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, Interdisciplinary Applications
Yayun Fu, Qianqian Zheng, Yanmin Zhao, Zhuangzhi Xu
Summary: A family of high-order linearly implicit exponential integrators conservative schemes is proposed for solving the multi-dimensional nonlinear fractional Schrodinger equation. By reformulating and discretizing the equation, energy-preserving schemes with high accuracy are constructed to efficiently perform long-time simulations.
FRACTAL AND FRACTIONAL
(2022)
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)
Article
Mathematics, Applied
Bin Wang, Xiaofei Zhao
Summary: In this paper, a class of highly oscillatory Hamiltonian systems with a scaling parameter ε in (0, 1] is considered. The problem arises from physical models in a limit parameter regime or time-compressed perturbations. Classical numerical methods are inefficient for solving the model due to the rapid temporal oscillations with O(1)-amplitude and O(1/ε)-frequency. Two new time-symmetric numerical integrators based on the two-scale formulation approach are proposed in this paper. The methods are proven to have uniform second order accuracy for all ε at finite times and exhibit near-conservation laws in long times through numerical experiments on various models.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2023)
Article
Computer Science, Interdisciplinary Applications
L. Minah Yang, Ian Grooms, Keith A. Julien
Summary: Exponential and IMEX integrators are not suitable for wave turbulence problems due to errors in simulating interactions between resonant waves and introducing dispersive and dissipative errors. Integrating Factor (IF) methods are the most suitable choice, and a novel near-minimax rational approximation of the matrix exponential is proposed for improved accuracy without the need to compute a matrix exponential.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Yayun Fu, Mengyue Shi
Summary: The paper presents a conservative Fourier pseudo-spectral scheme for solving conservative fractional partial differential equations. The scheme approximates the time direction using the exponential time difference averaged vector field method, and discretizes the fractional Laplacian operator using the Fourier pseudo-spectral method, enabling the use of the FFT technique to reduce computational complexity in long-time simulations. Additionally, the scheme can also be used for solving fractional Hamiltonian differential equations due to its construction based on the general Hamiltonian form of the equations. The conservation and accuracy of the scheme are demonstrated through the solution of the fractional Schrödinger equation.
Article
Mathematics, Applied
Gianluca Frasca-Caccia, Peter E. Hydon
Summary: A recently-introduced strategy uses symbolic algebra to construct finite difference schemes that preserve several local conservation laws of a given scalar PDE. By adapting this strategy to PDEs that are not in Kovalevskaya form and to systems of PDEs, the study demonstrates that the approach yields conservative and highly accurate schemes compared to existing methods.
APPLIED MATHEMATICS AND COMPUTATION
(2021)
Article
Mathematics, Applied
Zhuangzhi Xu, Wenjun Cai, Dongdong Hu, Yushun Wang
Summary: In this paper, a novel linearly implicit conservative scheme is developed for the two-dimensional nonlinear Schrodinger equation. Through rigorous analysis, it is proven that the proposed scheme can preserve energy conservation and has good numerical stability. Numerical experiments show that the proposed scheme has significant advantages in efficiency.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Ting Li, Bin Wang
Summary: This paper focuses on geometric exponential energy-preserving integrators for solving charged-particle dynamics in a magnetic field. It proposes two practical symmetric continuous-stage integrators and analyzes their energy-preserving property, symmetric conditions, and order conditions. The efficiency of the proposed methods is validated through numerical experiments, surpassing existing schemes in the literature.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Mathematics, Applied
Meiqiao Ai, Zhimin Zhang
Summary: This paper examines the valuation problem of life-contingent lookback options embedded in variable annuity with guaranteed minimum death benefit (GMDB). The underlying asset price process is assumed to be an exponential regime-switching Levy process, observed periodically. The Fourier cosine series expansion method is utilized to compute exponential moments of the discretely monitored maximum and minimum of the regime-switching Levy process. Additionally, explicit pricing formulas for the life-contingent lookback options embedded in GMDB products are derived. Numerical experiments confirm the accuracy and efficiency of the proposed method.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Optics
Phillip W. K. Jensen, Peter D. Johnson, Alexander A. Kunitsa
Summary: The study proposes a quantum algorithm to estimate the properties of molecules using near-term quantum devices. The method uses a recursive variational series estimation approach and evaluates each term in the expansion using a variational quantum algorithm. The algorithm is tested by computing the one-particle Green's function in the energy domain and the autocorrelation function in the time domain.
Article
Mathematics, Applied
Sergio Blanes, Fernando Casas, Cesareo Gonzalez, Mechthild Thalhammer
Summary: This work focuses on the convergence of high-order commutator-free quasi-Magnus (CFQM) exponential integrators for nonautonomous linear Schrodinger equations, providing detailed stability and local error analysis. CFQM exponential integrators preserve structural properties of the operator family, ensuring unconditional stability and full convergence order in the underlying Hilbert space under low regularity requirements on the initial state.
IMA JOURNAL OF NUMERICAL ANALYSIS
(2021)
Article
Mathematics, Applied
Ernst Hairer, Christian Lubich, Yanyan Shi
Summary: This study investigates the behavior of the Boris algorithm, a variational integrator, and a filtered variational integrator when numerically integrating the equations of motion for a charged particle in a mildly non-uniform strong magnetic field with large step sizes. The results show that satisfactory behavior is only achieved when the component of the initial velocity orthogonal to the magnetic field is filtered out.
