arXiv subjects

Katharina Schratz

Publications and source records attributed to Katharina Schratz.

At least 19 recordsLinked to original sources

arXiv2026

Computing nonlinear Schrödinger equations with Hermite functions beyond harmonic traps

Hermite basis functions are a classical tool for the spatial discretisation of Schrödinger equations with harmonic potential. In this work, we prove that their favourable stability properties extend to Schrödinger equations without a trap: the free Schrödinger flow is stable in the weighted Sobolev spaces $Σ^k$ which govern the convergence of Hermite spectral methods. This makes the Hermite basis a natural discretisation for a larger class of nonlinear Schrödinger equations posed on the full space $\mathbb{R}^d$, avoiding artificial periodisation and the associated distortion of the dynamics incurred by domain truncation in Fourier methods. Within this framework we provide a rigorous fully discrete convergence analysis of a splitting method for the cubic nonlinear Schrödinger equation. In addition, by combining the Hermite basis with a gauge transform, we introduce a novel, fully explicit, unconditionally stable numerical method for the derivative nonlinear Schrödinger equation. Our theoretical results are supported with numerical examples across various nonlinearities and dimensions, which showcase the accuracy, stability and resulting efficiency of this Hermite basis approach.

math.NA↗

arXiv2026

A splitting scheme for the wave maps equation at low regularity

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 discrete Bourgain spaces, as has proved fruitful for the low regularity analysis of the equation in the continuous setting. An important difficulty here is that the analysis of wave maps at low regularity requires the use of the null structure of the system, this structure thus has to be preserved at the discrete level to get an effective stable low regularity scheme. Since the null structure involves time derivatives, the scheme has to be designed carefully. The presence of time derivatives in the nonlinearity then constitutes the most significant source of numerical error. Nonetheless, we are able to prove convergence of the scheme for all subcritical initial data in $H^s$, $s>d/2$.

math.NA↗

arXiv2025

Low regularity error estimates for the time integration of 2D NLS

A filtered Lie splitting scheme is proposed for the time integration of the cubic nonlinear Schrödinger equation on the two-dimensional torus $\mathbb{T}^2$. The scheme is analyzed in a framework of discrete Bourgain spaces, which allows us to consider initial data with low regularity; more precisely initial data in $H^s(\mathbb{T}^2)$ with $s>0$. In this way, the usual stability restriction to smooth Sobolev spaces with index $s>1$ is overcome. Rates of convergence of order $τ^{s/2}$ in $L^2(\mathbb{T}^2)$ at this regularity level are proved. Numerical examples illustrate that these convergence results are sharp.

math.NA↗

arXiv2025

Low regularity full error estimates for the cubic nonlinear Schrödinger equation

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 scheme in time is considered. This scheme is shown to converge even for initial data with very low regularity. In particular, for data in $H^s(\mathbb T^2)$, where $s>0$, convergence of order $\mathcal O(τ^{s/2}+N^{-s})$ is proved in $L^2$. Here $τ$ denotes the time step size and $N$ the number of Fourier modes considered. The proof of this result is carried out in an abstract framework of discrete Bourgain spaces, the final convergence result, however, is given in $L^2$. The stated convergence behavior is illustrated by several numerical examples.

math.NA↗

arXiv2025

Time-relaxation structure-preserving explicit low-regularity integrators for the nonlinear Schrödinger equation

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 framework. The methods combine a resonance-based scheme for the twisted variable with a dynamically adjusted relaxation parameter that guarantees exact mass conservation. Unlike existing symmetric or structure-preserving low-regularity integrators, which are typically implicit and computationally expensive, the proposed methods are fully explicit, mass-conserving, and well-suited for solutions with low regularity. Furthermore, the schemes can be naturally extended to a broad class of evolution equations exhibiting the structure of strongly continuous contraction semigroups. Numerical results demonstrate the accuracy, robustness, and excellent long-time behavior of the methods under low-regularity conditions.

