Hilbert problems with theta-function asymptotics

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• Thomas Trogdon (University of Washington)
• Tuesday 25 July 2023, 11:30-12:30
• Seminar Room 1, Newton Institute.

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CATW04 - Complex analysis: techniques, applications and computations - perspectives in 2023

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.

This talk is part of the Isaac Newton Institute Seminar Series series.

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