54 citations · 160 across the 35 of their papers we have counts for
40 papers
A splitting scheme for the wave maps equation at low regularity
Katie Marsden, Frédéric Rousset, Katharina Schratz
We prove convergence of a filtered Lie splitting scheme for the wave maps equation with low regularity initial data in dimension 3. The convergence analysis is performed in discret…
Time-relaxation structure-preserving explicit low-regularity integrators for the nonlinear Schrödinger equation
Hang Li, Xicui Li, Katharina Schratz +1
We propose and rigorously analyze a novel family of explicit low-regularity exponential integrators for the nonlinear Schrödinger (NLS) equation, based on a time-relaxation framewo…
Resonances and computations
Yvain Bruned, Frédéric Rousset, Katharina Schratz
The computation of time dynamics arising in nonlinear time-dependent partial differential equations is an ongoing challenge in numerical analysis, especially once roughness comes i…
Computing rough solutions of the stochastic nonlinear wave equation
Jiachuan Cao, Buyang Li, Katharina Schratz
The regularity of solutions to the stochastic nonlinear wave equation plays a critical role in the accuracy and efficiency of numerical algorithms. Rough or discontinuous initial c…
Resonances as a computational tool
Frédéric Rousset, Katharina Schratz
A large toolbox of numerical schemes for dispersive equations has been established, based on different discretization techniques such as discretizing the variation-of-constants for…
Low regularity full error estimates for the cubic nonlinear Schrödinger equation
Lun Ji, Alexander Ostermann, Frédéric Rousset +1
For the numerical solution of the cubic nonlinear Schrödinger equation with periodic boundary conditions, a pseudospectral method in space combined with a filtered Lie splitting sc…
Symmetric resonance based integrators and forest formulae
Yvonne Alama Bronsard, Yvain Bruned, Georg Maierhofer +1
In the present work we introduce a unified framework that allows for the very first systematic construction of symmetric resonance-based integrators to approximate a wide class of…
Long-time error bounds of low-regularity integrators for nonlinear Schrödinger equations
Yue Feng, Georg Maierhofer, Katharina Schratz
We introduce a new non-resonant low-regularity integrator for the cubic nonlinear Schrödinger equation (NLSE) allowing for long-time error estimates which are optimal in the sense…
Low regularity error estimates for the time integration of 2D NLS
Lun Ji, Alexander Ostermann, Frédéric Rousset +1
A filtered Lie splitting scheme is proposed for the time integration of the cubic nonlinear Schrödinger equation on the two-dimensional torus . The scheme is analyzed…
Improved uniform error bounds on a Lawson-type exponential integrator for the long-time dynamics of sine--Gordon equation
Yue Feng, Katharina Schratz
We establish the improved uniform error bounds on a Lawson-type exponential integrator Fourier pseudospectral (LEI-FP) method for the long-time dynamics of sine-Gordon equation whe…
Low regularity integrators for semilinear parabolic equations with maximum bound principles
Cao-Kha Doan, Thi-Thao-Phuong Hoang, Lili Ju +1
This paper is concerned with conditionally structure-preserving, low regularity time integration methods for a class of semilinear parabolic equations of Allen-Cahn type. Important…
Numerical integration of Schrödinger maps via the Hasimoto transform
Valeria Banica, Georg Maierhofer, Katharina Schratz
We introduce a numerical approach to computing the Schrödinger map (SM) based on the Hasimoto transform which relates the SM flow to a cubic nonlinear Schrödinger (NLS) equation. I…
Bridging the gap: symplecticity and low regularity in Runge-Kutta resonance-based schemes
Georg Maierhofer, Katharina Schratz
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…
Approximations of dispersive PDEs in the presence of low-regularity randomness
Yvonne Alama Bronsard, Yvain Bruned, Katharina Schratz
We introduce a new class of numerical schemes which allow for low regularity approximations to the expectation , where denotes the -th Four…
A second-order low-regularity correction of Lie splitting for the semilinear Klein--Gordon equation
Buyang Li, Katharina Schratz, Franco Zivcovich
The numerical approximation of the semilinear Klein--Gordon equation in the -dimensional space, with , is studied by analyzing the consistency errors in approximating t…
Low regularity integrators via decorated trees
Yvonne Alama Bronsard, Yvain Bruned, Katharina Schratz
We introduce 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 pr…
Effective highly accurate time integrators for linear Klein-Gordon equations across the scales
Karolina Kropielnicka, Karolina Lademann, Katharina Schratz
We propose an efficient approach for time integration of Klein-Gordon equations with highly oscillatory in time input terms. The new methods are highly accurate in the entire range…
A semi-implicit low-regularity integrator for Navier-Stokes equations
Buyang Li, Shu Ma, Katharina Schratz
A new type of low-regularity integrator is proposed for Navier-Stokes equations, coupled with a stabilized finite element method in space. Unlike the other low-regularity integrato…
Uniformly accurate splitting schemes for the Benjamin-Bona-Mahony equation with dispersive parameter
María Cabrera Calvo, Katharina Schratz
We propose a new class of uniformly accurate splitting methods for the Benjamin-Bona-Mahony equation which converge uniformly in the dispersive parameter . The propose…
Time integrators for dispersive equations in the long wave regime
María Cabrera Calvo, Frédéric Rousset, Katharina Schratz
We introduce a novel class of time integrators for dispersive equations which allow us to reproduce the dynamics of the solution from the classical up to long wa…
Uniformly accurate low regularity integrators for the Klein--Gordon equation from the classical to non-relativistic limit regime
María Cabrera Calvo, Katharina Schratz
We propose a novel class of uniformly accurate integrators for the Klein--Gordon equation which capture classical as well as highly-oscillatory non-relativistic regimes $c\gg…
Convergence error estimates at low regularity for time discretizations of KdV
Frédéric Rousset, Katharina Schratz
We consider various filtered time discretizations of the periodic Korteweg--de Vries equation: a filtered exponential integrator, a filtered Lie splitting scheme as well as a filte…
Solving the linear semiclassical Schrödinger equation on the real line
Arieh Iserles, Karolina Kropielnicka, Katharina Schratz +1
The numerical solution of a linear Schrödinger equation in the semiclassical regime is very well understood in a torus . A raft of modern computational methods are pr…
Error estimates at low regularity of splitting schemes for NLS
Alexander Ostermann, Frédéric Rousset, Katharina Schratz
We study a filtered Lie splitting scheme for the cubic nonlinear Schrödinger equation. We establish error estimates at low regularity by using discrete Bourgain spaces. This allows…