57,135 research outputs found
Accurate macroscale modelling of spatial dynamics in multiple dimensions
Developments in dynamical systems theory provides new support for the
macroscale modelling of pdes and other microscale systems such as Lattice
Boltzmann, Monte Carlo or Molecular Dynamics simulators. By systematically
resolving subgrid microscale dynamics the dynamical systems approach constructs
accurate closures of macroscale discretisations of the microscale system. Here
we specifically explore reaction-diffusion problems in two spatial dimensions
as a prototype of generic systems in multiple dimensions. Our approach unifies
into one the modelling of systems by a type of finite elements, and the
`equation free' macroscale modelling of microscale simulators efficiently
executing only on small patches of the spatial domain. Centre manifold theory
ensures that a closed model exist on the macroscale grid, is emergent, and is
systematically approximated. Dividing space either into overlapping finite
elements or into spatially separated small patches, the specially crafted
inter-element/patch coupling also ensures that the constructed discretisations
are consistent with the microscale system/PDE to as high an order as desired.
Computer algebra handles the considerable algebraic details as seen in the
specific application to the Ginzburg--Landau PDE. However, higher order models
in multiple dimensions require a mixed numerical and algebraic approach that is
also developed. The modelling here may be straightforwardly adapted to a wide
class of reaction-diffusion PDEs and lattice equations in multiple space
dimensions. When applied to patches of microscopic simulations our coupling
conditions promise efficient macroscale simulation.Comment: some figures with 3D interaction when viewed in Acrobat Reader. arXiv
admin note: substantial text overlap with arXiv:0904.085
Multiscale modeling of rapid granular flow with a hybrid discrete-continuum method
Both discrete and continuum models have been widely used to study rapid
granular flow, discrete model is accurate but computationally expensive,
whereas continuum model is computationally efficient but its accuracy is
doubtful in many situations. Here we propose a hybrid discrete-continuum method
to profit from the merits but discard the drawbacks of both discrete and
continuum models. Continuum model is used in the regions where it is valid and
discrete model is used in the regions where continuum description fails, they
are coupled via dynamical exchange of parameters in the overlap regions.
Simulation of granular channel flow demonstrates that the proposed hybrid
discrete-continuum method is nearly as accurate as discrete model, with much
less computational cost
Combined Global and Local Search for the Falsification of Hybrid Systems
In this paper we solve the problem of finding a trajectory that shows that a
given hybrid dynamical system with deterministic evolution leaves a given set
of states considered to be safe. The algorithm combines local with global
search for achieving both efficiency and global convergence. In local search,
it exploits derivatives for efficient computation. Unlike other methods for
falsification of hybrid systems with deterministic evolution, we do not
restrict our search to trajectories of a certain bounded length but search for
error trajectories of arbitrary length
Eigenvalue Estimation of Differential Operators
We demonstrate how linear differential operators could be emulated by a
quantum processor, should one ever be built, using the Abrams-Lloyd algorithm.
Given a linear differential operator of order 2S, acting on functions
psi(x_1,x_2,...,x_D) with D arguments, the computational cost required to
estimate a low order eigenvalue to accuracy Theta(1/N^2) is
Theta((2(S+1)(1+1/nu)+D)log N) qubits and O(N^{2(S+1)(1+1/nu)} (D log N)^c)
gate operations, where N is the number of points to which each argument is
discretized, nu and c are implementation dependent constants of O(1). Optimal
classical methods require Theta(N^D) bits and Omega(N^D) gate operations to
perform the same eigenvalue estimation. The Abrams-Lloyd algorithm thereby
leads to exponential reduction in memory and polynomial reduction in gate
operations, provided the domain has sufficiently large dimension D >
2(S+1)(1+1/nu). In the case of Schrodinger's equation, ground state energy
estimation of two or more particles can in principle be performed with fewer
quantum mechanical gates than classical gates.Comment: significant content revisions: more algorithm details and brief
analysis of convergenc
Trapping and Steering on Lattice Strings: Virtual Slow Waves, Directional and Non-propagating Excitations
Using a lattice string model, a number of peculiar excitation situations
related to non-propagating excitations and non-radiating sources are
demonstrated. External fields can be used to trap excitations locally but also
lead to the ability to steer such excitations dynamically as long as the
steering is slower than the field's wave propagation. I present explicit
constructions of a number of examples, including temporally limited
non-propagating excitations, directional excitation and virtually slowed
propagation. Using these dynamical lattice constructions I demonstrate that
neither persistent temporal oscillation nor static localization are necessary
for non-propagating excitations to occur.Comment: 16 pages, 5 figures, RevTex4, references added, figure captions
improved, to appear in Physical Review
Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems
The present article presents a summarizing view at differential-algebraic
equations (DAEs) and analyzes how new application fields and corresponding
mathematical models lead to innovations both in theory and in numerical
analysis for this problem class. Recent numerical methods for nonsmooth
dynamical systems subject to unilateral contact and friction illustrate the
topicality of this development.Comment: Preprint of Book Chapte
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