Xiaofei Zhao

Weizhu Bao, Xuanchun Dong, Xiaofei Zhao

In this paper, two multiscale time integrators (MTIs), motivated from two types of multiscale decomposition by either frequency or frequency and amplitude, are proposed and analyzed for solving highly oscillatory second order differential equations with a dimensionless parameter $0<\varepsilon\le1$. In fact, the solution to this equation propagates waves with wavelength at $O(\varepsilon^2)$ when $0<\varepsilon\ll 1$, which brings significantly numerical burdens in practical computation. We rigorously establish two independent error bounds for the two MTIs at $O(τ^2/\varepsilon^2)$ and $O(\varepsilon^2)$ for $\varepsilon\in(0,1]$ with $τ>0$ as step size, which imply that the two MTIs converge uniformly with linear convergence rate at $O(τ)$ for $\varepsilon\in(0,1]$ and optimally with quadratic convergence rate at $O(τ^2)$ in the regimes when either $\varepsilon=O(1)$ or $0<\varepsilon\le τ$. Thus the meshing strategy requirement (or $\varepsilon$-scalability) of the two MTIs is $τ=O(1)$ for $0<\varepsilon\ll 1$, which is significantly improved from $τ=O(\varepsilon^3)$ and $τ=O(\varepsilon^2)$ requested by finite difference methods and exponential wave integrators to the equation, respectively. Extensive numerical tests and comparisons with those classical numerical integrators are reported, which gear towards better understanding on the convergence and resolution properties of the two MTIs. In addition, numerical results support the two error bounds very well.

Philippe Chartier, Nicolas Crouseilles, Mohammed Lemou et al.

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.

Mohammed Lemou, Florian Méhats, Xiaofei Zhao

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.

Yan Wang, Xiaofei Zhao

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.

Wei Liu, Tingfeng Wang, Yongjun Yuan et al.

This work focuses on the numerical computation of defocusing action ground states for rotating nonlinear Schrödinger equations (RNLS) using a direct gradient flow (DGF) method. We address theoretical gaps in the existing literature concerning the stability and convergence of this DGF scheme. Firstly, we prove the unconditional stability of the DGF scheme, demonstrating that the action functional is monotonically non-increasing along the discrete flow for arbitrary time step sizes. Secondly, we establish a rigorous convergence analysis, proving global convergence under minor assumptions and local exponential convergence to the action ground state under a reasonable non-degeneracy condition. The analysis relies on the uniform boundedness of sublevel sets of the action functional and introduces a tailored $H^1$-distance between phase-shift equivalence classes to handle complex-valued ground states with quantized vortices. A novel analytical framework is also developed to establish the exponential convergence rate. Numerical experiments are presented to validate the theoretical findings, demonstrating both the global migration towards a neighborhood of the ground state and subsequent exponential convergence.

Zhangyong Liang, Zhiping Mao, Xiaofei Zhao

In this paper, we propose a neural multiscale decomposition method (NeuralMD) for solving the nonlinear Klein-Gordon equation (NKGE) with a dimensionless parameter $\varepsilon\in(0,1]$ from the relativistic regime to the nonrelativistic limit regime. The solution of the NKGE propagates waves with wavelength at $O(1)$ and $O(\varepsilon^2)$ in space and time, respectively, which brings the oscillation in time. Existing collocation-based methods for solving this equation lead to spectral bias and propagation failure. To mitigate the spectral bias induced by high-frequency time oscillation, we employ a multiscale time integrator (MTI) to absorb the time oscillation into the phase. This decomposes the NKGE into a nonlinear Schrödinger equation with wave operator (NLSW) with well-prepared initial data and a remainder equation with small initial data. As $\varepsilon \to 0$, the NKGE converges to the NLSW at rate $O(\varepsilon^{2})$, and the contribution of the remainder equation becomes negligible. Furthermore, to alleviate propagation failure caused by medium-frequency time oscillation, we propose a gated gradient correlation correction strategy to enforce temporal coherence in collocation-based methods. As a result, the approximation of the remainder term is no longer affected by propagation failure. Comparative experiments with existing collocation-based methods demonstrate the superior performance of our method for solving the NKGE with various regularities of initial data over the whole regime.

Zhipeng Chang, Zhenye Wen, Xiaofei Zhao

Existing operator learning methods rely on supervised training with high-fidelity simulation data, introducing significant computational cost. In this work, we propose the deep Onsager operator learning (DOOL) method, a novel unsupervised framework for solving dissipative equations. Rooted in the Onsager variational principle (OVP), DOOL trains a deep operator network by directly minimizing the OVP-defined Rayleighian functional, requiring no labeled data, and then proceeds in time explicitly through conservation/change laws for the solution. Another key innovation here lies in the spatiotemporal decoupling strategy: the operator's trunk network processes spatial coordinates exclusively, thereby enhancing training efficiency, while integrated external time stepping enables temporal extrapolation. Numerical experiments on typical dissipative equations validate the effectiveness of the DOOL method, and systematic comparisons with supervised DeepONet and MIONet demonstrate its enhanced performance. Extensions are made to cover the second-order wave models with dissipation that do not directly follow OVP.

Chunmei Su, Xiaofei Zhao

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 $ε<γ\to 0^+$.