Article
Mathematics, Applied
Brian E. Moore
Summary: Exponential integrators based on discrete gradient methods are applied to non-canonical Hamiltonian systems with added linear forcing/damping terms, showing the ability to exactly preserve changes in dynamics in special circumstances. These methods are also symmetric, second order, and linearly stable, demonstrating advantages in accuracy and efficiency over other standard methods when applied to specific systems.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Victoria Wieloch, Martin Arnold
Summary: In this paper, BDF type multistep methods were applied to constrained systems with Lie group structure. The k-step Lie group integrator BLieDF, which avoids order reduction by perturbing the argument of the exponential map, was compared with multistep methods on Lie groups suggested by Faltinsen, Marthinsen and Munthe-Kaas, showing its advantages.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Physics, Fluids & Plasmas
Shaan A. Desai, Marios Mattheakis, Stephen J. Roberts
Summary: Recent advances in neural networks embedded with physics-informed priors have shown superior performance in learning and predicting the long-term dynamics of complex physical systems compared to vanilla neural networks. Researchers have generalized recent innovations into individual inductive bias segments to systematically investigate and experiment with different combinations of biases to improve predictive performance.
Article
Instruments & Instrumentation
Y. Q. Wei, B. N. Wan, B. Shen, L. Yang, F. Ji, Y. Wang, M. Chen, Z. J. Liu
Summary: This paper proposes an improved integration system to address the saturation and drift issues in integrators during long pulse operations. The system utilizes parallel analog integrators and digital rectification to compensate for drifts and controls them alternately using a microcontroller unit.
REVIEW OF SCIENTIFIC INSTRUMENTS
(2023)
Article
Automation & Control Systems
Xiang Xu, Lu Liu, Miroslav Krstic, Gang Feng
Summary: This article explores the control problem of linear strict-feedback systems with an infinite-delayed integrator, demonstrating the effectiveness of the predictor feedback combined with the backstepping design in solving control problems of systems with unbounded delays. The stability theorem developed for systems with unbounded delays is applied to prove the exponential stability of the resulting closed loop system. Simulation examples are provided to illustrate the effectiveness of the controller.
INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL
(2022)
Article
Mathematics, Applied
Stefan Hante, Martin Arnold
Summary: Variational integrators, known for their excellent numerical stability in long-term integration, are utilized in this paper for a novel second order variational integrator RATTLie for constrained systems on nonlinear configuration spaces with Lie group structure. The method exploits the linear structure of the Lie algebra, and is tested on a geometrically exact extensible Kirchhoff beam model.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Tomoki Ohsawa
Summary: We prove that the recently developed semiexplicit symplectic integrators preserve linear and quadratic invariants possessed by nonseparable Hamiltonian systems. These integrators share the structure-preserving properties with well-known symplectic Runge-Kutta methods and are shown to be symmetric and symplectic. The proof demonstrates how the extended Hamiltonian system inherits invariants in the extended phase space and how this inheritance preserves the original invariants in the original phase space. The paper also includes an analysis of the preservation/nonpreservation of invariants by other extended Hamiltonian systems and phase space integrators.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2023)
Article
Mathematics, Applied
Jeremy Chouchoulis, Jochen Schutz
Summary: This work presents an approximate family of implicit multiderivative Runge-Kutta time integrators for stiff initial value problems and investigates two different methods for computing higher order derivatives. Numerical results demonstrate that adding separate formulas yields better performance in dealing with stiff problems.
APPLIED NUMERICAL MATHEMATICS
(2024)
Article
Mathematics, Applied
Xinyuan Wu
Summary: We propose and analyze an exponential approach for solving highly ill-conditioned linear systems with a positive definite matrix. This approach also offers a new technique for finding the inverse of a given positive definite matrix through a one-stop procedure. The advantages of this approach include its simplicity and ease of implementation. Numerical experiments were conducted, and the results demonstrate the effectiveness and superiority of the exponential approach compared to standard MATLAB codes. (c) 2022 Elsevier Ltd. All rights reserved.
APPLIED MATHEMATICS LETTERS
(2023)
Article
Automation & Control Systems
Wenfeng Hu, Yi Cheng, Zhiyong Chen
Summary: This paper addresses the consensus problem for double-integrator multi-agent systems with a novel reset control protocol, revealing the conditions under which consensus is achieved and the role of the reset function. The proposed reset control provides a dynamically generated higher gain for better transient consensus performance.
Article
Computer Science, Interdisciplinary Applications
Molei Tao, Shi Jin
Summary: This article focuses on accurate and efficient numerical approximations of solutions for Hamiltonian mechanical systems with potential functions admitting jump discontinuities. It proposes several numerical methods and provides numerical evidence on their convergence, performance, and consistency.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Mathematics, Applied
F. S. Naranjo-Noda, J. C. Jimenez
Summary: This paper introduces a new class of Jacobian-free High Order Local Linearization (HOLL) methods for integrating large systems of initial value problems. These methods approximate a single phi-function times vector, eliminating the need for evaluating and storing Jacobian matrices, leading to more efficient computations and memory usage. The convergence rate and order preserving condition of the new methods are derived, and a novel Matrix-free Krylov-Pade approximation and an adaptive strategy for selecting the Krylov dimension and Pade order are proposed. Numerical simulations validate the theoretical findings and demonstrate the performance of the Jacobian-free integrators compared to others.
APPLIED NUMERICAL MATHEMATICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Lijie Mei, Li Huang, Xinyuan Wu
Summary: In this paper, two classes of energy-preserving functionally-fitted integrators for Poisson systems with highly oscillatory solutions are designed and analyzed using a new framework. The study shows that the order and stage order of the integrators may be affected by the used quadrature formula. Furthermore, the existence and uniqueness of the integrators, their implementation issues, and the conservation of Casimir functions are investigated. Numerical experiments demonstrate the remarkable accuracy and efficiency of the proposed high-order energy-preserving integrators.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Article
Engineering, Multidisciplinary
Giuseppe Capobianco, Jonas Harsch, Sigrid Leyendecker
Summary: This paper introduces a family of Lobatto IIIA-IIIB methods for simulating mechanical systems with frictional contact. The methods address both bilateral and unilateral constraints, as well as set-valued Coulomb friction. The discrete contact laws derived in this paper exhibit no contact penetration and satisfy the involved unilateral constraints. The behavior of these laws is showcased using benchmark examples.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2024)
Article
Chemistry, Analytical
Dong Eui Chang, Matthew Perlmutter, Joris Vankerschaver
Summary: The feedback integrators method, improved through the Dirac formula, is used to integrate equations of motion for mechanical systems with holonomic constraints. It produces numerical trajectories that stay within the constraint set and preserve the values of conserved quantities like energy. The method's excellent performance is demonstrated by implementing it on the spherical pendulum system and comparing it with other methods.
