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.
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 classi…
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 t…
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 fo…
In this paper, we propose a semi-discrete first-order low regularity exponential-type integrator (LREI) for the ``good" Boussinesq equation. It is shown that the method is convergent linearly in the space $H^r$ for solutions belonging to $H^{r+p(r)}$ where $0\le p(r)\le 1$ is non-increasing with respect to $r$, which means less additional derivatives might be needed when the numerical solution is measured in a more regular space. Particularly, the LREI presents the first-order accuracy in $H^{r}$ with no assumptions of additional derivatives when $r>5/2$. This is the first time to propose a lo…
A fully discrete and fully explicit low-regularity integrator is constructed for the one-dimensional periodic cubic nonlinear Schrödinger equation. The method can be implemented by using fast Fourier transform with $O(N\ln N)$ operations at every time level, and is proved to have an $L^2$-norm error bound of $O(τ\sqrt{\ln(1/τ)}+N^{-1})$ for $H^1$ initial data, without requiring any CFL condition, where $τ$ and $N$ denote the temporal stepsize and the degree of freedoms in the spatial discretisation, respectively.
Construction and analysis of an explicit fourth-order symmetric low-regularity integrator for the semilinear Klein-Gordon equation
School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）；School of Mathematics and Statistics, Northwestern Polytechnical University（西北工业大学数学与统计学院）
For the solution of the cubic nonlinear Schrödinger equation in one space dimension, we propose and analyse a fully discrete low-regularity integrator. The scheme is explicit and can easily be implemented using the fast Fourier transform with a complexity of $\mathcal{O}(N\log N)$ operations per time step, where $N$ denotes the degrees of freedom in the spatial discretisation. We prove that the new scheme provides an $\mathcal{O}(τ^{\frac32γ-\frac12-\varepsilon}+N^{-γ})$ error bound in $L^2$ for any initial data belonging to $H^γ$, $\frac12<γ\leq 1$, where $τ$ denotes the temporal step size. N…
In this paper, we introduce a novel class of embedded exponential-type low-regularity integrators (ELRIs) for solving the KdV equation and establish their optimal convergence results under rough initial data. The schemes are explicit and efficient to implement. By rigorous error analysis, we first show that the ELRI scheme provides the first order accuracy in $H^γ$ for initial data in $H^{γ+1}$ for $γ>\frac12$. Moreover, by adding two more correction terms to the first order scheme, we show a second order ELRI that provides the second order accuracy in $H^γ$ for initial data in $H^{γ+3}$ for $…
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-ord…
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 co…
In this paper, we establish the optimal convergence result of a second order exponential-type integrator from (136, Numer. Math., 2017) for solving the KdV equation under rough initial data. The scheme is explicit and efficient to implement. By rigorous error analysis, we show that the scheme provides the second order accuracy in $H^γ$ for initial data in $H^{γ+4}$ for any $γ\geq0$, where the regularity requirement is lower than the classical methods. The result is confirmed by numerical experiments and comparisons are made with the Strang splitting scheme.
