**ERC Starting Grant :**LAHACODE

Low-regularity and high oscillations: numerical analysis and

computation of dispersive evolution equations

PI: Katharina Schratz

Funding period: 2020 - 2025

(financed by the European Research Council)

https://cordis.europa.eu/project/id/850941

Partial differential equations (PDEs) play a central role in mathematics, allowing us to describe physical phenomena ranging from ultra-cold atoms (Bose-Einstein condensation) up to ultra-hot matter (nuclear fusion), from learning algorithms to fluids in the human brain. To understand nature we have to understand their qualitative behavior: existence and long time behavior of solutions, their geometric and dynamical properties, as well as to compute reliably their numerical solution. While linear problems and smooth solutions are nowadays well understood, a reliable description of non-smooth phenomena remains one of the most challenging open problems in computational mathematics since the underlying PDEs have very complicated solutions exhibiting high oscillations and loss of regularity. This leads to huge errors, massive computational costs and ultimately provokes the failure of classical schemes. Nevertheless, non-smooth phenomena play a fundamental role in modern physical modeling (e.g. blow-up phenomena, turbulences, high frequencies, low dispersion limits, etc.) which makes it an essential task to develop suitable numerical schemes. The overall ambition of LAHACODE is to make a crucial step towards closing this gap - addressing the fundamental question: How and to what extent can we reproduce the qualitative behavior of differential equations in a finite (discretized) world?

**Members :**

Yvonne Bronsard Alama

Albert Cohen

Katharina Schratz

**Former Members :**

Yue Feng

Georg Maierhofer

María Cabrera Calvo

Nikola Stoilov

Franco Zivcovich

**Publications :**

L. Ji, A. Ostermann, F. Rousset, K. Schratz,
*Low regularity error estimates for the time integration of 2D NLS*

https://arxiv.org/abs/2301.10639
(preprint 2023)

G. Maierhofer, K. Schratz,
*Bridging the gap: symplecticity and low regularity on the example of the KdV equation*

https://arxiv.org/abs/2205.05024
(preprint 2022)

Y. Feng, K. Schratz,
*Improved uniform error bounds on a Lawson-type exponential integrator for the long-time dynamics of sine-Gordon equation*

https://arxiv.org/abs/2211.09402 (preprint 2022)

Y. A. Bronsard, Y. Bruned, K. Schratz,

*Low regularity integrators via decorated trees*

https://arxiv.org/abs/2202.01171 (preprint 2022)

A. Iserles, K. Kropielnicka, K. Schratz, M. Webb.

*Solving the linear Schrödinger equation on the real line*

http://arxiv.org/abs/2102.00413 (preprint 2021)

Y. Feng, G. Maierhofer, K. Schratz,

*Long-time error bounds of low-regularity integrators for nonlinear Schrödinger equations*

https://arxiv.org/abs/2302.00383

**Math. Comp.**(to appear)

V. Banica, G. Maierhofer, K. Schratz,

*Numerical integration of Schrödinger maps via the Hasimoto transform*

https://arxiv.org/abs/2211.01282

**SIAM J. Numer. Anal**(to appear)

B. Li, K. Schratz, F. Zivcovich,

*A second-order low-regularity correction of Lie splitting for the semilinear Klein-Gordon equation*

http://arxiv.org/abs/2203.15539

**M2AN**57:899-919 (2023)

C.-K. Doan, T.-T.-P. Hoang, K. Schratz,

*Low regularity integrators for semilinear parabolic equations*

with maximum bound principles

with maximum bound principles

https://link.springer.com/article/10.1007/s10543-023-00946-2

**BIT Numer Math**(to appear)

M. Caliari, F. Cassini, F. Zivcovich,

*A μ-mode BLAS approach for multidimensional tensor-structured problems*

https://arxiv.org/abs/2112.11238

**Numerical Algorithms**92:2483--2508

M. Caliari, F. Cassini, L. Einkemmer, A. Ostermann, F. Zivcovich,

*A μ-mode integrator for solving evolution equations in Kronecker form*

https://arxiv.org/abs/2103.01691

**Journal of Computational Physics**110989

M. Cabrera Calvo

*Uniformly accurate integrators for Klein-Gordon-Schrödinger systems from the*

classical to non-relativistic limit regime

classical to non-relativistic limit regime

https://arxiv.org/abs/2201.05339

**J. Comp. App. Math**114756

Y. A. Bronsard, Y. Bruned, K. Schratz,

*Approximations of dispersive PDEs in the presence of low-regularity randomness*

https://arxiv.org/abs/2205.02156

**Found. Comput. Math.**(to appear)

Y. Alama Bronsard

*Error analysis of a class of semi-discrete schemes for solving the Gross-Pitaevskii equation at low regularity*