math.NA↗

arXiv2025

Symmetric resonance based integrators and forest formulae

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 nonlinear dispersive equations at low-regularity. The inclusion of symmetries in the construction of resonance-based schemes presents serious challenges and induces a need for a significant extension of prior approaches to allow for sufficient number of degrees of freedom in the resulting schemes while preserving the favorable low-regularity convergence properties of prior constructions. Motivated by recent work arXiv:2005.01649, we achieve this by introducing a novel formalism based on forest formulae that allows us to encode a wider range of possibilities of iterating Duhamel's formula and interpolatory approximations of lower order parts in the construction of these time-stepping methods. The forest formulae allow for a simple characterisation of symmetric schemes and provides a fascinating algebraic structure in its own right which echo those used in Quantum Field Theory for renormalising Feynman diagrams and those used for the renormalisation of singular SPDEs via the theory of Regularity Structures. Our constructions lead to the development of several new symmetric low regularity integrators that exhibit remarkable structure preservation and convergence properties which are witnessed in numerical experiments.

math.NA↗

arXiv2025

G-Adaptivity: optimised graph-based mesh relocation for finite element methods

We present a novel, and effective, approach to achieve optimal mesh relocation in finite element methods (FEMs). The cost and accuracy of FEMs is critically dependent on the choice of mesh points. Mesh relocation (r-adaptivity) seeks to optimise the mesh geometry to obtain the best solution accuracy at given computational budget. Classical r-adaptivity relies on the solution of a separate nonlinear "meshing" PDE to determine mesh point locations. This incurs significant cost at remeshing, and relies on estimates that relate interpolation- and FEM-error. Recent machine learning approaches have focused on the construction of fast surrogates for such classical methods. Instead, our new approach trains a graph neural network (GNN) to determine mesh point locations by directly minimising the FE solution error from the PDE system Firedrake to achieve higher solution accuracy. Our GNN architecture closely aligns the mesh solution space to that of classical meshing methodologies, thus replacing classical estimates for optimality with a learnable strategy. This allows for rapid and robust training and results in an extremely efficient and effective GNN approach to online r-adaptivity. Our method outperforms both classical, and prior ML, approaches to r-adaptive meshing. In particular, it achieves lower FE solution error, whilst retaining the significant speed-up over classical methods observed in prior ML work.

cs.LG↗

arXiv2025

Fully discrete backward error analysis for the midpoint rule applied to the nonlinear Schroedinger equation

The use of symplectic numerical schemes on Hamiltonian systems is widely known to lead to favorable long-time behaviour. While this phenomenon is thoroughly understood in the context of finite-dimensional Hamiltonian systems, much less is known in the context of Hamiltonian PDEs. In this work we provide the first dimension-independent backward error analysis for a Runge-Kutta-type method, the midpoint rule, which shows the existence of a modified energy for this method when applied to nonlinear Schroedinger equations regardless of the level of spatial discretisation. We use this to establish long-time stability of the numerical flow for the midpoint rule.

math.AP↗

arXiv2025

Resonances and computations

The computation of time dynamics arising in nonlinear time-dependent partial differential equations is an ongoing challenge in numerical analysis, especially once roughness comes into play. Classical numerical schemes in general fail to resolve the oscillatory behaviour in the solution which leads to numerical instabilities and loss of convergence. Dispersive equations, e.g., nonlinear Schrödinger, Korteweg--de Vries and wave equations, thereby pose in particular a big problem as in contrast to the parabolic setting, no strong smoothing can be expected, i.e., if the initial data is rough, the solution stays rough which makes their approximation a delicate task. In this review we give an overview on a new numerical ansatz which aims to tackle the time dynamics of nonlinear dispersive partial differential equations even for very rough data. This is achieved by a resonance analysis and decorated tree formalism that draws its inpiration from the combinatorics used in the theory of regularity structures for solving singular SPDEs. One can hope to see this formalism applied in other contexts for dispersive PDEs and beyond.

math.NA↗

arXiv2025

A fast neural hybrid Newton solver adapted to implicit methods for nonlinear dynamics

The use of implicit time-stepping schemes for the numerical approximation of solutions to stiff nonlinear time-evolution equations brings well-known advantages including, typically, better stability behaviour and corresponding support of larger time steps, and better structure preservation properties. However, this comes at the price of having to solve a nonlinear equation at every time step of the numerical scheme. In this work, we propose a novel deep learning based hybrid Newton's method to accelerate this solution of the nonlinear time step system for stiff time-evolution nonlinear equations. We propose a targeted learning strategy which facilitates robust unsupervised learning in an offline phase and provides a highly efficient initialisation for the Newton iteration leading to consistent acceleration of Newton's method. A quantifiable rate of improvement in Newton's method achieved by improved initialisation is provided and we analyse the upper bound of the generalisation error of our unsupervised learning strategy. These theoretical results are supported by extensive numerical results, demonstrating the efficiency of our proposed neural hybrid solver both in one- and two-dimensional cases.

