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Low-regularity and high oscillations: numerical analysis and computation of dispersive evolution equations

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

Partial differential equations (PDEs) play a central role in mathematics, allowing us to describe physical phenomena ranging from ultra-cold atoms up to ultra-hot matter, from learning algorithms to fluids in the human brain. To understand nature we have to understand their qualitative behavior and compute reliably their numerical approximation. While linear problems and smooth solutions are nowadays well understood, a reliable description of ‘non-smooth’ phenomena remains a challenging open problem. The overall ambition of the ERC-funded project LAHACODE is to make a crucial step towards closing this gap by deeply embedding the underlying structure of resonances into the numerical discretisation. This will allow us to link the finite dimensional discretisation to powerful existence results for nonlinear PDEs at low regularity.
We introduced a numerical framework for dispersive equations embedding their underlying resonance structure into the discretisation. This will allow us to resolve the nonlinear oscillations of the PDE and to approximate with high order accuracy a large class of equations under lower regularity assumptions than classical techniques require. The key idea to control the nonlinear frequency interactions in the system up to arbitrary high order thereby lies in a tailored decorated tree formalism. Our algebraic structures are close to the ones developed for singular SPDEs with Regularity Structures. We adapt them to the context of dispersive PDEs by using a novel class of decorations which encode the dominant frequencies. The structure proposed in this paper is new and gives a variant of the Butcher-Connes-Kreimer Hopf algebra on decorated trees. We observe a similar Birkhoff type factorisation as in SPDEs and perturbative quantum field theory. This factorisation allows us to single out oscillations and to optimise the local error by mapping it to the particular regularity of the solution. This use of the Birkhoff factorisation seems new in comparison to the literature. The field of singular SPDEs took advantage of numerical methods and renormalisation in perturbative quantum field theory by extending their structures via the adjunction of decorations and Taylor expansions. Now, through this work, Numerical Analysis is taking advantage of these extended structures and provides a new perspective on them. This result will appear in Forum of Mathematics, Pi.

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.
We aim to develop a general framework of low regularity integrators beyond periodic boundary conditions and beyond dispersive equations.