https://arxiv.org/abs/2201.13314

**J. Comp. App. Math**114632

B. Li, S. Ma, K. Schratz,

*A semi-implicit low-regularity integrator for Navier-Stokes equations*

http://arxiv.org/abs/2107.13427

**SIAM J. Numer. Anal.**60:2273-2292 (2022)

M. Cabrera Calvo, K. Schratz.

*Uniformly accurate splitting schemes for the Benjamin-Bona-Mahony*

equation with dispersive parameter

equation with dispersive parameter

http://arxiv.org/abs/2105.03732

**BIT Numer Math**60:888-912 (2022)

G. Maierhofer, D. Huybrechs,

*An analysis of least-squares oversampled collocation methods for*

compactly perturbed boundary integral equations in two dimensions

compactly perturbed boundary integral equations in two dimensions

https://arxiv.org/abs/2201.11683

**J. Comp. App. Math**114500

M. Cabrera Calvo, F. Rousset, K. Schratz.

*Time integrators for dispersive equations in the long wave regime*

http://arxiv.org/abs/2105.03731

**Math. Comp.**91:2197-2214 (2021)

M. Cabrera Calvo, K. Schratz.

*Uniformly accurate low regularity integrators for the Klein-Gordon equation*

from the classical to non-relativistic limit regime

from the classical to non-relativistic limit regime

http://arxiv.org/abs/2104.11672

**SIAM J. Numer. Anal.**60:888-912 (2022)

F. Rousset, K. Schratz.

*Convergence error estimates at low regularity for time discretizations of KdV*

https://arxiv.org/abs/2102.11125

**Pure and Applied Analysis**4:127-152 (2022)

Y. Bruned, K. Schratz.

*Resonance based schemes for dispersive equations via decorated trees*

doi:10.1017/fmp.2021.13

**Forum of Mathematics, Pi**10:e2 1-76 (2022)

A. Poulain, K. Schratz.

*Convergence, error analysis and longtime behavior of the Scalar Auxiliary*

Variable method for the nonlinear Schrödinger equation

Variable method for the nonlinear Schrödinger equation

https://arxiv.org/abs/2012.13943

**IMA J. Numer. Anal.**(to appear)

A. Ostermann, F. Rousset, K. Schratz.

*Error estimates at low regularity of splitting schemes for NLS*

https://arxiv.org/abs/2012.14146

**Math. Comp.**91:169-182 (2022)

F. Rousset, K. Schratz.

*A general framework of low regularity integrators*

http://arxiv.org/abs/2010.01640

**SIAM J. Numer. Anal.**59:1735-1768 (2021)

A. Ostermann, F. Rousset, K. Schratz.

*Fourier integrator for periodic NLS: low regularity estimates via discrete Bourgain spaces*

http://arxiv.org/abs/2006.12785

**J. Eur. Math. Soc. (JEMS)**25:3913-3952 (2022)

K. Schratz, Y. Wang, X. Zhao.

*Low-regularity integrators for nonlinear Dirac equations.*

https://arxiv.org/abs/1906.09413

**Math. Comp.**90:189-214 (2021)

A. Ostermann, F. Rousset, K. Schratz.

*Error estimates of a Fourier integrator for the cubic Schrödinger equation at low regularity.*

https://arxiv.org/abs/1902.06779

**Found. Comput. Math.**21:725-765 (2021)

S. Baumstark, K. Schratz.

*Asymptotic preserving integrators for the quantum Zakharov system.*

doi:10.1007/s10543-020-00815-2

**BIT Numer Math**(2020)

codes of my recent papers can be found at https://github.com/GeorgAUT/GLIMPSE