math.NA↗

arXiv2024

Computing rough solutions of the stochastic nonlinear wave equation

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 conditions pose significant challenges, often leading to a loss of accuracy and reduced computational efficiency in existing methods. In this study, we address these challenges by developing a novel and efficient numerical algorithm specifically designed for computing rough solutions of the stochastic nonlinear wave equation, while significantly relaxing the regularity requirements on the initial data. By leveraging the intrinsic structure of the stochastic nonlinear wave equation and employing advanced tools from harmonic analysis, we construct a time discretization method that achieves robust convergence for initial values $(u^{0}, v^{0}) \in H^γ \times H^{γ-1}$ for all $γ> 0$. Notably, our method attains an improved error rate of $O(τ^{2γ-})$ in one and two dimensions for $γ\in (0, \frac{1}{2}]$, and $O(τ^{\max(γ, 2γ- \frac{1}{2}-)})$ in three dimensions for $γ\in (0, \frac{3}{4}]$, where $τ$ denotes the time step size. These convergence rates surpass those of existing numerical methods under the same regularity conditions, underscoring the advantage of our approach. To validate the performance of our method, we present extensive numerical experiments that demonstrate its superior accuracy and computational efficiency compared to state-of-the-art methods. These results highlight the potential of our approach to enable accurate and efficient simulations of stochastic wave phenomena even in the presence of challenging initial conditions.

math.NA↗

arXiv2024

Bridging the gap: symplecticity and low regularity in Runge-Kutta resonance-based schemes

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 properties of the flow. This is particularly drastic in the case of the Korteweg-de Vries (KdV) equation and the nonlinear Schrödinger equation (NLSE) which are fundamental models in the broad field of dispersive infinite-dimensional Hamiltonian systems, possessing infinitely many conserved quantities, an important property which we wish to capture - at least up to some degree - also on the discrete level. Nowadays, a wide range of structure preserving integrators for Hamiltonian systems are available, however, typically these existing algorithms can only approximate highly regular solutions efficiently. State-of-the-art low-regularity integrators, on the other hand, poorly preserve the geometric structure of the underlying PDE. In this work we introduce a novel framework, so-called Runge-Kutta resonance-based methods, for a large class of dispersive nonlinear equations which incorporate a much larger amount of degrees of freedom than prior resonance-based schemes while featuring similarly favourable low-regularity convergence properties. In particular, for the KdV and NLSE case, we are able to bridge the gap between low regularity and structure preservation by characterising a large class of symplectic (in the Hamiltonian picture) resonance-based methods for both equations that allow for low-regularity approximations to the solution while preserving the underlying geometric structure of the continuous problem on the discrete level.

math.NA↗

arXiv2024

Resonances as a computational tool

A large toolbox of numerical schemes for dispersive equations has been established, based on different discretization techniques such as discretizing the variation-of-constants formula (e.g., exponential integrators) or splitting the full equation into a series of simpler subproblems (e.g., splitting methods). In many situations these classical schemes allow a precise and efficient approximation. This, however, drastically changes whenever non-smooth phenomena enter the scene such as for problems at low regularity and high oscillations. Classical schemes fail to capture the oscillatory nature of the solution, and this may lead to severe instabilities and loss of convergence. In this article we review a new class of resonance-based schemes. The key idea in the construction of the new schemes is to tackle and deeply embed the underlying nonlinear structure of resonances into the numerical discretization. As in the continuous case, these terms are central to structure preservation and offer the new schemes strong properties at low regularity.

