arXiv subjects

Katharina Schratz

Publications and source records attributed to Katharina Schratz.

At least 37 records · Page 2Linked to original sources

arXiv2022

Low regularity integrators for semilinear parabolic equations with maximum bound principles

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 properties of such equations include maximum bound principle (MBP) and energy dissipation law; for the former, that means the absolute value of the solution is pointwisely bounded for all the time by some constant imposed by appropriate initial and boundary conditions. The model equation is first discretized in space by the central finite difference, then by iteratively using Duhamel's formula, first- and second-order low regularity integrators (LRIs) are constructed for time discretization of the semi-discrete system. The proposed LRI schemes are proved to preserve the MBP and the energy stability in the discrete sense. Furthermore, their temporal error estimates are also successfully derived under a low regularity requirement that the exact solution of the semi-discrete problem is only assumed to be continuous in time. Numerical results show that the proposed LRI schemes are more accurate and have better convergence rates than classic exponential time differencing schemes, especially when the interfacial parameter approaches zero.

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arXiv2022

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

The numerical approximation of the semilinear Klein--Gordon equation in the $d$-dimensional space, with $d=1,2,3$, is studied by analyzing the consistency errors in approximating the solution. By discovering and utilizing a new cancellation structure in the semilinear Klein--Gordon equation, a low-regularity correction of the Lie splitting method is constructed, which can have second-order convergence in the energy space under the regularity condition $(u,\partial_tu)\in L^\infty(0,T;H^{1+\frac{d}{4}}\times H^{\frac{d}{4}})$, where $d=1,2,3$ denotes the dimension of space. In one dimension, the proposed method is shown to have a convergence order arbitrarily close to $\frac53$ in the energy space for solutions in the same space, i.e. no additional regularity in the solution is required. Rigorous error estimates are presented for a fully discrete spectral method with the proposed low-regularity time-stepping scheme. Numerical examples are provided to support the theoretical analysis and to illustrate the performance of the proposed method in approximating both nonsmooth and smooth solutions of the semilinear Klein--Gordon equation.

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arXiv2022

Low regularity integrators via decorated trees

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 problems, as well as dispersive equations, up to arbitrary high order on general domains. The structure of the local error of the new schemes is driven by nested commutators which in general require (much) lower regularity assumptions than classical methods do. Our main idea lies in embedding the central oscillations of the nonlinear PDE into the numerical discretisation. The latter is achieved by a novel decorated tree formalism inspired by singular SPDEs with Regularity Structures and allows us to control the nonlinear interactions in the system up to arbitrary high order on the infinite dimensional (continuous) as well as finite dimensional (discrete) level.

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arXiv2022

Solving the linear semiclassical Schrödinger equation on the real line

The numerical solution of a linear Schrödinger equation in the semiclassical regime is very well understood in a torus $\mathbb{T}^d$. A raft of modern computational methods are precise and affordable, while conserving energy and resolving high oscillations very well. This, however, is far from the case with regard to its solution in $\mathbb{R}^d$, a setting more suitable for many applications. In this paper we extend the theory of splitting methods to this end. The main idea is to derive the solution using a spectral method from a combination of solutions of the free Schrödinger equation and of linear scalar ordinary differential equations, in a symmetric Zassenhaus splitting method. This necessitates detailed analysis of certain orthonormal spectral bases on the real line and their evolution under the free Schrödinger operator.

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arXiv2022

Uniformly accurate low regularity integrators for the Klein--Gordon equation from the classical to non-relativistic limit regime

We propose a novel class of uniformly accurate integrators for the Klein--Gordon equation which capture classical $c=1$ as well as highly-oscillatory non-relativistic regimes $c\gg1$ and, at the same time, allow for low regularity approximations. In particular, the schemes converge with order $τ$ and $τ^2$, respectively, under lower regularity assumptions than classical schemes, such as splitting or exponential integrator methods, require. The new schemes in addition preserve the nonlinear Schrödinger (NLS) limit on the discrete level. More precisely, we will design our schemes in such a way that in the limit $c\to \infty$ they converge to a recently introduced class of low regularity integrators for NLS.

