Periodic Reporting for period 1 - GLIMPSE (Geometric and Low-regularity Integrators for the Matching and Preservation of Structure in the computation of dispersive Equations)
Berichtszeitraum: 2022-05-01 bis 2023-10-31
Some of the most intriguing phenomena in nature arise when the underlying physical laws can be described using nonlinear dispersive partial differential equations. This means that waves of different frequencies travel at different speeds – a mechanism that is, for instance, responsible for the breaking of ocean waves near the shore. When a computer is asked to approximate solutions that exhibit discontinuities (low-regularity), as is the case for instance in shock waves, these nonlinear frequency interactions pose a significant challenge which has recently been addressed by the development of so-called resonance-based numerical schemes. Despite their success, ‘resonance-based’ schemes are a relatively recent development and many interesting open problems are yet to be studied, including their (in-)ability to preserve geometric structure from the underlying model.
In this MSCA project the research fellow has, together with collaborators, successfully addressed this open problem through the construction of structure preserving low-regularity integrators. In particular, we have achieved the following major milestones:
(i) The construction of symmetric low-regularity integrators for specific applications (the nonlinear Schrödinger equation, the Korteweg–de Vries equation and the isotropic Landau–Lifschitz equation);
(ii) The classification of symmetric resonance-based schemes (low-regularity integrators) for a large class of dispersive nonlinear equations;
(iii) The construction of symplectic low-regularity integrators for the Korteweg–de Vries equation and the one-dimensional nonlinear Schrödinger equation;
(iv) The study of the long-time behaviour of symmetric resonance-based schemes for the nonlinear Schrödinger equation with weak nonlinearity and the design of a tailored integrator with significantly reduced error in the long-time regime;
(v) An application of these new schemes to the simulation of the evolution of vortex filaments in ideal fluids.
In this MSCA project the research fellow has, together with collaborators, successfully addressed this open problem through the construction of structure preserving low-regularity integrators. In particular, we have achieved the following major milestones:
(i) The construction of symmetric low-regularity integrators for specific applications (the nonlinear Schrödinger equation, the Korteweg–de Vries equation and the isotropic Landau–Lifschitz equation);
(ii) The classification of symmetric resonance-based schemes (low-regularity integrators) for a large class of dispersive nonlinear equations;
(iii) The construction of symplectic low-regularity integrators for the Korteweg–de Vries equation and the one-dimensional nonlinear Schrödinger equation;
(iv) The study of the long-time behaviour of symmetric resonance-based schemes for the nonlinear Schrödinger equation with weak nonlinearity and the design of a tailored integrator with significantly reduced error in the long-time regime;
(v) An application of these new schemes to the simulation of the evolution of vortex filaments in ideal fluids.
Over the course of this project the fellow, together with his collaborators Prof. Katharina Schratz, Prof. Valeria Banica, Prof. Yvain Bruned, Dr Yue Feng and Yvonne Alama Bronsard, (henceforth referred to as “we”) has developed structure preserving time-stepping methods that can effectively approximate low-regularity solutions of a large selection of dispersive nonlinear partial differential equations. The outcomes of this research are documented in detail in the following manuscripts (all of which are available on open access repositories):
[1] Maierhofer, G. and Schratz, K., “Bridging the gap: symplecticity and low regularity in Runge-Kutta resonance-based schemes”, 2022. arXiv.2205.05024.
[2] Alama Bronsard, Y., Bruned, Y., Maierhofer, G. and Schratz, K., “Symmetric resonance based integrators and forest formulae”, 2023. arXiv.2305.16737.
[3] Banica, V., Maierhofer, G. and Schratz, K., “Numerical integration of Schrödinger maps via the Hasimoto transform”, 2022. arXiv.2211.01282 to appear in SIAM J. Numer. Anal.
[4] Feng, Y., Maierhofer, G. and Schratz, K., “Long-time error bounds of low-regularity integrators for nonlinear Schrödinger equations”, 2023. arXiv.2302.00383 to appear in Math. Comput.
In particular the project was successful in: the development of symmetric low-regularity integrators for a large class of dispersive nonlinear equations with polynomial nonlinearities (cf. [2]) based on a new formalism bringing together algebraic structures and constructions of resonance-based methods; the analysis and development of symplectic resonance-based schemes for two of the central examples of such dispersive equations, the Korteweg-de Vries (KdV) equation and the nonlinear Schrödinger equation (NLSE) (cf. [1]). This work is complemented by applications of these schemes to long-time integration in the NLSE, together with rigorous analysis (cf. [4]), and to the approximation of the dynamics of vortex filaments in low-regularity regimes (cf. [3]).
[1] Maierhofer, G. and Schratz, K., “Bridging the gap: symplecticity and low regularity in Runge-Kutta resonance-based schemes”, 2022. arXiv.2205.05024.
[2] Alama Bronsard, Y., Bruned, Y., Maierhofer, G. and Schratz, K., “Symmetric resonance based integrators and forest formulae”, 2023. arXiv.2305.16737.
[3] Banica, V., Maierhofer, G. and Schratz, K., “Numerical integration of Schrödinger maps via the Hasimoto transform”, 2022. arXiv.2211.01282 to appear in SIAM J. Numer. Anal.
[4] Feng, Y., Maierhofer, G. and Schratz, K., “Long-time error bounds of low-regularity integrators for nonlinear Schrödinger equations”, 2023. arXiv.2302.00383 to appear in Math. Comput.
In particular the project was successful in: the development of symmetric low-regularity integrators for a large class of dispersive nonlinear equations with polynomial nonlinearities (cf. [2]) based on a new formalism bringing together algebraic structures and constructions of resonance-based methods; the analysis and development of symplectic resonance-based schemes for two of the central examples of such dispersive equations, the Korteweg-de Vries (KdV) equation and the nonlinear Schrödinger equation (NLSE) (cf. [1]). This work is complemented by applications of these schemes to long-time integration in the NLSE, together with rigorous analysis (cf. [4]), and to the approximation of the dynamics of vortex filaments in low-regularity regimes (cf. [3]).
The outcomes of this action have contributed a new avenue in the study of low-regularity numerical schemes and phenomena by closing the gap between geometric numerical integration and prior designs of resonance-based schemes. This constitutes an important contribution to the scientific field of computational mathematics whose impact can be measured by both the number of publications associated with this research (two have already been accepted to major scientific journals in this field, two further publications are submitted and at least one further work is in the process of being compiled into a manuscript) and the platforms which the fellow, supervisor and collaborators have been invited to present this work (invited presentations at some of the major conferences in this field, several invited technical seminars at universities in and outside of Europe and at several smaller technical workshops). In addition is it worth noting that, while achieving the objectives set for this action, the insights gained over the course of this project form an important basis for further work by both the fellow and the collaborators involved with the action, as well as further collaborations that have started to develop since the end of this action. Code associated with this action can be found at https://github.com/GeorgAUT/GLIMPSE(öffnet in neuem Fenster).