NUMERISCHE MATHEMATIK
(2022)
Article
Mathematics, Applied
Dajana Conte, Gianluca Frasca-Caccia
Summary: The exponential fitting technique is effective for oscillatory solutions and outperforms standard methods in the large time-step regime. This paper explores exponentially fitted Runge-Kutta methods and their ability to preserve local conservation laws of linear and quadratic quantities, with numerical testing showing better performance and exact satisfaction of discrete conservation laws for mass and momentum in various fields such as fluid dynamics and quantum physics.
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
(2022)
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
David Cohen, Lluis Quer-Sardanyons
IMA JOURNAL OF NUMERICAL ANALYSIS
(2016)
Article
Mathematics, Applied
Rikard Anton, David Cohen, Stig Larsson, Xiaojie Wang
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2016)
Article
Mathematics, Applied
Yuto Miyatake, David Cohen, Daisuke Furihata, Takayasu Matsuo
JAPAN JOURNAL OF INDUSTRIAL AND APPLIED MATHEMATICS
(2017)
Article
Mathematics, Applied
Yoshio Komori, David Cohen, Kevin Burrage
SIAM JOURNAL ON SCIENTIFIC COMPUTING
(2017)
Article
Mathematics, Applied
Rikard Anton, David Cohen
JOURNAL OF COMPUTATIONAL MATHEMATICS
(2018)
Article
Mathematics, Applied
Rikard Anton, David Cohen, Lluis Quer-Sardanyons
IMA JOURNAL OF NUMERICAL ANALYSIS
(2020)
Article
Mathematics, Applied
Chuchu Chen, David Cohen, Raffaele D'Ambrosio, Annika Lang
ADVANCES IN COMPUTATIONAL MATHEMATICS
(2020)
Article
Mathematics, Applied
David Cohen, Kristian Debrabant, Andreas Roessler
APPLIED NUMERICAL MATHEMATICS
(2020)
Article
Mathematics, Applied
David Cohen, Gilles Vilmart
Summary: This paper presents a numerical analysis of a class of randomly perturbed Hamiltonian systems and Poisson systems, showing the long-time behavior of energy and quadratic Casimirs for the considered additive noise perturbation. A drift-preserving splitting scheme with exact drift preservation of energy and quadratic Casimirs, mean-square order of convergence 1, and weak order of convergence 2 is proposed and analyzed, with illustration through numerical experiments.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2021)
Article
Mathematics, Interdisciplinary Applications
Charles-Edouard Brehier, David Cohen
Summary: We analyze a splitting integrator for the time discretization of the Schrodinger equation with nonlocal interaction cubic nonlinearity and white noise dispersion. We prove that this time integrator has first-order convergence in the pth mean sense, for any p greater than or equal to 1 in some Sobolev spaces. We demonstrate that the splitting schemes preserve the L-2-norm, which is essential for proving the strong convergence result. Finally, numerical experiments are conducted to illustrate the performance of the proposed numerical scheme.
SIAM-ASA JOURNAL ON UNCERTAINTY QUANTIFICATION
(2022)
Article
Mathematics, Applied
David Cohen, Annika Lang
Summary: Solutions to the stochastic wave equation on the unit sphere are approximated using spectral methods. Strong, weak, and almost sure convergence rates for the proposed numerical schemes are provided, depending on the smoothness of the driving noise and the initial conditions. Numerical experiments confirm the theoretical rates. The developed numerical method is extended to stochastic wave equations on higher-dimensional spheres and to the free stochastic Schrodinger equation on the unit sphere.
Article
Mathematics, Applied
Charles-Edouard Brehier, David Cohen
Summary: This paper analyzes the qualitative properties and order of convergence of a splitting scheme for a class of nonlinear stochastic Schrodinger equations. The numerical solution is shown to be symplectic and preserves the expected mass. Exponential moment bounds are proved for the exact and numerical solutions, enabling us to determine strong orders of convergence as well as orders of convergence in probability and almost surely. Extensive numerical experiments demonstrate the performance of the proposed numerical scheme.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Mathematics, Applied
David Cohen, Gilles Vilmart
Summary: We conduct a numerical analysis on a class of randomly perturbed Hamiltonian systems and Poisson systems. For such systems with additive noise perturbation, we examine the long-term behavior of energy and quadratic Casimirs for the exact solutions. We then propose and analyze a drift-preserving splitting scheme with properties including exact preservation of energy and quadratic Casimirs, mean-square convergence order of 1, and weak convergence order of 2. These properties are demonstrated through numerical experiments.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2022)
Proceedings Paper
Mathematics, Applied
David Cohen
INTERNATIONAL CONFERENCE OF NUMERICAL ANALYSIS AND APPLIED MATHEMATICS (ICNAAM 2017)
(2018)
Article
Mathematics, Applied
David Cohen, Guillaume Dujardin
STOCHASTICS AND PARTIAL DIFFERENTIAL EQUATIONS-ANALYSIS AND COMPUTATIONS
(2017)
