arXiv subjects

Lun Ji

Publications and source records attributed to Lun Ji.

10 recordsLinked to original sources

arXiv2026

Explicit Fourier Integrator for the Periodic dNLS via Gauge Transformation: Low-Regularity Estimates in Discrete Bourgain Spaces

The derivative nonlinear Schr\"odinger equation is a fundamental model for the propagation of nonlinear dispersive waves in, for example, plasma physics and nonlinear optics. In this work, we consider this model on the one-dimensional torus and study a filtered explicit Fourier integrator for the corresponding periodic problem. After applying a periodic gauge transformation, we consider a frequency-truncated model and its filtered exponential-Euler discretization. The main difficulty comes from the derivative cubic nonlinearity in the periodic setting, since local smoothing is unavailable and resonant interactions are stronger than in the non-periodic case. To address this issue, we develop a discrete Bourgain-space framework adapted to the gauge-transformed equation. For initial data $u_0 \in H^s(\mathbb{T})$ with $1/2 < s \le 5/2$, we prove that the numerical error is of order $\mathcal{O}(\tau^{s/2-1/4})$ in $H^{1/2}(\mathbb{T})$, where $\tau$ denotes the employed time step size. Numerical experiments confirm the predicted convergence behavior and demonstrate the effectiveness of the filtered scheme for rough solutions.

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arXiv2026

A low regularity exponential-type integrator for the derivative nonlinear Schrödinger equation

In this work, we present a first-order unfiltered exponential integrator for the one-dimensional derivative nonlinear Schrödinger equation with low regularity. Our analysis shows that for any $s>\frac12$, the method converges with first-order in $H^s(\mathbb{T})$ for initial data $u_0\in H^{s+1}(\mathbb{T})$. Moreover, we constructed a symmetrized version of this method that performs better in terms of both global error and conservation behavior. To the best of our knowledge, these are the first low regularity integrators for the derivative nonlinear Schrödinger equation. Numerical experiments illustrate our theoretical findings.

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arXiv2026

Low-regularity error estimates of a filtered Lie-Trotter splitting scheme for the Zakharov system in arbitrary dimensions

In this paper, we establish error estimates for a fully discrete, filtered Lie splitting scheme applied directly to the Zakharov system -- a model whose solutions may exhibit extremely low regularity in arbitrary dimensions. Remarkably, we find that the scheme exhibits an \emph{approximately structure-preserving} behavior in the fully discrete setting. Our error analysis relies on multilinear estimates developed within the framework of discrete Bourgain spaces. Specifically, we prove that if the exact solution $(E,z,z_t)$ belongs to $H^{s+r+1/2}\times H^{s+r}\times H^{s+r-1}$, then the numerical error measured in the norm $H^{r+1/2}\times H^{r}\times H^{r-1}$ is of order $\mathcal{O}(τ^{s/2}+N^{-s})$ for $s\in(0,2]$, where $r=\max(0,\tfrac d2-1)$ and $N$ denotes the number of spatial grid points. To the best of our knowledge, this is the first rigorous error estimate for splitting methods applied directly to the original Zakharov system -- without introducing auxiliary variables for reformulating the equations. Such reformulations typically compromise the system's intrinsic geometric structure, whereas our approach preserves it approximately by operating on the system in its native form. Finally, we present numerical experiments that corroborate and illustrate the theoretical convergence rates.

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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.

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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.

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arXiv2025

Error estimates of time-splitting schemes for nonlinear Klein--Gordon equation with rough data

In this work, we consider the convergence analysis of time-splitting schemes for the nonlinear Klein--Gordon/wave equation under rough initial data. The optimal error bounds of the Lie splitting and the Strang splitting are established with sharp dependence on the regularity index of the solution from a wide range that is approaching the lower bound for well-posedness. Particularly for very rough data, the technique of discrete Bourgain space is utilized and developed, which can apply for general second-order wave models. Numerical verifications are provided.

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arXiv2024

Filtered Lie-Trotter splitting for the "good" Boussinesq equation: low regularity error estimates

We investigate a filtered Lie-Trotter splitting scheme for the ``good" Boussinesq equation and derive an error estimate for initial data with very low regularity. Through the use of discrete Bourgain spaces, our analysis extends to initial data in $H^{s}$ for $0 1/2$ imposed by the bilinear estimate in smooth Sobolev spaces. We establish convergence rates of order $τ^{s/2}$ in $L^2$ for such levels of regularity. Our analytical findings are supported by numerical experiments.

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arXiv2023

Low regularity error estimates for high dimensional nonlinear Schrödinger equations

The filtered Lie splitting scheme is an established method for the numerical integration of the periodic nonlinear Schrödinger equation at low regularity. Its temporal convergence was recently analyzed in a framework of discrete Bourgain spaces in one and two space dimensions for initial data in $H^s$ with $0<s\leq 2$. Here, this analysis is extended to dimensions $d=3, 4, 5$ for data satisfying $d/2-1 < s \leq 2$. In this setting, convergence of order $s/2$ in $L^2$ is proven. Numerical examples illustrate these convergence results.

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arXiv2022

Explicit K-symplectic methods for nonseparable non-canonical Hamiltonian systems

We propose efficient numerical methods for nonseparable non-canonical Hamiltonian systems which are explicit, K-symplectic in the extended phase space with long time energy conservation properties. They are based on extending the original phase space to several copies of the phase space and imposing a mechanical restraint on the copies of the phase space. Explicit K-symplectic methods are constructed for three non-canonical Hamiltonian systems. Numerical results show that they outperform the higher order Runge-Kutta methods in preserving the phase orbit and the energy of the system over long time.

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arXiv2022

Poisson Integrators based on splitting method for Poisson systems

We propose Poisson integrators for the numerical integration of separable Poisson systems. We analyze three situations in which the Poisson systems are separated in three ways and the Poisson integrators can be constructed by using the splitting method. Numerical results show that the Poisson integrators outperform the higher order non-Poisson integrators in phase orbit tracking, long-term energy conservation and efficiency.

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