arXiv subjects

Xiaofei Zhao

Publications and source records attributed to Xiaofei Zhao.

At least 37 records · Page 2Linked to original sources

arXiv2024

Fourier pseudospectral methods for the spatial variable-order fractional wave equations

In this paper, we propose Fourier pseudospectral methods to solve the variable-order space fractional wave equation and develop an accelerated matrix-free approach for its effective implementation. In constant-order cases, our methods can be efficiently implemented via the (inverse) fast Fourier transforms, and the computational cost at each time step is ${\mathcal O}(N\log N)$ with $N$ the total number of spatial points. However, this fast algorithm fails in the variable-order cases due to the spatial dependence of the Fourier multiplier. On the other hand, the direct matrix-vector multiplication approach becomes impractical due to excessive memory requirements. To address this challenge, we proposed an accelerated matrix-free approach for the efficient computation of variable-order cases. The computational cost is ${\mathcal O}(MN\log N)$ and storage cost ${\mathcal O}(MN)$, where $M \ll N$. Moreover, our method can be easily parallelized to further enhance its efficiency. Numerical studies show that our methods are effective in solving the variable-order space fractional wave equations, especially in high-dimensional cases. Wave propagation in heterogeneous media is studied in comparison to homogeneous counterparts. We find that wave dynamics in fractional cases become more intricate due to nonlocal interactions. Specifically, dynamics in heterogeneous media are more complex than those in homogeneous media.

math.NA↗

arXiv2023

Error estimate of a quasi-Monte Carlo time-splitting pseudospectral method for nonlinear Schrodinger equation with random potentials

In this paper, we consider the numerical solution of a nonlinear Schrodinger equation with spatial random potential. The randomly shifted quasi-Monte Carlo (QMC) lattice rule combined with the time-splitting pseudospectral discretization is applied and analyzed. The nonlinearity in the equation induces difficulties in estimating the regularity of the solution in random space. By the technique of weighted Sobolev space, we identify the possible weights and show the existence of QMC that converges optimally at the almost-linear rate without dependence on dimensions. The full error estimate of the scheme is established. We present numerical results to verify the accuracy and investigate the wave propagation.

math.NA↗

arXiv2023

Engineering Perovskite Emissions via Optical Quasi-Bound-States-in-the-Continuum

Metal halide perovskite quantum dots (PQDs) have emerged as promising materials due to their exceptional photoluminescence (PL) properties. A wide range of applications could benefit from adjustable luminescence properties, while preserving the physical and chemical properties of the PQDs. Therefore, post-synthesis engineering has gained attention recently, involving the use of ion-exchange or external stimuli, such as extreme pressure, magnetic and electric fields. Nevertheless, these methods typically suffer from spectrum broadening, intensity quenching or yield multiple bands. Alternatively, photonic antennas can modify the radiative decay channel of perovskites via the Purcell effect, with the largest wavelength shift being 8 nm to date, at an expense of 5-fold intensity loss. Here, we present an optical nanoantenna array with polarization-controlled quasi-bound-states-in-the-continuum (q-BIC) resonances, which can engineer and shift the photoluminescence wavelength over a ~39 nm range and confers a 21-fold emission enhancement of FAPbI3 perovskite QDs. The spectrum is engineered in a non-invasive manner via lithographically defined antennas and the pump laser polarization at ambient conditions. Our research provides a path towards advanced optoelectronic devices, such as spectrally tailored quantum emitters and lasers.

physics.optics↗

arXiv2023

Computing the action ground state for the rotating nonlinear Schrödinger equation

We consider the computations of the action ground state for a rotating nonlinear Schrödinger equation. It reads as a minimization of the action functional under the Nehari constraint. In the focusing case, we identify an equivalent formulation of the problem which simplifies the constraint. Based on it, we propose a normalized gradient flow method with asymptotic Lagrange multiplier and establish the energy-decaying property. Popular optimization methods are also applied to gain more efficiency. In the defocusing case, we prove that the ground state can be obtained by the unconstrained minimization. Then the direct gradient flow method and unconstrained optimization methods are applied. Numerical experiments show the convergence and accuracy of the proposed methods in both cases, and comparisons on the efficiency are discussed. Finally, the relation between the action and the energy ground states are numerically investigated.

math.NA↗

arXiv2021

Geometric two-scale integrators for highly oscillatory system: uniform accuracy and near conservations

In this paper, we consider a class of highly oscillatory Hamiltonian systems which involve a scaling parameter $\varepsilon\in(0,1]$. The problem arises from many physical models in some limit parameter regime or from some time-compressed perturbation problems. The solution of the model exhibits rapid temporal oscillations with $\mathcal{O}(1)$-amplitude and $\mathcal{O}(1/\varepsilon)$-frequency, which makes classical numerical methods inefficient. We apply the two-scale formulation approach to the problem and propose two new time-symmetric numerical integrators. The methods are proved to have the uniform second order accuracy for all $\varepsilon$ at finite times and some near-conservation laws in long times. Numerical experiments on a Hénon-Heiles model, a nonlinear Schrödinger equation and a charged-particle system illustrate the performance of the proposed methods over the existing ones.

