Adaptivity with moving grids

CJ Budd, W Huang, RD Russell - Acta Numerica, 2009 - cambridge.org
In this article we survey r-adaptive (or moving grid) methods for solving time-dependent
partial differential equations (PDEs). Although these methods have received much less …

[LIBRO][B] A concise introduction to geometric numerical integration

S Blanes, F Casas - 2017 - books.google.com
Discover How Geometric Integrators Preserve the Main Qualitative Properties of Continuous
Dynamical Systems A Concise Introduction to Geometric Numerical Integration presents the …

Backward error analysis for numerical integrators

S Reich - SIAM Journal on Numerical Analysis, 1999 - SIAM
Backward error analysis has become an important tool for understanding the long time
behavior of numerical integration methods. This is true in particular for the integration of …

Understanding self-attention mechanism via dynamical system perspective

Z Huang, M Liang, J Qin, S Zhong… - Proceedings of the …, 2023 - openaccess.thecvf.com
The self-attention mechanism (SAM) is widely used in various fields of artificial intelligence
and has successfully boosted the performance of different models. However, current …

Implementing few-body algorithmic regularization with post-Newtonian terms

S Mikkola, D Merritt - The Astronomical Journal, 2008 - iopscience.iop.org
We discuss the implementation of a new regular algorithm for simulation of the gravitational
few-body problem. The algorithm uses components from earlier methods, including the …

Geometric integration and its applications

CJ Budd, MD Piggott - 2001 - ems.press
Geometric integration is the general term for a set of numerical algorithms for solving
differential equations that aim to reproduce qualitative features in the solution. These can be …

A class of symplectic integrators with adaptive time step for separable Hamiltonian systems

M Preto, S Tremaine - The Astronomical Journal, 1999 - iopscience.iop.org
Symplectic integration algorithms are well suited for long-term integrations of Hamiltonian
systems, because they preserve the geometric structure of the Hamiltonian flow. However …

Symplectic local time-step** in non-dissipative DGTD methods applied to wave propagation problems

S Piperno - ESAIM: Mathematical Modelling and Numerical …, 2006 - numdam.org
The Discontinuous Galerkin Time Domain (DGTD) methods are now popular for the solution
of wave propagation problems. Able to deal with unstructured, possibly locally-refined …

On fast simulation of dynamical system with neural vector enhanced numerical solver

Z Huang, S Liang, H Zhang, H Yang, L Lin - Scientific reports, 2023 - nature.com
The large-scale simulation of dynamical systems is critical in numerous scientific and
engineering disciplines. However, traditional numerical solvers are limited by the choice of …

Variable time step integration with symplectic methods

E Hairer - Applied Numerical Mathematics, 1997 - Elsevier
Symplectic methods for Hamiltonian systems are known to have favorable properties
concerning long-time integrations (no secular terms in the error of the energy integral, linear …