Journal of Computational Physics最新文献_第4页

Generalized upwind summation-by-parts operators and their application to nodal discontinuous Galerkin methods

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2025-02-15 DOI: 10.1016/j.jcp.2025.113841

Jan Glaubitz , Hendrik Ranocha , Andrew R. Winters , Michael Schlottke-Lakemper , Philipp Öffner , Gregor Gassner

High-order numerical methods for conservation laws are highly sought after due to their potential efficiency. However, it is challenging to ensure their robustness, particularly for under-resolved flows. Baseline high-order methods often incorporate stabilization techniques that must be applied judiciously—sufficient to ensure simulation stability but restrained enough to prevent excessive dissipation and loss of resolution. Recent studies have demonstrated that combining upwind summation-by-parts (USBP) operators with flux vector splitting can increase the robustness of finite difference (FD) schemes without introducing excessive artificial dissipation. This work investigates whether the same approach can be applied to nodal discontinuous Galerkin (DG) methods. To this end, we demonstrate the existence of USBP operators on arbitrary grid points and provide a straightforward procedure for their construction. Our discussion encompasses a broad class of USBP operators, not limited to equidistant grid points, and enables the development of novel USBP operators on Legendre–Gauss–Lobatto (LGL) points that are well-suited for nodal DG methods. We then examine the robustness properties of the resulting DG-USBP methods for challenging examples of the compressible Euler equations, such as the Kelvin–Helmholtz instability. Similar to high-order FD-USBP schemes, we find that combining flux vector splitting techniques with DG-USBP operators does not lead to excessive artificial dissipation. Furthermore, we find that combining lower-order DG-USBP operators on three LGL points with flux vector splitting indeed increases the robustness of nodal DG methods. However, we also observe that higher-order USBP operators offer less improvement in robustness for DG methods compared to FD schemes. We provide evidence that this can be attributed to USBP methods adding dissipation only to unresolved modes, as FD schemes typically have more unresolved modes than nodal DG methods.

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引用次数: 0

PF-PINNs: Physics-informed neural networks for solving coupled Allen-Cahn and Cahn-Hilliard phase field equations

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2025-02-14 DOI: 10.1016/j.jcp.2025.113843

Nanxi Chen , Sergio Lucarini , Rujin Ma , Airong Chen , Chuanjie Cui

Physics-informed neural networks (PINNs) have emerged as a promising tool for effectively resolving diverse partial differential equations. Despite the numerous recent advances, PINNs often encounter significant challenges when dealing with complex nonlinear systems, such as the coupling Allen-Cahn (AC) and Cahn-Hilliard (CH) equations for phase field interfacial problems. In this work, we present an enhanced PINN framework, termed PF-PINNs, for the robust and efficient resolution of AC-CH coupled PDEs. Key features of the PF-PINNs framework include: (1) a normalisation and de-normalisation method to bridge the disparity in temporal and spatial scales in real-world physical problems, (2) an advanced sampling strategy designed to efficiently diffuse the initial interface and dynamically monitor its evolution throughout the training process, and (3) an NTK-based adaptive weighting strategy with random-batch method to balance the complex loss terms associated with phase field governing equations. We conduct extensive benchmarks on electrochemical corrosion, to showcase the accuracy and efficiency of the proposed PF-PINNs framework. The comparison of our results with reference solutions from FEniCS demonstrates that our PF-PINNs framework is a versatile and powerful tool for a wide range of AC-CH phase field applications.

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引用次数: 0

A precise conformally mapped method for water waves in complex transient environments

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2025-02-14 DOI: 10.1016/j.jcp.2025.113848

Andreas H. Akselsen

A two-dimensional water wave model based on conformal mapping is presented. The model is exact in the sense that it does not rely on truncated series expansions, nor suffer any numerical diffusion. Additionally, it is computationally highly efficient as it numerically evaluates only the surface line while using a fixed number of FFT operations per time step. A double layered mapping enforces prescribed outer boundaries without iteration. The model also supports transient boundaries, including walls. Mapping models are presented that support smooth bathymetries and angled overhanging geometries. An exact piston-type wavemaker model demonstrates the method's potential as a numerical wave tank. The model is tested and validated through a number of examples covering shallow water waves, wavemaker generation, rising bathymetry shelves, and wave reflection from slanting structures. A paddle-type wavemaker model, developed from the present theory, will be detailed in a forthcoming paper.