math.NA↗

arXiv2021

Pseudospectral methods with PML for nonlinear Klein-Gordon equations in classical and non-relativistic regimes

Two different Perfectly Matched Layer (PML) formulations with efficient pseudo-spectral numerical schemes are derived for the standard and non-relativistic nonlinear Klein-Gordon equations (NKGE). A pseudo-spectral explicit exponential integrator scheme for a first-order formulation and a linearly implicit preconditioned finite-difference scheme for a second-order formulation are proposed and analyzed. To obtain a high spatial accuracy, new regularized Bermúdez type absorption profiles are introduced for the PML. It is shown that the two schemes are efficient, but the linearly implicit scheme should be preferred for accuracy purpose when used within the framework of pseudo-spectral methods combined with the regularized Bermúdez type functions. In addition, in the non-relativistic regime, numerical examples lead to the conclusion that the error related to regularized Bermúdez type absorption functions is insensitive to the small parameter $\varepsilon$ involved in the NKGE. The paper ends by a two-dimensional example showing that the strategy extends to the rotating NKGE where the vortex dynamics is very well-reproduced.

math.NA↗

arXiv2020

Embedded exponential-type low-regularity integrators for KdV equation under rough data

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 $γ\ge0$. The proposed ELRIs further reduce the regularity requirement of existing methods so far for optimal convergence. The theoretical results are confirmed by numerical experiments, and comparisons with existing methods illustrate the efficiency of the new methods.

math.NA↗

arXiv2020

Optimal convergence of a second order low-regularity integrator for the KdV equation

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.

math.NA↗

arXiv2020

Numerical integrators for continuous disordered nonlinear Schrödinger equation

In this paper, we consider the numerical solution of the continuous disordered nonlinear Schrödinger equation, which contains a spatial random potential. We address the finite time accuracy order reduction issue of the usual numerical integrators on this problem, which is due to the presence of the random/rough potential. By using the recently proposed low-regularity integrator (LRI) from (33, SIAM J. Numer. Anal., 2019), we show how to integrate the potential term by losing two spatial derivatives. Convergence analysis is done to show that LRI has the second order accuracy in $L^2$-norm for potentials in $H^2$. Numerical experiments are done to verify this theoretical result. More numerical results are presented to investigate the accuracy of LRI compared with classical methods under rougher random potentials from applications.

math.NA↗

arXiv2020

Error estimates of some splitting schemes for charged-particle dynamics under strong magnetic field

In this work, we consider the error estimates of some splitting schemes for the charged-particle dynamics under a strong magnetic field. We first propose a novel energy-preserving splitting scheme with computational cost per step independent from the strength of the magnetic field. Then under the maximal ordering scaling case, we establish for the scheme and in fact for a class of Lie-Trotter type splitting schemes, a uniform (in the strength of the magnetic field) and optimal error bound in the position and in the velocity parallel to the magnetic field. For the general strong magnetic field case, the modulated Fourier expansions of the exact and the numerical solutions are constructed to obtain a convergence result. Numerical experiments are presented to illustrate the error and energy behaviour of the splitting schemes.

math.NA↗

arXiv2019

Uniformly accurate methods for three dimensional Vlasov equations under strong magnetic field with varying direction

In this paper, we consider the three dimensional Vlasov equation with an inhomogeneous, varying direction, strong magnetic field. Whenever the magnetic field has constant intensity, the oscillations generated by the stiff term are periodic. The homogenized model is then derived and several state-of-the-art multiscale methods, in combination with the Particle-In-Cell discretisation, are proposed for solving the Vlasov-Poisson equation. Their accuracy as much as their computational cost remain essentially independent of the strength of the magnetic field. The proposed schemes thus allow large computational steps, while the full gyro-motion can be restored by a linear interpolation in time. In the linear case, extensions are introduced for general magnetic field (varying intensity and direction). Eventually, numerical experiments are exposed to illustrate the efficiency of the methods and some long-term simulations are presented.

math.NA↗

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.

math.NA↗

arXiv2019

A uniformly and optimally accurate method for the Klein-Gordon-Zakharov system in simultaneous high-plasma-frequency and subsonic limit regime

