arXiv subjects

Changjian Xie

Publications and source records attributed to Changjian Xie.

At least 19 recordsLinked to original sources

arXiv2026

Efficient Hermitian and skew-Hermitian splitting methods for linear systems in micromagnetic simulations

For the Landau-Lifshitz equation, the discrete linear systems obtained by our semi-implicit method possess the following properties: they are large-sparse systems with non-Hermitian yet positive-definite coefficient matrices. To solve these systems efficiently, we apply the Hermitian/skew-Hermitian splitting (HSS) method and its inexact variant (IHSS). Numerical experiments in one and three dimensions show that the spectral radius of the HSS iteration remains below its theoretical upper bound and strictly below one for the tested grid resolutions and damping parameters. Moreover, the theoretical bound closely follows the actual spectral radius, providing an accurate estimate of the convergence behavior. The IHSS results demonstrate effective convergence for the tested cases and show that its efficiency is sensitive to the splitting parameter. Overall, the two semi-implicit schemes exhibit comparable convergence behavior.

math.NA↗

arXiv2026

Efficient Multigrid Methods for Semi-implicit Landau-Lifshitz Schemes in Micromagnetic Simulations

An efficient aggregation-based multigrid approach is developed in this work to solve linear algebraic systems generated by discretizing the Landau-Lifshitz equation in micromagnetics. The discretization couples the first-order backward differentiation formula with first-order extrapolation, and the underlying equation describes magnetization dynamics within micromagnetic simulations. For these linear systems, standard iterative methods and conventional multigrid solvers intended for symmetric problems suffer from deteriorated convergence under mesh refinement and poor computational performance. Existing algorithms struggle to handle the system's intrinsic features, namely its sparse spectral properties and non-symmetric matrix structure. To overcome these limitations, a multigrid framework is constructed, where smoothing and coarse-grid correction operators are customized for the target linear system. Numerical tests confirm the robustness and mesh-independent convergence of the resulting method. Compared with well-established Krylov-subspace solvers and conventional multigrid techniques, the aggregation-based multigrid solver cuts down iteration counts and overall computational cost considerably, while producing faithful representations of magnetization dynamics.

math.NA↗

arXiv2026

Unexpected Collisional Rotational Excitation via Long-Range Capture and Orbiting

Collisional rotational excitation is a fundamental process in many gaseous environments. The textbook hard-sphere model stipulates that high rotational excitation results from head-on collisions, leading primarily to backward scattering, whereas long-range glancing collisions in the forward direction are inefficient for rotational energy transfer. Here, we report rotational state resolved product imaging for a system with strong attractive interaction, the charge-transfer collision between spin-orbit selected Ar+(2P3/2) ions and para/ortho-H2 molecules. Surprisingly, the H2+ products are rotationally excited and dominated by forward scattering, in sharp contrast to conventional wisdom. Quantum dynamical calculations on a first-principles diabatic potential energy matrix reproduce the observations. Trajectory surface hopping analysis further reveals that rotational excitation occurs mostly with large impact parameters, and the captured complex undergoes orbiting motion owing to the strong attractive interaction between the two collision partners before they break up. This novel mechanism should be general for collisional systems featuring strong attractive interactions, which undermine the hard-sphere assumption.

physics.chem-ph↗

arXiv2026

Improved Energy Stable Symmetric Gauss-Seidel Projection Method for Micromagnetics Simulations