引用次数: 0

Numerical simulation of tokamak plasma equilibrium evolution

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2025-02-14 DOI: 10.1016/j.jcp.2025.113849

G. Gros , B. Faugeras , C. Boulbe , J.-F. Artaud , R. Nouailletas , F. Rapetti

This paper focuses on the numerical methods recently developed in order to simulate the time evolution of a tokamak plasma equilibrium at the resistive diffusion time scale. Starting from the method proposed by Heumann in 2021 for the coupling of magnetic equilibrium and current diffusion, we introduce a new space discretization for the poloidal flux using coupled

C^{0}

and

C^{1}

finite elements. This, together with the use of cubic spline functions to represent the poloidal current function in the resistive diffusion equation, enables to restrain numerical oscillations which can occur with the original method. In order to compute consistently the plasma resistivity and the non-inductive bootstrap current terms needed in the resistive diffusion equation we add to the model an evolution equation for electron temperature in the plasma. It is also used to evolve the pressure term in the simulation. These numerical methods are implemented in the plasma equilibrium code NICE. A free plasma displacement is simulated and comparison with experimental results from the WEST tokamak are used to validate the simulation. The code is also coupled to a magnetic feedback controller making it possible to simulate a prescribed plasma scenario. The results for an X-point formation scenario in the WEST tokamak are presented as an illustration of the efficiency of the developed numerical methods.

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引用次数: 0

Local randomized neural networks with finite difference methods for interface problems

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2025-02-13 DOI: 10.1016/j.jcp.2025.113847

Yunlong Li , Fei Wang

Accurate modeling of complex physical problems, such as fluid-structure interaction, requires multiphysics coupling across the interface, which often has intricate geometry and dynamic boundaries. Conventional numerical methods face challenges in handling interface conditions. Deep neural networks offer a mesh-free and flexible alternative, but they suffer from drawbacks such as time-consuming optimization and local optima. In this paper, we propose a mesh-free approach based on Randomized Neural Networks (RaNNs) and finite difference methods (FDM), which avoid optimization solvers during training, making them more efficient than traditional deep neural networks. Our approach, called Local Randomized Neural Networks with finite difference methods (LRaNN-FDM), uses different RaNNs to approximate solutions in different subdomains. We discretize the interface problem into a linear system at randomly sampled points across the domain, boundary, and interface using a finite difference scheme, and then solve it by a least-square method. Unlike automatic differentiation for partial derivative calculations, the finite difference approach offers significantly faster computation. For time-dependent interface problems, we use a space-time approach based on LRaNNs. We show the effectiveness and robustness of the LRaNN-FDM through numerical examples of elliptic and parabolic interface problems. We also demonstrate that our approach can handle high-dimension interface problems. Compared to conventional numerical methods, our approach achieves higher accuracy with fewer degrees of freedom, eliminates the need for complex interface meshing and fitting, and significantly reduces training time, outperforming deep neural networks.

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引用次数: 0

On the preprocessing of physics-informed neural networks: How to better utilize data in fluid mechanics

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2025-02-13 DOI: 10.1016/j.jcp.2025.113837

Shengfeng Xu , Yuanjun Dai , Chang Yan , Zhenxu Sun , Renfang Huang , Dilong Guo , Guowei Yang

Physics-Informed Neural Networks (PINNs) serve as a flexible alternative for tackling forward and inverse problems in differential equations, displaying impressive advancements in diverse areas of applied mathematics. Despite integrating both data and underlying physics to enrich the neural network's understanding, concerns regarding the effectiveness and practicality of PINNs persist. Over the past few years, extensive efforts in the current literature have been made to enhance this evolving method, by drawing inspiration from both machine learning algorithms and numerical methods. Despite notable progressions in PINNs algorithms, the important and fundamental field of data preprocessing remain unexplored, limiting the applications of PINNs especially in solving inverse problems. Therefore in this paper, a concise yet potent data preprocessing method focusing on data normalization was proposed. By applying a linear transformation to both the data and corresponding equations concurrently, the normalized PINNs approach was evaluated on the task of reconstructing flow fields in four turbulent cases. The results illustrate that by adhering to the data preprocessing procedure, PINNs can robustly achieve higher prediction accuracy for all flow quantities under different hyperparameter setups, without incurring extra computational cost, distinctly improving the utilization of limited training data. Though mainly verified in Navier-Stokes (NS) equations, this method holds potential for application to various other equations.

