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Low-regularity time integrators

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GCS - Geometry, compatibility and structure preservation in computational differential equations

Nonlinear Schrödinger equations are usually solved by pseudo-spectral methods, where the time integration is performed by splitting schemes or exponential integrators. Notwithstanding the benefits of this approach, its successful application requires additional regularity of the solution. For instance, second-order Strang splitting requires four additional derivatives for the solution of the cubic nonlinear Schrödinger equation. Similar statements can be made about other dispersive equations like the Korteweg-de Vries or the Boussinesq equation.

In this talk, we introduce low-regularity Fourier integrators as an alternative. They are obtained from Duhamel's formula in the following way: first, a Lawson-type transformation eliminates the leading linear term and second, the dominant nonlinear terms are integrated exactly in Fourier space. For cubic nonlinear Schrödinger equations, first-order convergence of such methods only requires the boundedness of one additional derivative of the solution, and second-order convergence the boundedness of two derivatives. Similar improvements can also be obtained for other dispersive problems.

This is joint work with Frédéric Rousset (Université Paris-Sud), Katharina Schratz (Hariot-Watt, UK), and Chunmei Su (Technical University of Munich).

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

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