The Gauss-Seidel projection method (GSPM) constitutes an efficient and numerically stable numerical framework for micromagnetic simulations of ferromagnetic media. This scheme attains first-order temporal accuracy and second-order spatial accuracy. Fast Fourier transform (FFT) techniques can be incorporated to accelerate both the solution of the arising linear algebraic systems and the evaluation of stray magnetic fields. The conventional GSPM relies on a single-sided Gauss-Seidel iteration, which leverages the latest updated state variables associated with the heat-diffusion subproblem. In this work, we develop a symmetric Gauss-Seidel projection method (SGSPM) that retains first-order temporal accuracy and second-order spatial consistency. The proposed symmetric variant exhibits superior stability properties relative to the standard GSPM. Specifically, SGSPM adopts a two-pass symmetric Gauss-Seidel iteration, where updated information from the heat-diffusion stage is fully exploited to rigorously guarantee discrete energy stability. We validate the performance of the devised scheme through numerical investigations of magnetization dynamic evolution and magnetic domain-wall propagation. Numerical evidence demonstrates that the improved symmetric scheme delivers enhanced stability for capturing magnetization motion dynamics.

math.NA↗

arXiv2026

A Comparative Study of Projected and Unprojected Schemes for Micromagnetic Simulations

In micromagnetic simulations, the constant magnitude of the magnetization can be derived from the continuity equation. Since the time evolution of the magnetization in the continuity equation is perpendicular to the plane determined by the magnetization and the effective field, taking the inner product of both sides of the model with the magnetization shows that the evolution rate of the magnitude of the magnetization is zero, thus keeping the magnitude constant. From this perspective, the equation itself can maintain the constraint of constant magnetization magnitude. We discretized the continuity equation and compared two first-order semi-implicit strategies in time: one is the implicit Gauss-Seidel method, and the other is the semi-implicit Backward Differentiation Formula (BDF) method. We considered the comparison between these two schemes with and without the projection step. The results of micromagnetic simulations show that when the dissipation coefficient is large, the implicit Gauss-Seidel method without the projection step has significant differences from the method with the projection step in both the achieved steady state and domain wall motion. When an appropriate dissipation coefficient is selected, the difference between the two narrows, and both the steady state and domain wall motion can be simulated. For the other method, BDF1, whether the dissipation coefficient is large or small, the results with and without the projection step are quite consistent, and it can effectively simulate the domain wall motion.

cond-mat.mtrl-sci↗

arXiv2026

A New Fractional Step Structure Preserving Method for The Landau-Lifshitz-Gilbert Equation

In this paper, we propose a structure preserving method using a Crank-Nicolson's type method with an implicit Gauss-Seidel fractional iteration. Such a method is of first-order accuracy in time and second-order accuracy in space, stable and length preserving. Such a proposed method brings great benefits for the theoretical analysis. The numerical accuracy, norm preserving and stability are verified for 1D and 3D tests.

math.NA↗

arXiv2026

Semi-implicit Structure Preserving Method for The Landau-Lifshitz Equation

A critical challenge inherent to the projection method applied to the Landau-Lifshitz equation is the deficiency of rigorous theoretical justifications for the stability of its projection step. To mitigate this limitation, we introduce a semi-implicit numerical scheme, which is formulated on the basis of the first-order Backward Differentiation Formula (BDF) incorporated with one-sided extrapolation and a Crank-Nicolson-type norm-preserving procedure. This proposed scheme exhibits three fundamental characteristics: structure preservation, numerical stability, and first-order accuracy in time. In practical implementations, the scheme not only ensures stable computation and adheres to the norm constraint but also guarantees the uniqueness of the numerical solution, thereby providing substantial facilitation for the theoretical analysis of the normalizing step.

math.NA↗

arXiv2026

An Efficient Energy Stable Structure Preserving Method for The Landau-Lifshitz Equation

One of the main difficulties in micromagnetics simulation is the norm preserving constraints $\|\mathbf{m}\|=1$ at the continuous or the discrete level. Another difficulty is the stability with the time step constraint. Using standard explicit integrators leads to a physical time step of sub-pico seconds, which is often two orders of magnitude smaller than the fastest physical time scales. Direct implicit integrators require solving complicated, coupled systems. Another major difficulty with the projection method in this field is the lack of rigorous theoretical guarantees regarding its stability of the projection step. In this paper, we introduce a first order method. Such a method is structure preserving based on a combination of a Gauss-Seidel iteration, a double diffusion iteration and a Crank-Nicolson iteration to preserve the norm constraints.