物理信息神经网络（PINNs）是解决微分方程正演和反演问题的灵活选择，在应用数学的各个领域都取得了令人瞩目的进展。尽管整合了数据和基础物理学来丰富神经网络的理解，但人们对 PINNs 的有效性和实用性的担忧依然存在。在过去几年中，现有文献通过从机器学习算法和数值方法中汲取灵感，为增强这种不断发展的方法做出了大量努力。尽管 PINNs 算法取得了显著进展，但数据预处理这一重要的基础领域仍未得到开发，限制了 PINNs 的应用，尤其是在解决逆问题方面。因此，本文提出了一种简洁而有效的数据预处理方法，重点是数据归一化。通过同时对数据和相应方程进行线性变换，对归一化 PINNs 方法在四种湍流情况下重建流场的任务进行了评估。结果表明，通过遵守数据预处理程序，归一化 PINNs 可以在不同超参数设置下稳健地获得更高的所有流动量预测精度，而不会产生额外的计算成本，明显提高了有限训练数据的利用率。虽然该方法主要在纳维-斯托克斯（NS）方程中得到验证，但也有可能应用于其他各种方程。

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引用次数: 0

A new boundary condition for the nonlinear Poisson-Boltzmann equation in electrostatic analysis of proteins

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2025-02-13 DOI: 10.1016/j.jcp.2025.113844

Sylvia Amihere , Yiming Ren , Weihua Geng , Shan Zhao

As a well-established implicit solvent model, the Poisson-Boltzmann equation (PBE) models the electrostatic interactions between a solute biomolecule and its surrounding solvent environment over an unbounded domain. One numerical challenge in solving the nonlinear PBE lies in the boundary treatment. Physically, the boundary condition of this solute solvent system is defined at infinity where the electrostatic potential decays to zero. Computationally, a finite domain has to be employed in grid-based numerical algorithms. However, the Dirichlet boundary conditions commonly used in protein simulations are known to produce unphysical solutions in some cases. This motivates the development of a few asymptotic conditions in the PBE literature, which are global boundary conditions and have to resort to iterative algorithms for calculating volume integrals from the previous step. To overcome these limitations, a modified Robin condition is proposed in this work as a local boundary condition for the nonlinear PBE, which can be implemented in any finite difference or finite element method. The derivation is based on the facts that away from the biomolecule, the asymptotic decaying pattern of the nonlinear PBE is essentially the same as that of the linearized PBE, and the monopole term will dominate other terms in the multipole expansion. Asymptotic analysis has been carried out to validate the application range and robustness of the proposed Robin condition. Moreover, a second order boundary implementation by means of a matched interface and boundary (MIB) scheme has been constructed for three-dimensional biomolecular simulations. Extensive numerical experiments have been conducted to examine the robustness, accuracy, and efficiency of the new boundary treatment for calculating electrostatic free energies of Kirkwood spheres and various protein systems.

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引用次数: 0

High-order exponential time differencing multi-resolution alternative finite difference WENO methods for nonlinear degenerate parabolic equations

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2025-02-12 DOI: 10.1016/j.jcp.2025.113838

Ziyao Xu, Yong-Tao Zhang

In this paper, we focus on the finite difference approximation of nonlinear degenerate parabolic equations, a special class of parabolic equations where the viscous term vanishes in certain regions. This vanishing gives rise to additional challenges in capturing sharp fronts, beyond the restrictive CFL conditions commonly encountered with explicit time discretization in parabolic equations. To resolve the sharp front, we adopt the high-order multi-resolution alternative finite difference WENO (A-WENO) methods for the spatial discretization, which is designed to effectively suppress oscillations in the presence of large gradients and achieve nonlinear stability. To alleviate the time step restriction from the nonlinear stiff diffusion terms, we employ the exponential time differencing Runge-Kutta (ETD-RK) methods, a class of efficient and accurate exponential integrators, for the time discretization. However, for highly nonlinear spatial discretizations such as high-order WENO schemes, it is a challenging problem how to efficiently form the linear stiff part in applying the exponential integrators, since direct computation of a Jacobian matrix for high-order WENO discretizations of the nonlinear diffusion terms is very complicated and expensive. Here we propose a novel and effective approach of replacing the exact Jacobian of high-order multi-resolution A-WENO scheme with that of the corresponding high-order linear scheme in the ETD-RK time marching, based on the fact that in smooth regions the nonlinear weights closely approximate the corresponding linear weights, while in non-smooth regions the stiff diffusion degenerates. The algorithm is described in detail, and numerous numerical experiments are conducted to demonstrate the effectiveness of such a treatment and the good performance of our method. The stiffness of the nonlinear parabolic partial differential equations (PDEs) is resolved well, and large time-step size computations of

Δ t \sim O (Δ x)

are achieved.

