83 research outputs found
Spectral methods in time for hyperbolic equations
A pseudospectral numerical scheme for solving linear, periodic, hyperbolic problems is described. It has infinite accuracy both in time and in space. The high accuracy in time is achieved without increasing the computational work and memory space which is needed for a regular, one step explicit scheme. The algorithm is shown to be optimal in the sense that among all the explicit algorithms of a certain class it requires the least amount of work to achieve a certain given resolution. The class of algorithms referred to consists of all explicit schemes which may be represented as a polynomial in the spatial operator
The eigenvalues of the pseudospectral Fourier approximation to the operator sin (2x) d/dx
It is shown that the eigenvalues Z sub i of the pseudospectral Fourier approximation to the operator sin(2x) curly d/curly dx satisfy (R sub e) (Z sub i) = + or - 1 or (R sub e)(Z sub I) = 0. Whereas this does not prove stability for the Fourier method, applied to the hyperbolic equation U sub t = sin (2x)(U sub x) - pi x pi; it indicates that the growth in time of the numerical solution is essentially the same as that of the solution to the differential equation
A pseudospectral Legendre method for hyperbolic equations with an improved stability condition
A new pseudospectral method is introduced for solving hyperbolic partial differential equations. This method uses different grid points than previously used pseudospectral methods: in fact the grid are related to the zeroes of the Legendre polynomials. The main advantage of this method is that the allowable time step is proportional to the inverse of the number of grid points 1/N rather than to 1/n(2) (as in the case of other pseudospectral methods applied to mixed initial boundary value problems). A highly accurate time discretization suitable for these spectral methods is discussed
New, Highly Accurate Propagator for the Linear and Nonlinear Schr\"odinger Equation
A propagation method for the time dependent Schr\"odinger equation was
studied leading to a general scheme of solving ode type equations. Standard
space discretization of time-dependent pde's usually results in system of ode's
of the form u_t -Gu = s where G is a operator (matrix) and u is a
time-dependent solution vector. Highly accurate methods, based on polynomial
approximation of a modified exponential evolution operator, had been developed
already for this type of problems where G is a linear, time independent matrix
and s is a constant vector. In this paper we will describe a new algorithm for
the more general case where s is a time-dependent r.h.s vector. An iterative
version of the new algorithm can be applied to the general case where G depends
on t or u. Numerical results for Schr\"odinger equation with time-dependent
potential and to non-linear Schr\"odinger equation will be presented.Comment: 14 page
An efficient scheme for numerical simulations of the spin-bath decoherence
We demonstrate that the Chebyshev expansion method is a very efficient
numerical tool for studying spin-bath decoherence of quantum systems. We
consider two typical problems arising in studying decoherence of quantum
systems consisting of few coupled spins: (i) determining the pointer states of
the system, and (ii) determining the temporal decay of quantum oscillations. As
our results demonstrate, for determining the pointer states, the
Chebyshev-based scheme is at least a factor of 8 faster than existing
algorithms based on the Suzuki-Trotter decomposition. For the problems of
second type, the Chebyshev-based approach has been 3--4 times faster than the
Suzuki-Trotter-based schemes. This conclusion holds qualitatively for a wide
spectrum of systems, with different spin baths and different Hamiltonians.Comment: 8 pages (RevTeX), 3 EPS figure
Quantum Dynamics of Spin Wave Propagation Through Domain Walls
Through numerical solution of the time-dependent Schrodinger equation, we
demonstrate that magnetic chains with uniaxial anisotropy support stable
structures, separating ferromagnetic domains of opposite magnetization. These
structures, domain walls in a quantum system, are shown to remain stable if
they interact with a spin wave. We find that a domain wall transmits the
longitudinal component of the spin excitations only. Our results suggests that
continuous, classical spin models described by LLG equation cannot be used to
describe spin wave-domain wall interaction in microscopic magnetic systems
Homeomorphisms generated from overlapping affine iterated function systems
We develop the theory of fractal homeomorphisms generated from pairs of
overlapping affine iterated function systems
Origin of the Canonical Ensemble: Thermalization with Decoherence
We solve the time-dependent Schrodinger equation for the combination of a
spin system interacting with a spin bath environment. In particular, we focus
on the time development of the reduced density matrix of the spin system. Under
normal circumstances we show that the environment drives the reduced density
matrix to a fully decoherent state, and furthermore the diagonal elements of
the reduced density matrix approach those expected for the system in the
canonical ensemble. We show one exception to the normal case is if the spin
system cannot exchange energy with the spin bath. Our demonstration does not
rely on time-averaging of observables nor does it assume that the coupling
between system and bath is weak. Our findings show that the canonical ensemble
is a state that may result from pure quantum dynamics, suggesting that quantum
mechanics may be regarded as the foundation of quantum statistical mechanics.Comment: 12 pages, 4 figures, accepted for publication by J. Phys. Soc. Jp
Updating Maryland\u27s Sea-level Rise Projections
With its 3,100 miles of tidal shoreline and low-lying rural and urban lands, The Free State is one of the most vulnerable to sea-level rise. Historically, Marylanders have long had to contend with rising water levels along its Chesapeake Bay and Atlantic Ocean and coastal bay shores. Shorelines eroded and low-relief lands and islands, some previously inhabited, were inundated. Prior to the 20th century, this was largely due to the slow sinking of the land since Earth’s crust is still adjusting to the melting of large masses of ice following the last glacial period. Over the 20th century, however, the rate of rise of the average level of tidal waters with respect to land, or relative sea-level rise, has increased, at least partially as a result of global warming. Moreover, the scientific evidence is compelling that Earth’s climate will continue to warm and its oceans will rise even more rapidly.
Recognizing the scientific consensus around global climate change, the contribution of human activities to it, and the vulnerability of Maryland’s people, property, public investments, and natural resources, Governor Martin O’Malley established the Maryland Commission on Climate Change on April 20, 2007. The Commission produced a Plan of Action1 that included a comprehensive climate change impact assessment, a greenhouse gas reduction strategy, and strategies for reducing Maryland’s vulnerability to climate change. The Plan has led to landmark legislation to reduce the state’s greenhouse gas emissions and a variety of state policies designed to reduce energy consumption and promote adaptation to climate change
Finite Element Modeling of Ultrasonic Waves in Viscoelastic Media
Linear viscoelasticity theory offers a minimal framework within which to construct a consistent, linear and causal model of mechanical wave dispersion. The term dispersion is used here to imply temporal wave spreading and amplitude reduction due to absorptive material properties rather than due to geometrical wave spreading. Numerical modeling of wave propagation in absorptive media has been the subject of recent research in such areas as material property measurement [1] [2], seismology [3] [4] [5] and medical ultrasound [6] [7]. Previously, wave attenuation has been included in transient finite element formulations via a constant damping matrix [8] or functionally in terms of a power law relation [9]. The formulation presented here is based on representing the viscoelastic shear and bulk moduli of the medium as either a discrete or continuous spectrum of decaying exponentials [10]. As a first test of the correctness of the viscoelastic finite element formulation, the finite element results for a simple hypothetical medium are compared with an equivalent Laplace-Hankel transform domain solution.</p
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