math.NA↗

arXiv2025

Hardware-Software Collaborative Computing of Photonic Spiking Reinforcement Learning for Robotic Continuous Control

Robotic continuous control tasks impose stringent demands on the energy efficiency and latency of computing architectures due to their high-dimensional state spaces and real-time interaction requirements. Conventional electronic computing platforms face computational bottlenecks, whereas the fusion of photonic computing and spiking reinforcement learning (RL) offers a promising alternative. Here, we propose a novel computing architecture based on photonic spiking RL, which integrates the Twin Delayed Deep Deterministic policy gradient (TD3) algorithm with spiking neural network (SNN). The proposed architecture employs an optical-electronic hybrid computing paradigm wherein a silicon photonic Mach-Zehnder interferometer (MZI) chip executes linear matrix computations, while nonlinear spiking activations are performed in the electronic domain. Experimental validation on the Pendulum-v1 and HalfCheetah-v2 benchmarks demonstrates the system capability for software-hardware co-inference, achieving a control policy reward of 5831 on HalfCheetah-v2, a 23.33% reduction in convergence steps, and an action deviation below 2.2%. Notably, this work represents the first application of a programmable MZI photonic computing chip to robotic continuous control tasks, attaining an energy efficiency of 1.39 TOPS/W and an ultralow computational latency of 120 ps. Such performance underscores the promise of photonic spiking RL for real-time decision-making in autonomous and industrial robotic systems.

cs.RO↗

arXiv2025

A novel third-order accurate and stable scheme for micromagnetic simulations

High-fidelity numerical simulation serves as a cornerstone for exploring magnetization dynamics in micromagnetics. This work introduces a novel third-order temporally accurate and stable numerical scheme for the Landau-Lifshitz-Gilbert (LLG) equation, aiming to address the limitations in accuracy and efficiency often encountered with conventional approaches. Validation via nanostrip simulations confirms two principal advantages of the proposed method: it attains strict third-order temporal accuracy, surpassing many current techniques, and it offers superior computational efficiency, enabling rapid convergence without sacrificing numerical precision. For Gilbert damping coefficients $α$ ranging from $0.1$ to values below $10$, the scheme preserves strong stability and effectively avoids non-physical magnetization states. The magnetic microstructures predicted by this method are in excellent agreement with those from established benchmark methods, affirming its reliability for quantitative physical analysis. Salient distinctions between the proposed scheme and an existing third-order semi-implicit method include: (1) Solving the linear system associated with the existing scheme demands substantially greater computational time, underscoring the need for highly efficient solvers; (2) Although the proposed method shows increased sensitivity to damping parameters, it reliably converges to stable physical states and is effective in simulating magnetic domain wall motion, producing outcomes consistent with prior validated studies; (3) The energy levels computed by the proposed method are significantly lower than those obtained via the existing third-order scheme.

math-ph↗

arXiv2025

Enhancing Micromagnetics Simulations with a Third-Order Semi-Implicit Projection Method

Micromagnetics depends on high-fidelity numerical methods for magnetization dynamics. This work proposes a third-order temporal accuracy scheme for the Landau-Lifshitz-Gilbert equation, addressing accuracy-efficiency trade-offs in existing methods. Validated via nanostrip simulations (representative of real devices), the scheme offers two key advantages: rigorous third-order accuracy (surpassing existing simulation methods) and higher computational efficiency, ensuring fast convergence without precision loss. It maintains stability for Gilbert damping $α$ from $0.1$ to $10$, avoiding non-physical states. The magnetic microstructures it captures are consistent with established methods, confirming reliability for physical analysis.

math-ph↗

arXiv2025

Convergence analysis of a third order semi-implicit projection method for Landau-Lifshitz-Gilbert equation

