Article
Mathematics, Applied
Yanrong Zhang, Jie Shen
Summary: We propose three numerical methods for solving the Klein-Gordon-Schrodinger (KGS) equations with/without damping terms, based on the SAV approach, Lagrange multiplier SAV approach, and Lagrange multiplier approach. These methods differ in how they preserve the Hamiltonian and wave energy, but all are validated through numerical tests for efficiency and accuracy in solving the KGS equations.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Jiaojiao Guo, Qingqu Zhuang
Summary: Three efficient energy stable schemes are proposed to solve the Klein-Gordon-Zakharov equations, based on the traditional scalar auxiliary variable (SAV) method, the exponential SAV (ESAV) method, and the Lagrange multiplier method. These schemes lead to linear equations with constant coefficients to be solved in each time step, preserving modified energies or original energy conservation property. Numerical examples are provided to verify the accuracy, efficiency, and energy stability of the schemes.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2023)
Article
Computer Science, Interdisciplinary Applications
Longbin Wu, Qiang Ma, Xiaohua Ding
Summary: This paper presents an energy-preserving scheme for the nonlinear fractional Klein-Gordon Schrodinger equation using the scalar auxiliary variable approach. By introducing a scalar variable, the system is transformed into a new equivalent system, and a linear implicit energy-preserving scheme is obtained by applying the extrapolated Crank-Nicolson method in the temporal direction and Fourier pseudospectral method in the spatial direction.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2021)
Review
Mathematics, Applied
Yayun Fu, Xuelong Gu, Yushun Wang, Wenjun Cai
Summary: We present a class of arbitrarily high-order conservative schemes for the Klein-Gordon Schrodinger equations, which combine the symplectic Runge-Kutta method with the quadratic auxiliary variable approach and can effectively preserve the conservation of energy and mass.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Computer Science, Interdisciplinary Applications
Yantao Guo, Yayun Fu
Summary: This paper constructs two efficient exponential energy-preserving schemes for solving the fractional Klein-Gordon Schrodinger equation, which are built upon the newly proposed partitioned averaged vector field method and exponential time difference technique. The schemes also apply the Fourier pseudo-spectral method to discretize the fractional Laplacian operator and utilize the FFT technique to reduce computational complexity. Numerical experiments demonstrate that the proposed schemes are efficient, conserve energy, and exhibit better numerical stability results compared to traditional schemes.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2023)
Article
Computer Science, Interdisciplinary Applications
Y. H. Youssri
Summary: This paper focuses on developing spectral solutions for the nonlinear fractional Klein-Gordon equation. The typical collocation method and the tau method are used, along with a new operational matrix of fractional derivatives of Fibonacci polynomials. The resulting system of algebraic equations is solved to obtain a semi-analytic solution, and convergence and error analysis are discussed.
INTERNATIONAL JOURNAL OF MODERN PHYSICS C
(2022)
Article
Mathematics, Applied
Junqing Jia, Huanying Xu, Xiaoyun Jiang
Summary: This paper presents a fast algorithm to solve the two-dimensional nonlinear coupled time-space fractional Klein-Gordon-Zakharov (KGZ) equations, utilizing an efficient sum-of-exponentials (SOE) approximation and a Fourier spectral method to approximate the time and space directions, and leveraging previous time levels to handle nonlinear terms. A numerical example demonstrates that the numerical method achieves second order accuracy in time, spectral accuracy in space, and the fast algorithm is effective.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Mathematics, Applied
Junjie Wang
Summary: In this paper, a symplectic-preserving Fourier spectral scheme is introduced for solving space fractional Klein-Gordon-Schrodinger equations involving fractional Laplacian. By using the midpoint rule, the semi-discrete system is transformed into a symplectic approximation scheme, and its convergence is proven. The splitting idea further reduces computational cost.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2021)
Article
Physics, Multidisciplinary
Pablo Raban, Renato Alvarez-Nodarse, Niurka R. Quintero
Summary: The stability of topological solitary waves and pulses in one-dimensional nonlinear Klein-Gordon systems is investigated. The linearized equation leads to a Sturm-Liouville problem, which is systematically solved for the l(l+1)sech(2)(x) potential, demonstrating the orthogonality and completeness of the solution set. Two families of novel nonlinear Klein-Gordon potentials are introduced, and their exact solutions are calculated even when the nonlinear potential is unknown. It is found that the kinks of the novel models are stable, while the pulses are unstable unless certain spatial inhomogeneities are introduced.
