Periodic Reporting for period 1 - LAHACODE (Low-regularity and high oscillations: numerical analysis and computation of dispersive evolution equations)
Reporting period: 2020-05-01 to 2021-10-31
We developed a general framework of low regularity integrators for an abstract evolution equation which involves not only dispersive type equations, but also hyperbolic problems and parabolic problems (such as the wave equation, nonlinear heat equation, the complex Ginzburg-Landau equation, the half wave and Klein--Gordon equations). The main idea is based on new oscillatory integrators which allow us to embed the underlying oscillations of the partial differential equation into the numerical discretisation. This result was published in SIAM J. Num. Analysis. Furthermore, we could develop a new type of low-regularity integrator for Navier-Stokes equations, coupled with a stabilized finite element method in space. Unlike the other low-regularity integrators for nonlinear dispersive equations, which are all fully explicit in time, the proposed method is semi-implicit in time in order to preserve the energy-decay structure of NS equations. First-order convergence of the proposed method is established independent of the viscosity coefficient μ, under weaker regularity conditions than other existing numerical methods, including the semi-implicit Euler method and classical exponential integrators.
We introduced discrete Bourgain spaces, which allow us to establish low regularity error estimates for the nonlinear Schrödinger equation under periodic boundary condition. This result will appear in the Journal of the European Mathematical Society (JEMS). We could furthermore establish sharp L^2 error estimates for the periodic Korteweg--de Vries equation and, with the aid of discrete Strichartz type estimates, for the nonlinear Schrödinger equation on the full space in up to three spatial dimensions. The latter results are published in Foundations of Computational Mathematics and Mathematics of Computations.