The convergence analysis of a third-order scheme for the highly nonlinear Landau-Lifshitz-Gilbert equation with a non-convex constraint is considered. In this paper, we first present a fully discrete semi-implicit method for solving the Landau-Lifshitz-Gilbert equation based on the third-order backward differentiation formula and the one-sided extrapolation (using previous time-step numerical values). A projection step is further used to preserve the length of the magnetization. We provide a rigorous convergence analysis for the fully discrete numerical solution by the introduction of two sets of approximated solutions where one set of solutions solves the Landau-Lifshitz-Gilbert equation and the other is projected onto the unit sphere. Third-order accuracy in time and fourth order accuracy in space is obtained provided that the spatial step-size is the same order as the temporal step-size and slightly large damping parameter $α$ (greater than $\sqrt{2}/2$). And also, the unique solvability of the numerical solution without any assumption for the step-size in both time and space is theoretically justified, using a monotonicity analysis. All these theoretical results are guaranteed by numerical examples in both 1D and 3D spaces.

math.NA↗

arXiv2025

Efficient And Stable Third-order Method for Micromagnetics Simulations

To address the magnetization dynamics in ferromagnetic materials described by the Landau-Lifshitz-Gilbert equation under large damping parameters, a third-order accurate numerical scheme is developed by building upon a second-order method \cite{CaiChenWangXie2022} and leveraging its efficiency. This method boasts two key advantages: first, it only involves solving linear systems with constant coefficients, enabling the use of fast solvers and thus significantly enhancing numerical efficiency over existing first or second-order approaches. Second, it achieves third-order temporal accuracy and fourth-order spatial accuracy, while being unconditionally stable for large damping parameters. Numerical tests in 1D and 3D scenarios confirm both its third-order accuracy and efficiency gains. When large damping parameters are present, the method demonstrates unconditional stability and reproduces physically plausible structures. For domain wall dynamics simulations, it captures the linear relationship between wall velocity and both the damping parameter and external magnetic field, outperforming lower-order methods in this regard.

math.NA↗

arXiv2025

Error Analysis of Third-Order in Time and Fourth-Order Linear Finite Difference Scheme for Landau-Lifshitz-Gilbert Equation under Large Damping Parameters

This work proposes and analyzes a fully discrete numerical scheme for solving the Landau-Lifshitz-Gilbert (LLG) equation, which achieves fourth-order spatial accuracy and third-order temporal accuracy.Spatially, fourth-order accuracy is attained through the adoption of a long-stencil finite difference method, while boundary extrapolation is executed by leveraging a higher-order Taylor expansion to ensure consistency at domain boundaries. Temporally, the scheme is constructed based on the third-order backward differentiation formula (BDF3), with implicit discretization applied to the linear diffusion term for numerical stability and explicit extrapolation employed for nonlinear terms to balance computational efficiency. Notably, this numerical method inherently preserves the normalization constraint of the LLG equation, a key physical property of the system.Theoretical analysis confirms that the proposed scheme exhibits optimal convergence rates under the $\ell^{\infty}([0,T],\ell^2)$ and $\ell^2([0,T],H_h^1)$ norms. Finally, numerical experiments are conducted to validate the correctness of the theoretical convergence results, demonstrating good agreement between numerical observations and analytical conclusions.

math.NA↗

arXiv2025

Instability of Numerical Method for Micromagnetics Simulations with Large Damping Parameters

We propose and implement a third-order accurate numerical scheme for the Landau-Lifshitz-Gilbert equation, which describes magnetization dynamics in ferromagnetic materials under large damping parameters. This method offers two key advantages: (1) It solves only constant-coefficient linear systems, enabling fast solvers and thus achieving much higher numerical efficiency than existing second-order methods. (2) It attains third-order temporal accuracy and fourth-order spatial accuracy, and is unconditionally stable for large damping parameters. Numerical examples in 1D and 3D simulations verify both its third-order accuracy and efficiency gains. However, when large damping parameters and pre-projection solutions are involved, both this proposed method and a second-order method of the same style fail to capture reasonable physical structures, despite extensive theoretical analyses. Additionally, comparisons of domain wall dynamics among BDF2, BDF3, and BDF1 show that BDF2 and BDF3 yield failed simulations, while BDF1 performs marginally better.