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arXiv2021

Resonance based schemes for dispersive equations via decorated trees

We introduce a numerical framework for dispersive equations embedding their underlying resonance structure into the discretisation. This will allow us to resolve the nonlinear oscillations of the PDE and to approximate with high order accuracy a large class of equations under lower regularity assumptions than classical techniques require. The key idea to control the nonlinear frequency interactions in the system up to arbitrary high order thereby lies in a tailored decorated tree formalism. Our algebraic structures are close to the ones developed for singular SPDEs with Regularity Structures. We adapt them to the context of dispersive PDEs by using a novel class of decorations {which encode the dominant frequencies}. The structure proposed in this paper is new and gives a variant of the Butcher-Connes-Kreimer Hopf algebra on decorated trees. We observe a similar Birkhoff type factorisation as in SPDEs and perturbative quantum field theory. This factorisation allows us to single out oscillations and to optimise the local error by mapping it to the particular regularity of the solution. This use of the Birkhoff factorisation seems new in comparison to the literature. The field of singular SPDEs took advantage of numerical methods and renormalisation in perturbative quantum field theory by extending their structures via the adjunction of decorations and Taylor expansions. Now, through this work, Numerical Analysis is taking advantage of these extended structures and provides a new perspective on them.

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arXiv2021

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

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 integrators for nonlinear dispersive equations, which are all fully explicit in time, the proposed method is semi-implicit in time in order to preserve the energy-decay structure of NS equations. First-order convergence of the proposed method is established independent of the viscosity coefficient $μ$, under weaker regularity conditions than other existing numerical methods, including the semi-implicit Euler method and classical exponential integrators. Numerical results show that the proposed method is more accurate than the semi-implicit Euler method in the viscous case $μ=O(1)$, and more accurate than the classical exponential integrator in the inviscid case $μ\rightarrow 0$.

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arXiv2021

Convergence, error analysis and longtime behavior of the Scalar Auxiliary Variable method for the nonlinear Schrödinger equation

We carry out the convergence analysis of the Scalar Auxiliary Variable (SAV) method applied to the nonlinear Schrödinger equation which preserves a modified Hamiltonian on the discrete level. We derive a weak and strong convergence result, establish second-order global error bounds and present long time error estimates on the modified Hamiltonian. In addition, we illustrate the favorable energy conservation of the SAV method compared to classical splitting schemes in certain applications.

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arXiv2021

Time integrators for dispersive equations in the long wave regime

We introduce a novel class of time integrators for dispersive equations which allow us to reproduce the dynamics of the solution from the classical $ \varepsilon = 1$ up to long wave limit regime $ \varepsilon \ll 1 $ on the natural time scale of the PDE $t= \mathcal{O}(\frac{1}{\varepsilon})$. Most notably our new schemes converge with rates at order $τ\varepsilon$ over long times $t= \frac{1}{\varepsilon}$.

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arXiv2021

Uniformly accurate splitting schemes for the Benjamin-Bona-Mahony equation with dispersive parameter

We propose a new class of uniformly accurate splitting methods for the Benjamin-Bona-Mahony equation which converge uniformly in the dispersive parameter $\varepsilon$. The proposed splitting schemes are furthermore asymptotic convergent and preserve the KdV limit. We carry out a rigorous convergence analysis of the splitting schemes exploiting the smoothing properties in the system. This will allow us to establish improved error bounds with gain either in regularity (for non smooth solutions) or in the dispersive parameter $\varepsilon$. The latter will be interesting in regimes of a small dispersive parameter. We will in particular show that in the classical BBM case $P(\partial_x) = \partial_x$ our Lie splitting does not require any spatial regularity, i.e, first order time convergence holds in $H^{r}$ for solutions in $H^{r}$ without any loss of derivative. This estimate holds uniformly in $\varepsilon$. In regularizing regimes $\varepsilon=\mathcal{O}(1) $ we even gain a derivative with our time discretisation at the cost of loosing in terms of $\frac{1}{\varepsilon}$. Numerical experiments underline our theoretical findings.

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arXiv2021

Convergence error estimates at low regularity for time discretizations of KdV

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 filtered resonance based discretisation and establish convergence error estimates at low regularity. Our analysis is based on discrete Bourgain spaces and allows to prove convergence in $L^2$ for rough data $u_{0} \in H^s,$ $s>0$ with an explicit convergence rate.

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arXiv2021

A general framework of low regularity integrators

We introduce a new general framework for the approximation of evolution equations at low regularity and develop a new class of schemes for a wide range of equations under lower regularity assumptions than classical methods require. In contrast to previous works, our new framework allows a unified practical formulation and the construction of the new schemes does not rely on any Fourier based expansions. This allows us for the first time to overcome the severe restriction to periodic boundary conditions, to embed in the same framework parabolic and dispersive equations and to handle nonlinearities that are not polynomial. In particular, as our new formalism does no longer require periodicity of the problem, one may couple the new time discretisation technique not only with spectral methods, but rather with various spatial discretisations. We apply our general theory to the time discretization of various concrete PDEs, such as the nonlinear heat equation, the nonlinear Schrödinger equation, the complex Ginzburg-Landau equation, the half wave and Klein--Gordon equations, set in $Ω\subset \mathbb{R}^d$, $d \leq 3$ with suitable boundary conditions.