math.NA↗

arXiv2023

Numerical integration of Schrödinger maps via the Hasimoto transform

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. In exploiting this nonlinear transform we are able to introduce the first fully explicit unconditionally stable symmetric integrators for the SM equation. Our approach consists of two parts: an integration of the NLS equation followed by the numerical evaluation of the Hasimoto transform. Motivated by the desire to study rough solutions to the SM equation, we also introduce a new symmetric low-regularity integrator for the NLS equation. This is combined with our novel fast low-regularity Hasimoto (FLowRH) transform, based on a tailored analysis of the resonance structures in the Magnus expansion and a fast realisation based on block-Toeplitz partitions, to yield an efficient low-regularity integrator for the SM equation. This scheme in particular allows us to obtain approximations to the SM in a more general regime (i.e. under lower regularity assumptions) than previously proposed methods. The favorable properties of our methods are exhibited both in theoretical convergence analysis and in numerical experiments.

math.NA↗

arXiv2023

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

We introduce a new class of numerical schemes which allow for low regularity approximations to the expectation $ \mathbb{E}(|u_{k}(τ, v^η)|^2)$, where $u_k$ denotes the $k$-th Fourier coefficient of the solution $u$ of the dispersive equation and $ v^η(x) $ the associated random initial data. 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 (see arXiv:2005.01649) 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.

math.PR↗

arXiv2023

Effective highly accurate time integrators for linear Klein-Gordon equations across the scales

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, from slowly varying up to highly oscillatory regimes. Our approach is based on splitting methods tailored to the structure of the input term which allows us to resolve the oscillations in the system uniformly in all frequencies, while the error constant does not grow as the oscillations increase. Numerical experiments highlight our theoretical findings and demonstrate the efficiency of the new schemes.

math.NA↗

arXiv2023

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

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 of the underlying PDE. The main idea thereby lies in treating the zeroth mode exactly within the discretization. For long-time error estimates, we rigorously establish the long-time error bounds of different low-regularity integrators for the nonlinear Schrödinger equation (NLSE) with small initial data characterized by a dimensionless parameter $\varepsilon \in (0, 1]$. We begin with the low-regularity integrator for the quadratic NLSE in which the integral is computed exactly and the improved uniform first-order convergence in $H^r$ is proven at $O(\varepsilon τ)$ for solutions in $H^r$ with $r > 1/2$ up to the time $T_{\varepsilon } = T/\varepsilon $ with fixed $T > 0$. Then, the improved uniform long-time error bound is extended to a symmetric second-order low-regularity integrator in the long-time regime. For the cubic NLSE, we design new non-resonant first-order and symmetric second-order low-regularity integrators which treat the zeroth mode exactly and rigorously carry out the error analysis up to the time $T_{\varepsilon } = T/\varepsilon ^2$. With the help of the regularity compensation oscillation (RCO) technique, the improved uniform error bounds are established for the new non-resonant low-regularity schemes, which further reduce the long-time error by a factor of $\varepsilon^2$ compared with classical low-regularity integrators for the cubic NLSE. Numerical examples are presented to validate the error estimates and compare with the classical time-splitting methods in the long-time simulations.

math.NA↗

arXiv2022

Improved uniform error bounds on a Lawson-type exponential integrator for the long-time dynamics of sine--Gordon equation

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 where the amplitude of the initial data is $O(\varepsilon)$ with $0 < \varepsilon \ll 1$ a dimensionless parameter up to the time at $O(1/\varepsilon^2)$. The numerical scheme combines a Lawson-type exponential integrator in time with a Fourier pseudospectral method for spatial discretization, which is fully explicit and efficient in practical computation thanks to the fast Fourier transform. By separating the linear part from the sine function and employing the regularity compensation oscillation (RCO) technique which is introduced to deal with the polynomial nonlinearity by phase cancellation, we carry out the improved error bounds for the semi-discreization at $O(\varepsilon^2τ)$ instead of $O(τ)$ according to classical error estimates and at $O(h^m+\varepsilon^2τ)$ for the full-discretization up to the time $T_{\varepsilon} = T/\varepsilon^2$ with $T>0$ fixed. This is the first work to establish the improved uniform error bound for the long-time dynamics of the NKGE with non-polynomial nonlinearity. The improved error bound is extended to an oscillatory sine-Gordon equation with $O(\varepsilon^2)$ wavelength in time and $O(\varepsilon^{-2})$ wave speed, which indicates that the temporal error is independent of $\varepsilon$ when the time step size is chosen as $O(\varepsilon^2)$. Finally, numerical examples are shown to confirm the improved error bounds and to demonstrate that they are sharp.

math.NA↗