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2022)
Article
Mathematics, Applied
Guoyu Zhang, Chengming Huang, Mingfa Fei, Nan Wang
Summary: In this study, a linearized finite element method for solving two-dimensional fractional Klein-Gordon equations was proposed, which demonstrated bounded numerical solution with second-order accuracy. The convergence of the numerical solution was proved using mathematical induction.
NUMERICAL ALGORITHMS
(2021)
Article
Mathematics, Applied
Dingwen Deng, Qihong Wang
Summary: Recently, Xiaofeng Yang's group introduced invariant energy-quadratization methods (IEQMs) to develop linear and energy-dissipation-preserving methods for nonlinear energy-dissipation systems. They first introduced two auxiliary functions to rewrite the sine-Gordon equation (SGE) and coupled sine-Gordon equations (CSGEs) into equivalent systems, and then suggested two energy-preserving Du Fort-Frankel-type finite difference methods (EP-DFFT-FDMs) for them. The discrete energy conservative laws and convergence rates in the H-1-norm were derived using discrete energy methods, demonstrating the stability and performance of the proposed methods.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2023)
Article
Mathematics, Applied
Jingye Yan, Xu Qian, Hong Zhang, Songhe Song
Summary: The paper introduces two regularized finite difference methods that preserve the energy of the logarithmic Klein-Gordon equation, and proposes a regularized logarithmic Klein-Gordon equation with a small regulation parameter to approximate the LogKGE. By combining the energy method, inverse inequality, and cut-off technique of the nonlinearity, the error bound of the two schemes is derived with an error estimate reported for the mesh size, time step, and parameter. Numerical results are provided to support the conclusions.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Interdisciplinary Applications
Lewa Alzaleq, Valipuram Manoranjan
Summary: In this paper, a numerical scheme is developed to solve the Klein-Gordon equation with cubic nonlinearity while conserving the discrete energy. The theoretical proof demonstrates that the scheme also conserves other energy-like discrete quantities. Furthermore, the convergence and stability of the scheme are proven. Numerical simulations are presented to showcase the performance of the energy-conserving scheme.
FRACTAL AND FRACTIONAL
(2022)
Article
Mathematics, Applied
Dingwen Deng, Jingliang Chen, Qihong Wang
Summary: In this study, two classes of weighted energy-preserving Du Fort-Frankel finite difference methods are proposed for numerical simulations of sine-Gordon equations and nonlinear coupled sine-Gordon equations. It is shown that these methods satisfy the discrete energy conservative laws and converge to exact solutions in the H-1 norm. Furthermore, our methods are unconditionally stable in the L-2 norm despite being explicit schemes. Numerical results confirm the accuracy of theoretical findings and demonstrate the superiority of our algorithms in terms of computational efficiency and energy conservation.
NUMERICAL ALGORITHMS
(2023)
Article
Mathematics
C. Buriol, L. G. Delatorre, V. H. Gonzalez Martinez, D. C. Soares, E. H. G. Tavares
Summary: The study deals with a nonlinear Klein-Gordon system in an inhomogeneous medium, with local damping distributed around the boundary according to the Geometric Control Condition. It is shown that the energy of the system exponentially goes to zero for initial data within bounded sets of finite energy phase-space.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematics, Applied
Bin Wang
Summary: In this paper, a novel exponential energy-preserving method is proposed for solving charged-particle dynamics in a strong magnetic field. The method can exactly preserve the energy of the dynamics and maintain the near conservation of the magnetic moment over a long period. The numerical experiment demonstrates the long-time behavior of the proposed method.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Ting Li, Bin Wang
APPLIED MATHEMATICS LETTERS
(2020)
Article
Mathematics, Applied
Ernst Hairer, Christian Lubich, Bin Wang
NUMERISCHE MATHEMATIK
(2020)
Article
Mathematics, Applied
Xinyuan Wu, Bin Wang, Lijie Mei
Summary: In this paper, the oscillation-preserving behavior of existing RKN-type methods is analyzed from the perspective of geometric integration. It is found that if both the internal stages and updates of an RKN-type method respect the characteristics of oscillatory solutions, then the method is oscillation preserving. Other concerns relating to oscillation preservation and the importance of this property for solving highly oscillatory systems are also discussed.