Two exponential wave integrator Fourier pseudospectral (EWI-FP) methods are presented and analyzed for the long-time dynamics of the Dirac equation with small potentials characterized by $\varepsilon \in (0, 1]$ a dimensionless parameter. Based on the (symmetric) exponential wave integrator for temporal derivatives in phase space followed by applying the Fourier pseudospectral discretization for spatial derivatives, the EWI-FP methods are explicit and of spectral accuracy in space and second-order accuracy in time for any fixed $\varepsilon = \varepsilon_0$. Uniform error bounds are rigorously…
In this paper, we analyse a new exponential-type integrator for the nonlinear cubic Schrödinger equation on the $d$ dimensional torus $\mathbb T^d$. The scheme has recently also been derived in a wider context of decorated trees in [Y. Bruned and K. Schratz, arXiv:2005.01649]. It is explicit and efficient to implement. Here, we present an alternative derivation, and we give a rigorous error analysis. In particular, we prove second-order convergence in $H^γ(\mathbb T^d)$ for initial data in $H^{γ+2}(\mathbb T^d)$ for any $γ> d/2$. This improves the previous work in [Knöller, A. Ostermann, and K…
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 nonlinearit…
This article is concerned with the construction and analysis of new time discretizations for the KdV equation on a torus for low-regularity solutions below $H^1$. New harmonic analysis tools, including new averaging approximations to the exponential phase functions, new frequency decomposition techniques, and new trilinear estimates of the KdV operator, are established for the construction and analysis of time discretizations with higher convergence orders under low-regularity conditions. In addition, new techniques are introduced to establish stability estimates of time discretizations under …
Maximum bound principle for Q-tensor gradient flow with low regularity integrators
Beijing Normal University（北京师范大学）；Laboratory of Mathematics and Complex Systems, Ministry of Education and School of Mathematical Sciences, Beijing Normal University（教育部数学与复杂系统重点实验室，北京师范大学数学科学学院）
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 tre…
Uniformly accurate exponential-type integrators for klein-gordon equations with asymptotic convergence to classical splitting schemes in the nonlinear schroedinger limit
Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）；INRIA & ENS Rennes（法国国家信息与自动化研究所与雷恩高等师范学院）
Optimal error bounds on an exponential wave integrator Fourier spectral method for fractional nonlinear Schrödinger equations with low regularity potential and nonlinearity
In this paper, we propose a first-order Fourier integrator for solving the cubic nonlinear Schrödinger equation in one dimension. The scheme is explicit and can be implemented using the fast Fourier transform. By a rigorous analysis, we prove that the new scheme provides the first order accuracy in $H^γ$ for any initial data belonging to $H^γ$, for any $γ>\frac32$. That is, up to some fixed time $T$, there exists some constant $C=C(\|u\|_{L^\infty([0,T]; H^γ)})>0$, such that $$ \|u^n-u(t_n)\|_{H^γ(\mathbb T)}\le C τ, $$ where $u^n$ denotes the numerical solution at $t_n=nτ$. Moreover, the mass…
A uniformly accurate (UA) multiscale time integrator pseudospectral method for the Dirac equation in the nonrelativistic limit regime
National University of Singapore（新加坡国立大学）；Beijing Computational Science Research Center（北京计算科学研究中心）；Purdue University（普渡大学）；Université de Lorraine（洛林大学）；Inria Nancy-Grand Est（南锡大东部国家研究学院）
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.
We study the elliptic equation with a line Dirac delta function as the source term subject to the Dirichlet boundary condition in a two-dimensional domain. Such a line Dirac measure causes different types of solution singularities in the neighborhood of the line fracture. We establish new regularity results for the solution in a class of weighted Sobolev spaces and propose finite element algorithms that approximate the singular solution at the optimal convergence rate. Numerical tests are presented to justify the theoretical findings.
In this paper, we propose an efficient exponential integrator finite element method for solving a class of semilinear parabolic equations in rectangular domains. The proposed method first performs the spatial discretization of the model equation using the finite element approximation with continuous multilinear rectangular basis functions, and then takes the explicit exponential Runge-Kutta approach for time integration of the resulting semi-discrete system to produce fully-discrete numerical solution. Under certain regularity assumptions, error estimates measured in $H^1$-norm are successfull…
In this note we describe how some objects from generalized geometry appear in the qualitative analysis and numerical simulation of mechanical systems. In particular we discuss double vector bundles and Dirac structures. It turns out that those objects can be naturally associated to systems with constraints -- we recall the mathematical construction in the context of so called implicit Lagrangian systems. We explain how they can be used to produce new numerical methods, that we call Dirac integrators. On a test example of a simple pendulum in a gravity field we compare the Dirac integrators wit…
In this work, we provide the first strong convergence result of numerical approximation of a general second order semilinear stochastic fractional order evolution equation involving a Caputo derivative in time of order $α\in(\frac 34, 1)$ and driven by Gaussian and non-Gaussian noises simultaneously more useful in concrete applications. The Gaussian noise considered here is a Hilbert space valued Q-Wiener process and the non-Gaussian noise is defined through compensated Poisson random measure associated to a Lévy process. The linear operator is not necessary self-adjoint. The fractional stocha…