math-ph↗

arXiv2021

Convergence Analysis of A Second-order Accurate, Linear Numerical Scheme for The Landau-Lifshitz Equation with Large Damping Parameters

A second order accurate, linear numerical method is analyzed for the Landau-Lifshitz equation with large damping parameters. This equation describes the dynamics of magnetization, with a non-convexity constraint of unit length of the magnetization. The numerical method is based on the second-order backward differentiation formula in time, combined with an implicit treatment of the linear diffusion term and explicit extrapolation for the nonlinear terms. Afterward, a projection step is applied to normalize the numerical solution at a point-wise level. This numerical scheme has shown extensive advantages in the practical computations for the physical model with large damping parameters, which comes from the fact that only a linear system with constant coefficients (independent of both time and the updated magnetization) needs to be solved at each time step, and has greatly improved the numerical efficiency. Meanwhile, a theoretical analysis for this linear numerical scheme has not been available. In this paper, we provide a rigorous error estimate of the numerical scheme, in the discrete $\ell^{\infty}(0,T; \ell^2) \cap \ell^2(0,T; H_h^1)$ norm, under suitable regularity assumptions and reasonable ratio between the time step-size and the spatial mesh-size. In particular, the projection operation is nonlinear, and a stability estimate for the projection step turns out to be highly challenging. Such a stability estimate is derived in details, which will play an essential role in the convergence analysis for the numerical scheme, if the damping parameter is greater than 3.

math.NA↗

arXiv2021

A Machine-Learning Method for Time-Dependent Wave Equations over Unbounded Domains

Time-dependent wave equations represent an important class of partial differential equations (PDE) for describing wave propagation phenomena, which are often formulated over unbounded domains. Given a compactly supported initial condition, classical numerical methods reduce such problems to bounded domains using artificial boundary condition (ABC). In this work, we present a machine-learning method to solve this type of equations as an alternative to ABCs. Specifically, the mapping from the initial conditions to the PDE solution is represented by a neural network, trained using wave packets that are parameterized by their band width and wave numbers. The accuracy is tested for both the second-order wave equation and the Schrodinger equation, including the nonlinear Schrodinger equation. We examine the accuracy from both interpolations and extrapolations. For initial conditions lying in the training set, the learned map has good interpolation accuracy, due to the approximation property of deep neural networks. The learned map also exhibits some good extrapolation accuracy. We also demonstrate the effectiveness of the method for problems in irregular domains. Overall, the proposed method provides an interesting alternative for finite-time simulation of wave propagation.

math.NA↗

arXiv2021

A second-order numerical method for Landau-Lifshitz-Gilbert equation with large damping parameters

A second order accurate numerical scheme is proposed and implemented for the Landau-Lifshitz-Gilbert equation, which models magnetization dynamics in ferromagnetic materials, with large damping parameters. The main advantages of this method are associated with the following features: (1) It only solves linear systems of equations with constant coefficients where fast solvers are available, so that the numerical efficiency has been greatly improved, in comparison with the existing Gauss-Seidel project method. (2) The second-order accuracy in time is achieved, and it is unconditionally stable for large damping parameters. Moreover, both the second-order accuracy and the great efficiency improvement will be verified by several numerical examples in the 1D and 3D simulations. In the presence of large damping parameters, it is observed that this method is unconditionally stable and finds physically reasonable structures while many existing methods have failed. For the domain wall dynamics, the linear dependence of wall velocity with respect to the damping parameter and the external magnetic field will be obtained through the reported simulations.

physics.comp-ph↗