{"title":"High-order exponential time differencing multi-resolution alternative finite difference WENO methods for nonlinear degenerate parabolic equations","authors":"Ziyao Xu, Yong-Tao Zhang","doi":"10.1016/j.jcp.2025.113838","DOIUrl":"10.1016/j.jcp.2025.113838","url":null,"abstract":"<div><div>In this paper, we focus on the finite difference approximation of nonlinear degenerate parabolic equations, a special class of parabolic equations where the viscous term vanishes in certain regions. This vanishing gives rise to additional challenges in capturing sharp fronts, beyond the restrictive CFL conditions commonly encountered with explicit time discretization in parabolic equations. To resolve the sharp front, we adopt the high-order multi-resolution alternative finite difference WENO (A-WENO) methods for the spatial discretization, which is designed to effectively suppress oscillations in the presence of large gradients and achieve nonlinear stability. To alleviate the time step restriction from the nonlinear stiff diffusion terms, we employ the exponential time differencing Runge-Kutta (ETD-RK) methods, a class of efficient and accurate exponential integrators, for the time discretization. However, for highly nonlinear spatial discretizations such as high-order WENO schemes, it is a challenging problem how to efficiently form the linear stiff part in applying the exponential integrators, since direct computation of a Jacobian matrix for high-order WENO discretizations of the nonlinear diffusion terms is very complicated and expensive. Here we propose a novel and effective approach of replacing the exact Jacobian of high-order multi-resolution A-WENO scheme with that of the corresponding high-order linear scheme in the ETD-RK time marching, based on the fact that in smooth regions the nonlinear weights closely approximate the corresponding linear weights, while in non-smooth regions the stiff diffusion degenerates. The algorithm is described in detail, and numerous numerical experiments are conducted to demonstrate the effectiveness of such a treatment and the good performance of our method. The stiffness of the nonlinear parabolic partial differential equations (PDEs) is resolved well, and large time-step size computations of <span><math><mi>Δ</mi><mi>t</mi><mo>∼</mo><mi>O</mi><mo>(</mo><mi>Δ</mi><mi>x</mi><mo>)</mo></math></span> are achieved.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"529 ","pages":"Article 113838"},"PeriodicalIF":3.8,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143429044","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

An asymptotic-preserving conservative semi-Lagrangian scheme for the Vlasov-Maxwell system in the quasi-neutral limit

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2025-02-12 DOI: 10.1016/j.jcp.2025.113840

Hongtao Liu , Xiaofeng Cai , Yong Cao , Giovanni Lapenta

In this paper, we present an asymptotic-preserving conservative Semi-Lagrangian (CSL) scheme for the Vlasov-Maxwell system in the quasi-neutral limit, where the Debye length is negligible compared to the macroscopic scales of interest. The proposed method relies on two key ingredients: the CSL scheme and a reformulated Maxwell equation (RME). The CSL scheme is employed for the phase space discretization of the Vlasov equation, ensuring mass conservation and removing the Courant-Friedrichs-Lewy restriction, thereby enhancing computational efficiency. To efficiently calculate the electromagnetic field in both non-neutral and quasi-neutral regimes, the RME is derived by semi-implicitly coupling the Maxwell equation and the moments of the Vlasov equation. Furthermore, we apply Gauss's law correction to the electric field derived from the RME to prevent unphysical charge separation. The synergy of the CSL and RME enables the proposed method to provide reliable solutions, even when the spatial and temporal resolution might not fully resolve the Debye length and plasma period. As a result, the proposed method offers a unified and accurate numerical simulation approach for complex electromagnetic plasma physics while maintaining efficiency in both quasi-neutral and non-quasi-neutral regimes. Several numerical experiments, ranging from 3D to 5D simulations, are presented to demonstrate the accuracy, stability, and efficiency of the proposed method.

