Review
Mathematics, Applied
Duy Phan, Alexander Ostermann
Summary: This paper considers two types of second-order in time partial differential equations, namely semilinear wave equations and semilinear beam equations. To solve these equations with exponential integrators, an approach to efficiently compute the action of the matrix exponential and related matrix functions is presented. Various numerical simulations are provided to illustrate the effectiveness of this approach.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Matthieu Brachet, Laurent Debreu, Christopher Eldred
Summary: The choice of time integration scheme is crucial in the development of an ocean model. This paper investigates various time integration schemes applied to the shallow water equations. Different algorithms for time stepping in linearized shallow water equations are analyzed, and a detailed comparison between classical schemes and exponential integrators is proposed.
APPLIED NUMERICAL MATHEMATICS
(2022)
Article
Chemistry, Physical
Julien Roulet, Jiri Vanicek
Summary: The explicit split-operator algorithm is not suitable for all types of nonlinear Schrodinger equations, leading to the proposal of high-order geometric integrators suitable for general time-dependent nonlinear Schrodinger equations. These integrators, based on the symmetric compositions of the implicit midpoint method, are both norm-conserving and time-reversible, showing good performance in numerical experiments.
JOURNAL OF CHEMICAL PHYSICS
(2021)
Article
Mathematics, Applied
Dongping Li, Xiuying Zhang, Renyun Liu
Summary: This paper discusses numerical integration for large-scale systems of stiff Riccati differential equations using exponential Rosenbrock-type integrators, addressing implementation issues and utilizing low-rank approximations based on high quality numerical algebra codes. Numerical comparisons demonstrate the high accuracy and efficiency of exponential integrators for solving large-scale systems of stiff Riccati differential equations.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
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
Fabio Pusateri, Israel Michael Sigal
Summary: The density functional theory (DFT) is a successful theory in describing the electronic structure of matter, with the Kohn-Sham (KS) equation as its foundation. The paper focuses on the time-dependent behavior of the KS equation, proving global existence and scattering in the full short-range regime with new and simple techniques that are compatible with the structure of DFT. These techniques involve commutator vector fields and non-abelian versions of Sobolev-Klainerman-type spaces and inequalities.
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
(2021)
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
Tommaso Buvoli, Michael L. Minion
Summary: This paper investigates the stability and efficiency of exponential integrators on non-diffusive equations and proposes a simple repartitioning approach to improve their stability. The effectiveness of the approach is validated through numerical experiments, and it is found that the repartitioning method does not require the use of high-order spatial derivatives unlike the approach of adding hyperviscosity.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Quantum Science & Technology
Dong An, Di Fang, Lin Lin
Summary: This study demonstrates that, under suitable assumptions, using Trotter type methods, the computational cost in quantum simulation may not increase at all as the norm of the Hamiltonian increases when measuring the error in terms of the vector norm. This result outperforms previous error bounds in the quantum simulation literature and clarifies the importance of commutator scalings in time-dependent Hamiltonian simulations.
Review
Physics, Mathematical
Kenji Yajima
Summary: The paper reports recent results on the existence and uniqueness of unitary propagators for N-particle Schrodinger equations, which can be applied to most interesting problems in physics.
REVIEWS IN MATHEMATICAL PHYSICS
(2021)
Article
Computer Science, Theory & Methods
Francois Golse, Shi Jin, Thierry Paul
Summary: The research demonstrates the uniform convergence of time splitting algorithms for the von Neumann equation of quantum dynamics in the Planck constant h, and provides explicit error estimates uniform in h for the first-order Lie-Trotter and second-order Strang splitting methods.
FOUNDATIONS OF COMPUTATIONAL MATHEMATICS
(2021)
Article
Chemistry, Physical
Shreyan Ganguly, Ramesh Ramachandran
Summary: This report examines the nuances of analytical methods used in the derivation of evolution operators in periodically driven quantum systems based on the Floquet theorem. By employing generalized multimodal expansion, time-propagators are derived in the finite-dimensional Hilbert space. While Floquet methods in the infinite-dimensional extended Hilbert space remain the method of choice for non-stroboscopic time-evolution, the discussed expansion schemes present an attractive option for similar studies in the standard Hilbert space. Nevertheless, the convergence criteria and suitability of such methods require formal validation in experimentally relevant problems.
PHYSICAL CHEMISTRY CHEMICAL PHYSICS
(2023)
Article
Chemistry, Physical
Maxim Secor, Alexander Soudackov, Sharon Hammes-Schiffer
Summary: The utilization of artificial neural networks (ANNs) as propagators of the time-dependent Schrodinger equation accelerates molecular simulations, successfully applied to systems with time-dependent potentials such as proton transfer. Trained ANN propagators map wavepackets from a given time to a future time, enabling simulation over long time scales, and have the potential for diverse quantum dynamical simulations of chemical and biological processes.
JOURNAL OF PHYSICAL CHEMISTRY LETTERS
(2021)
Article
Mathematics, Applied
Yijin Gao, Jay Mayfield, Songting Luo
Summary: In this study, operator splitting-based numerical methods are introduced to solve the time-dependent Schrodinger equation with a position-dependent effective mass. The wavefunction is propagated either by the Krylov subspace method-based exponential integration or by an asymptotic Green's function-based time propagator. These methods have complexity O(NlogN) per step with appropriate algebraic manipulations and fast Fourier transform, where N is the number of spatial points.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
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.
