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

Periodic Reporting for period 2 - LAHACODE (Low-regularity and high oscillations: numerical analysis and computation of dispersive evolution equations)

Période du rapport: 2021-11-01 au 2023-04-30

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.

Recent years have seen an increasing amount of research devoted to the development of so-called resonance-based methods for dispersive nonlinear partial differential equations. In many situations, this new class of methods allows for approximations in a much more general setting (e.g. for rough data) than, for instance, classical splitting or exponential integrator methods. However, they lack one important property: the preservation of geometric structures. This is particularly drastic in the case of the Korteweg--de Vries (KdV) equation which is a fundamental model in the broad field of dispersive equations that is completely integrable, possessing infinitely many conserved quantities, an important property which we wish to capture -- at least up to some degree -- also on the discrete level. A revolutionary step in this direction was set by the theory of geometric numerical integration resulting in the development of a wide range of structure-preserving algorithms for Hamiltonian systems. However, in general, these methods rely heavily on highly regular solutions. State-of-the-art low-regularity integrators, on the other hand, poorly preserve the geometric structure of the underlying PDE. We made a first step towards bridging the gap between low regularity and structure preservation. We introduced a novel symplectic (in the Hamiltonian picture) resonance-based method on the example of the KdV equation that allows for low-regularity approximations to the solution while preserving the underlying geometric structure of the continuous problem on the discrete level. This is a joint work with Georg Maierhofer, see also arxiv

We introduced a general framework of low regularity integrators which allows us to approximate the time dynamics of a large class of equations, including parabolic and hyperbolic problems, as well as dispersive equations, up to arbitrary high order on general domains. The structure of the local error of the new schemes is driven by nested commutators which in general require (much) lower regularity assumptions than classical methods do. Our main idea lieds in embedding the central oscillations of the nonlinear PDE into the numerical discretisation. The latter is achieved by a novel decorated tree formalism inspired by singular SPDEs with Regularity Structures and allows us to control the nonlinear interactions in the system up to arbitrary high order on the infinite dimensional (continuous) as well as finite dimensional (discrete) level. This is a joint work with Yvonne Alama Bronsard and Yvain Bruned, see also

We introduced a new class of numerical schemes which allow for low regularity approximations to the expectation. This quantity plays an important role in physics, in particular in the study of wave turbulence where one needs to adopt a statistical approach in order to obtain deep insight into the generic long-time behaviour of solutions to dispersive equations. Our new class of schemes is based on Wick's theorem and Feynman diagrams together with a resonance based discretisation set in a more general context: we introduce a novel combinatorial structure called paired decorated forests which are two decorated trees whose decorations on the leaves come in pair. The character of the scheme draws its inspiration from the treatment of singular stochastic partial differential equations via Regularity Structures. In contrast to classical approaches, we do not discretize the PDE itself, but rather its expectation. This allows us to heavily exploit the optimal resonance structure and underlying gain in regularity on the finite dimensional (discrete) level. This is a joint work with Yvonne Alama Bronsard and Yvain Bruned, see also
We aim to develop a general framework of low regularity integrators beyond periodic boundary conditions and beyond dispersive equations. We aim to bridge the gap between low regularity and structure preservation when approximating non-linear PDEs.