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arXiv2020

Error estimates at low regularity of splitting schemes for NLS

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 us to handle data in $H^s$ with $0 1/2$ . More precisely, we prove convergence rates of order $τ^{s/2}$ in $L^2$ at this level of regularity.

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arXiv2020

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

In this paper, we propose a new scheme for the integration of the periodic nonlinear Schrödinger equation and rigorously prove convergence rates at low regularity. The new integrator has decisive advantages over standard schemes at low regularity. In particular, it is able to handle initial data in $H^s$ for $0 < s\le 1$. The key feature of the integrator is its ability to distinguish between low and medium frequencies in the solution and to treat them differently in the discretization. This new approach requires a well-balanced filtering procedure which is carried out in Fourier space. The convergence analysis of the proposed scheme is based on discrete (in time) Bourgain space estimates which we introduce in this paper. A numerical experiment illustrates the superiority of the new integrator over standard schemes for rough initial data.

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arXiv2019

Low-regularity integrators for nonlinear Dirac equations

In this work, we consider the numerical integration of the nonlinear Dirac equation and the Dirac-Poisson system (NDEs) under rough initial data. We propose a ultra low-regularity integrator (ULI) for solving the NDEs which enables optimal first-order time convergence in $H^r$ for solutions in $H^{r}$, i.e., without requiring any additional regularity on the solution. In contrast to classical methods, ULI overcomes the numerical loss of derivatives and is therefore more efficient and accurate for approximating low regular solutions. Convergence theorems and the extension of ULI to second order are established. Numerical experiments confirm the theoretical results and underline the favourable error behaviour of the new method at low regularity compared to classical integration schemes.

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arXiv2019

A Fourier integrator for the cubic nonlinear Schrödinger equation with rough initial data

Standard numerical integrators suffer from an order reduction when applied to nonlinear Schrödinger equations with low-regularity initial data. For example, standard Strang splitting requires the boundedness of the solution in $H^{r+4}$ in order to be second-order convergent in $H^r$, i.e., it requires the boundedness of four additional derivatives of the solution. We present a new type of integrator that is based on the variation-of-constants formula and makes use of certain resonance based approximations in Fourier space. The latter can be efficiently evaluated by fast Fourier methods. For second-order convergence, the new integrator requires two additional derivatives of the solution in one space dimension, and three derivatives in higher space dimensions. Numerical examples illustrating our convergence results are included. These examples demonstrate the clear advantage of the Fourier integrator over standard Strang splitting for initial data with low regularity.

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arXiv2019

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

We present a new filtered low-regularity Fourier integrator for the cubic nonlinear Schrödinger equation based on recent time discretization and filtering techniques. For this new scheme, we perform a rigorous error analysis and establish better convergence rates at low regularity than known for classical schemes in the literature so far. In our error estimates, we combine the better local error properties of the new scheme with a stability analysis based on general discrete Strichartz-type estimates. The latter allow us to handle a much rougher class of solutions as the error analysis can be carried out directly at the level of $L^2$ compared to classical results \black in dimension $d$, \black which are limited to higher-order (sufficiently smooth) Sobolev spaces $H^s$ with $s>d/2$. In particular, we are able to establish a global error estimate in $L^2$ for $H^1$ solutions which is roughly of order $τ^{ {1\over 2} + { 5-d \over 12} }$ in dimension $d \leq 3$ ($τ$ denoting the time discretization parameter). This breaks the "natural order barrier" of $τ^{1/2}$ for $H^1$ solutions which holds for classical numerical schemes (even in combination with suitable filter functions).

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arXiv2018

Uniformly accurate oscillatory integrators for the Klein-Gordon-Zakharov system from low- to high-plasma frequency regimes

We present a novel class of oscillatory integrators for the Klein-Gordon-Zakharov system which are uniformly accurate with respect to the plasma frequency $c$. Convergence holds from the slowly-varying low-plasma up to the highly oscillatory high-plasma frequency regimes without any step size restriction and, especially, uniformly in $c$. The introduced schemes are moreover asymptotic consistent and approximates the solutions of the corresponding Zakharov limit system in the high-plasma frequency limit ($c \to \infty$). We in particular present the construction of the first- and second-order uniformly accurate oscillatory integrators and establish rigorous, uniform error estimates. Numerical experiments underline our theoretical convergence results.

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