We present a uniformly and optimally accurate numerical method for solving the Klein-Gordon-Zakharov (KGZ) system with two dimensionless parameters $0<ε\le1$ and $0<γ\le 1$, which are inversely proportional to the plasma frequency and the acoustic speed, respectively. In the simultaneous high-plasma-frequency and subsonic limit regime, i.e. $ε<γ\to 0^+$, the KGZ system collapses to a cubic Schrödinger equation, and the solution propagates waves with $O(ε^2)$-wavelength in time and meanwhile contains rapid outgoing initial layers with speed $O(1/γ)$ in space due to the incompatibility of the initial data. By presenting a multiscale decomposition of the KGZ system, we propose a multiscale time integrator Fourier pseduospectral method which is explicit, efficient and uniformly accurate for solving the KGZ system for all $0<ε<γ\leq1$. Numerical results are reported to show the efficiency and accuracy of scheme. Finally, the method is applied to investigate the convergence rates of the KGZ system to its limiting models when $ε<γ\to 0^+$.

math.NA↗

arXiv2018

Uniformly accurate methods for Vlasov equations with non-homogeneous strong magnetic field

In this paper, we consider the numerical solution of highly-oscillatory Vlasov and Vlasov-Poisson equations with non-homogeneous magnetic field. Designed in the spirit of recent uniformly accurate methods, our schemes remain insensitive to the stiffness of the problem, in terms of both accuracy and computational cost. The specific difficulty (and the resulting novelty of our approach) stems from the presence of a non-periodic oscillation, which necessitates a careful ad-hoc reformulation of the equations. Our results are illustrated numerically on several examples.

math.NA↗

arXiv2016

Uniformly accurate numerical schemes for the nonlinear Dirac equation in the nonrelativistic limit regime

We apply the two-scale formulation approach to propose uniformly accurate (UA) schemes for solving the nonlinear Dirac equation in the nonrelativistic limit regime. The nonlinear Dirac equation involves two small scales $\varepsilon$ and $\varepsilon^2$ with $\varepsilon\to0$ in the nonrelativistic limit regime. The small parameter causes high oscillations in time which brings severe numerical burden for classical numerical methods. We transform our original problem as a two-scale formulation and present a general strategy to tackle a class of highly oscillatory problems involving the two small scales $\varepsilon$ and $\varepsilon^2$. Suitable initial data for the two-scale formulation is derived to bound the time derivatives of the augmented solution. Numerical schemes with uniform (with respect to $\varepsilon\in (0,1]$) spectral accuracy in space and uniform first order or second order accuracy in time are proposed. Numerical experiments are done to confirm the UA property.

math.NA↗

arXiv2016

Symmetric high order Gautschi-type exponential wave integrators pseudospectral method for the nonlinear Klein-Gordon equation in the nonrelativistic limit regime

A group of high order Gautschi-type exponential wave integrators (EWIs) Fourier pseudospectral method are proposed and analyzed for solving the nonlinear Klein-Gordon equation (KGE) in the nonrelativistic limit regime, where a parameter $0<\varepsilon\ll1$ which is inversely proportional to the speed of light, makes the solution propagate waves with wavelength $O(\varepsilon^2)$ in time and $O(1)$ in space. With the Fourier pseudospectral method to discretize the KGE in space, we propose a group of EWIs with designed Gautschi's type quadratures for the temporal integrations, which can offer any intended even order of accuracy provided that the solution is smooth enough, while all the current existing EWIs offer at most second order accuracy. The scheme is explicit, time symmetric and rigorous error estimates show the meshing strategy of the proposed method is time step $τ=O(\varepsilon^2)$ and mesh size $h=O(1)$ as $0<\varepsilon\ll1$, which is `optimal' among all classical numerical methods towards solving the KGE directly in the limit regime, and which also distinguish our methods from other high order approaches such as Runge-Kutta methods which require $τ=O(\varepsilon^3)$. Numerical experiments with comparisons are done to confirm the error bound and show the superiority of the proposed methods over existing classical numerical methods.

math.NA↗

arXiv2016

Modulation equations approach for solving vortex and radiation in nonlinear Schrodinger equation

We apply the modulation theory to study the vortex and radiation solution in the two-dimensional nonlinear Schrödinger equation. The full modulation equations which describe the dynamics of the vortex and radiation separately are derived. A general algorithm is proposed to efficiently and accurately find vortices with different values of energy and spin index. The modulation equations are solved by accurate numerical method. Numerical tests and simulations of scattering are given.

math-ph↗

arXiv2016

A modulation equations approach for numerically solving the moving soliton and radiation solutions of NLS

Based on our previous work for solving the nonlinear Schrodinger equation with multichannel dynamics that is given by a localized standing wave and radiation, in this work we deal with the multichannel solution which consists of a moving soliton and radiation. We apply the modulation theory to give a system of ODEs coupled to the radiation term for describing the solution, which is valid for all times. The modulation equations are solved accurately by the proposed numerical method. The soliton and radiation are captured separately in the computation, and they are solved on the translated domain that is moving with them. Thus for a fixed finite physical domain in the lab frame, the multichannel solution can pass through the boundary naturally, which can not be done by imposing any existing boundary conditions. We comment on the differences of this method from the collective coordinates.

nlin.PS↗