logoljll LOGO_ERC
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)

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
Math. Comp. (to appear)
V. Banica, G. Maierhofer, K. Schratz, Numerical integration of Schrödinger maps via the Hasimoto transform
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
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

BIT Numer Math (to appear)
M. Caliari, F. Cassini, F. Zivcovich,
A μ-mode BLAS approach for multidimensional tensor-structured problems
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
Journal of Computational Physics 110989
M. Cabrera Calvo Uniformly accurate integrators for Klein-Gordon-Schrödinger systems from the
classical to non-relativistic limit regime

J. Comp. App. Math 114756
Y. A. Bronsard, Y. Bruned, K. Schratz, Approximations of dispersive PDEs in the presence of low-regularity randomness
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
J. Comp. App. Math 114632
B. Li, S. Ma, K. Schratz, A semi-implicit low-regularity integrator for Navier-Stokes equations
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

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

J. Comp. App. Math 114500
M. Cabrera Calvo, F. Rousset, K. Schratz. Time integrators for dispersive equations in the long wave regime
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

SIAM J. Numer. Anal. 60:888-912 (2022)
F. Rousset, K. Schratz. Convergence error estimates at low regularity for time discretizations of KdV
Pure and Applied Analysis 4:127-152 (2022)
Y. Bruned, K. Schratz. Resonance based schemes for dispersive equations via decorated trees
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

IMA J. Numer. Anal. (to appear)
A. Ostermann, F. Rousset, K. Schratz. Error estimates at low regularity of splitting schemes for NLS
Math. Comp. 91:169-182 (2022)
F. Rousset, K. Schratz. A general framework of low regularity integrators
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
J. Eur. Math. Soc. (JEMS) 25:3913-3952 (2022)
K. Schratz, Y. Wang, X. Zhao. Low-regularity integrators for nonlinear Dirac equations.

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.
Found. Comput. Math. 21:725-765 (2021)
S. Baumstark, K. Schratz. Asymptotic preserving integrators for the quantum Zakharov system.
BIT Numer Math (2020)

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