4,644 research outputs found
Spectral methods for multiscale stochastic differential equations
This paper presents a new method for the solution of multiscale stochastic
differential equations at the diffusive time scale. In contrast to
averaging-based methods, e.g., the heterogeneous multiscale method (HMM) or the
equation-free method, which rely on Monte Carlo simulations, in this paper we
introduce a new numerical methodology that is based on a spectral method. In
particular, we use an expansion in Hermite functions to approximate the
solution of an appropriate Poisson equation, which is used in order to
calculate the coefficients of the homogenized equation. Spectral convergence is
proved under suitable assumptions. Numerical experiments corroborate the theory
and illustrate the performance of the method. A comparison with the HMM and an
application to singularly perturbed stochastic PDEs are also presented
Quantization in classical mechanics and reality of Bohm's psi-field
Based on the Chetaev theorem on stable dynamical trajectories in the presence
of dissipative forces, we obtain the generalized condition for stability of
Hamilton systems in the form of the Schrodinger equation. It is shown that the
energy of dissipative forces, which generate the Chetaev generalized condition
of stability, coincides exactly with the Bohm "quantum" potential. Within the
framework of Bohmian quantum mechanics supplemented by the generalized Chetaev
theorem and on the basis of the principle of least action for dissipative
forces, we show that the squared amplitude of a wave function in the
Schrodinger equation is equivalent semantically and syntactically to the
probability density function for the number of particle trajectories, relative
to which the velocity and the position of the particle are not hidden
parameters. The conditions for the correctness of the Bohm-Chetaev
interpretation of quantum mechanics are discussed.Comment: 16 pages, significant improvement after 0806.4050 and 0804.1427. (v2)
revised and reconsidered conclusion
Sequences of Exact Analytical Solutions for Plane-Waves in Graded Media
We present a new method for building sequences of solvable profiles of the
electromagnetic (EM) admittance in lossless isotropic materials with 1D graded
permittivity and permeability (in particular profiles of the optical
refractive-index). These solvable profiles lead to analytical closed-form
expressions of the EM fields, for both TE and TM modes. The Property-and-Field
Darboux Transformations method, initially developed for heat diffusion
modelling, is here transposed to the Maxwell equations in the optical-depth
space. Several examples are provided, all stemming from a constant
seed-potential, which makes them based on elementary functions only. Solvable
profiles of increasingly complex shape can be obtained by iterating the process
or by assembling highly flexible canonical profiles. Their implementation for
modelling optical devices like matching layers, rugate filters, Bragg gratings,
chirped mirrors or 1D photonic crystals, offers an exact and cost-effective
alternative to the classical approachesComment: 74 pages, 20 figures, Corrected typos in Annex
Solving periodic semilinear stiff PDEs in 1D, 2D and 3D with exponential integrators
Dozens of exponential integration formulas have been proposed for the
high-accuracy solution of stiff PDEs such as the Allen-Cahn, Korteweg-de Vries
and Ginzburg-Landau equations. We report the results of extensive comparisons
in MATLAB and Chebfun of such formulas in 1D, 2D and 3D, focusing on fourth and
higher order methods, and periodic semilinear stiff PDEs with constant
coefficients. Our conclusion is that it is hard to do much better than one of
the simplest of these formulas, the ETDRK4 scheme of Cox and Matthews
Accelerating the Fourier split operator method via graphics processing units
Current generations of graphics processing units have turned into highly
parallel devices with general computing capabilities. Thus, graphics processing
units may be utilized, for example, to solve time dependent partial
differential equations by the Fourier split operator method. In this
contribution, we demonstrate that graphics processing units are capable to
calculate fast Fourier transforms much more efficiently than traditional
central processing units. Thus, graphics processing units render efficient
implementations of the Fourier split operator method possible. Performance
gains of more than an order of magnitude as compared to implementations for
traditional central processing units are reached in the solution of the time
dependent Schr\"odinger equation and the time dependent Dirac equation
The quark-gluon plasma, turbulence, and quantum mechanics
Quark-gluon plasmas formed in heavy ion collisions at high energies are well
described by ideal classical fluid equations with nearly zero viscosity. It is
believed that a similar fluid permeated the entire universe at about three
microseconds after the big bang. The estimated Reynolds number for this
quark-gluon plasma at 3 microseconds is approximately 10^19. The possibility
that quantum mechanics may be an emergent property of a turbulent proto-fluid
is tentatively explored. A simple relativistic fluid equation which is
consistent with general relativity and is based on a cosmic dust model is
studied. A proper time transformation transforms it into an inviscid Burgers
equation. This is analyzed numerically using a spectral method. Soliton-like
solutions are demonstrated for this system, and these interact with the known
ergodic behavior of the fluid to yield a stochastic and chaotic system which is
time reversible. Various similarities to quantum mechanics are explored.Comment: 41 pages. Content changes in the azimuthal soliton sectio
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