NUMERICAL ALGORITHMS
(2021)
Article
Mathematics, Applied
Bin Wang, Xinyuan Wu, Yonglei Fang
NUMERICAL MATHEMATICS-THEORY METHODS AND APPLICATIONS
(2020)
Article
Mathematics, Applied
Bin Wang, Xinyuan Wu
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2020)
Article
Mathematics, Applied
Bin Wang, Xinyuan Wu, Yonglei Fang
Article
Computer Science, Software Engineering
Bin Wang, Xinyuan Wu
Summary: This paper presents a long-term analysis of one-stage extended Runge-Kutta-Nystrom integrators for highly oscillatory Hamiltonian systems, showing the near conservation of total and oscillatory energy over a long term for both symmetric and symplectic integrators. By establishing a relationship between ERKN integrators and trigonometric integrators, explicit integrators were proven to have near energy conservation. The technology of modulated Fourier expansion was used for the long-term analysis of implicit integrators, deriving near energy conservation through adaptations of this technology.
BIT NUMERICAL MATHEMATICS
(2021)
Article
Mathematics, Applied
Yonglei Fang, Ting Huang, Xiong You, Juan Zheng, Bin Wang
Summary: This paper presents two new symmetric linear multi-step methods for initial-value problems with two principal frequencies. These methods can integrate the problem without truncation error and have been analyzed for stability and phase lags. Numerical experiments show high effectiveness and robustness compared to some well-known one-frequency trigonometrically/exponentially fitted symmetric multi-step methods in recent literature.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Xicui Li, Bin Wang
Summary: In this letter, energy-preserving (EP) splitting methods for solving charged-particle dynamics in normal or strong magnetic fields are studied. Two novel EP splitting methods are formulated and their energy-preserving property and accuracy are analyzed. A numerical experiment is conducted to demonstrate the error and energy behavior of the splitting methods.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Mathematics, Applied
Bin Wang, Yaolin Jiang
Summary: This paper formulates and analyzes exponential integrations applied to nonlinear Schrodinger equations in a normal or highly oscillatory regime. It introduces exponential integrators that have energy preservation, optimal convergence, and long-time near conservations of density, momentum, and actions. The paper presents continuous-stage exponential integrators that can exactly preserve the energy of Hamiltonian systems and establishes their optimal convergence and near conservations of density, momentum, and actions over long times.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Bin Wang, Xinyuan Wu
Summary: This paper investigates the long-time near conservation of oscillatory energy for the adapted average vector field (AAVF) method applied to highly oscillatory Hamiltonian systems. The conservation of oscillatory energy is achieved through constructing a modulated Fourier expansion of the AAVF method, and a similar result in the multi-frequency case is also presented in this study.
JOURNAL OF COMPUTATIONAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Ting Li, Bin Wang
Summary: In this letter, explicit exponential algorithms for two-dimensional charged-particle dynamics with non-homogeneous electromagnetic fields are proposed and studied. The reformulation of the considered system allows for the derivation of three practical algorithms for charged-particle dynamics. The convergence of these algorithms is established and proved, and numerical tests demonstrate their excellent behavior and support the convergence result.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Mathematics, Applied
Ting Li, Changying Liu, Bin Wang
Summary: In this paper, one-stage explicit trigonometric integrators for solving quasilinear wave equations are formulated and studied. The trigonometric integrators are used as the semidiscretization in time, while a spectral Galerkin method is employed for the discretization in space. It is shown that these integrators have second-order convergence in time and the result holds for the fully discrete scheme without requiring any CFL-type coupling.
Article
Mathematics, Applied
Ting Li, Changying Liu, Bin Wang
Summary: In this paper, the long-time numerical conservation of energy and kinetic energy for highly oscillatory conservative systems is investigated using widely used exponential integrators. Modulated Fourier expansions of two types of exponential integrators are constructed, and the long-time numerical conservation of energy and kinetic energy is obtained by deriving two almost-invariants of the expansions. Practical examples and numerical experiments confirm and demonstrate the theoretical results.
NUMERICAL MATHEMATICS-THEORY METHODS AND APPLICATIONS
(2022)