{"title":"An asymptotic-preserving conservative semi-Lagrangian scheme for the Vlasov-Maxwell system in the quasi-neutral limit","authors":"Hongtao Liu , Xiaofeng Cai , Yong Cao , Giovanni Lapenta","doi":"10.1016/j.jcp.2025.113840","DOIUrl":"10.1016/j.jcp.2025.113840","url":null,"abstract":"<div><div>In this paper, we present an asymptotic-preserving conservative Semi-Lagrangian (CSL) scheme for the Vlasov-Maxwell system in the quasi-neutral limit, where the Debye length is negligible compared to the macroscopic scales of interest. The proposed method relies on two key ingredients: the CSL scheme and a reformulated Maxwell equation (RME). The CSL scheme is employed for the phase space discretization of the Vlasov equation, ensuring mass conservation and removing the Courant-Friedrichs-Lewy restriction, thereby enhancing computational efficiency. To efficiently calculate the electromagnetic field in both non-neutral and quasi-neutral regimes, the RME is derived by semi-implicitly coupling the Maxwell equation and the moments of the Vlasov equation. Furthermore, we apply Gauss's law correction to the electric field derived from the RME to prevent unphysical charge separation. The synergy of the CSL and RME enables the proposed method to provide reliable solutions, even when the spatial and temporal resolution might not fully resolve the Debye length and plasma period. As a result, the proposed method offers a unified and accurate numerical simulation approach for complex electromagnetic plasma physics while maintaining efficiency in both quasi-neutral and non-quasi-neutral regimes. Several numerical experiments, ranging from 3D to 5D simulations, are presented to demonstrate the accuracy, stability, and efficiency of the proposed method.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"528 ","pages":"Article 113840"},"PeriodicalIF":3.8,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143419232","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Adaptive time-stepping Hermite spectral scheme for nonlinear Schrödinger equation with wave operator: Conservation of mass, energy, and momentum

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2025-02-12 DOI: 10.1016/j.jcp.2025.113842

Shimin Guo , Zhengqiang Zhang , Liquan Mei

The aim of this paper is to establish an efficient numerical scheme for nonlinear Schrödinger equation with wave operator (NLSW) on unbounded domains to simultaneously conserve the first three kinds of invariants, namely the mass, the energy, and the momentum conservation laws. Regarding the mass and momentum conservation laws as the globally physical constraints, we elaborately combine the exponential scalar auxiliary variable (ESAV) method with Lagrange multiplier approach to build up the algorithm-friendly reformulation which links between the invariants and existing numerical methods. We employ the Crank-Nicolson and Hermite-Galerkin spectral methods for temporal discretization and spatial approximation, respectively. Additionally, we design a new adaptive time-stepping strategy based on the variation of the solution to improve the efficiency of our scheme. At each time level, we only need to solve a linear system plus a set of quadratic algebraic equations which can be efficiently solved by Newton's method. To enhance the applicability of the proposed scheme, we extend our methodology to N-coupled NLSW system where the mass, the energy, and the momentum are simultaneously conserved at the fully-discrete level. Numerical experiments are provided to show the convergence rates, the efficiency, and the conservation properties of the proposed scheme. In addition, the nonlinear dynamics of 2D/3D solitons are simulated to deepen the understanding of NLSW model.

{"title":"Adaptive time-stepping Hermite spectral scheme for nonlinear Schrödinger equation with wave operator: Conservation of mass, energy, and momentum","authors":"Shimin Guo , Zhengqiang Zhang , Liquan Mei","doi":"10.1016/j.jcp.2025.113842","DOIUrl":"10.1016/j.jcp.2025.113842","url":null,"abstract":"<div><div>The aim of this paper is to establish an efficient numerical scheme for nonlinear Schrödinger equation with wave operator (NLSW) on unbounded domains to simultaneously conserve the first three kinds of invariants, namely the mass, the energy, and the momentum conservation laws. Regarding the mass and momentum conservation laws as the globally physical constraints, we elaborately combine the exponential scalar auxiliary variable (ESAV) method with Lagrange multiplier approach to build up the algorithm-friendly reformulation which links between the invariants and existing numerical methods. We employ the Crank-Nicolson and Hermite-Galerkin spectral methods for temporal discretization and spatial approximation, respectively. Additionally, we design a new adaptive time-stepping strategy based on the variation of the solution to improve the efficiency of our scheme. At each time level, we only need to solve a linear system plus a set of quadratic algebraic equations which can be efficiently solved by Newton's method. To enhance the applicability of the proposed scheme, we extend our methodology to <em>N</em>-coupled NLSW system where the mass, the energy, and the momentum are simultaneously conserved at the fully-discrete level. Numerical experiments are provided to show the convergence rates, the efficiency, and the conservation properties of the proposed scheme. In addition, the nonlinear dynamics of 2D/3D solitons are simulated to deepen the understanding of NLSW model.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"528 ","pages":"Article 113842"},"PeriodicalIF":3.8,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143419227","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0