1,798 research outputs found
Adaptive Energy Preserving Methods for Partial Differential Equations
A method for constructing first integral preserving numerical schemes for
time-dependent partial differential equations on non-uniform grids is
presented. The method can be used with both finite difference and partition of
unity approaches, thereby also including finite element approaches. The schemes
are then extended to accommodate -, - and -adaptivity. The method is
applied to the Korteweg-de Vries equation and the Sine-Gordon equation and
results from numerical experiments are presented.Comment: 27 pages; some changes to notation and figure
A Space-time Smooth Artificial Viscosity Method For Nonlinear Conservation Laws
We introduce a new methodology for adding localized, space-time smooth,
artificial viscosity to nonlinear systems of conservation laws which propagate
shock waves, rarefactions, and contact discontinuities, which we call the
-method. We shall focus our attention on the compressible Euler equations in
one space dimension. The novel feature of our approach involves the coupling of
a linear scalar reaction-diffusion equation to our system of conservation laws,
whose solution is the coefficient to an additional (and artificial)
term added to the flux, which determines the location, localization, and
strength of the artificial viscosity. Near shock discontinuities, is
large and localized, and transitions smoothly in space-time to zero away from
discontinuities. Our approach is a provably convergent, spacetime-regularized
variant of the original idea of Richtmeyer and Von Neumann, and is provided at
the level of the PDE, thus allowing a host of numerical discretization schemes
to be employed. We demonstrate the effectiveness of the -method with three
different numerical implementations and apply these to a collection of
classical problems: the Sod shock-tube, the Osher-Shu shock-tube, the
Woodward-Colella blast wave and the Leblanc shock-tube. First, we use a
classical continuous finite-element implementation using second-order
discretization in both space and time, FEM-C. Second, we use a simplified WENO
scheme within our -method framework, WENO-C. Third, we use WENO with the
Lax-Friedrichs flux together with the -equation, and call this WENO-LF-C.
All three schemes yield higher-order discretization strategies, which provide
sharp shock resolution with minimal overshoot and noise, and compare well with
higher-order WENO schemes that employ approximate Riemann solvers,
outperforming them for the difficult Leblanc shock tube experiment.Comment: 34 pages, 27 figure
On a selection principle for multivalued semiclassical flows
We study the semiclassical behaviour of solutions of a Schr ̈odinger equation with a scalar po- tential displaying a conical singularity. When a pure state interacts strongly with the singularity of the flow, there are several possible classical evolutions, and it is not known whether the semiclassical limit cor- responds to one of them. Based on recent results, we propose that one of the classical evolutions captures the semiclassical dynamics; moreover, we propose a selection principle for the straightforward calculation of the regularized semiclassical asymptotics. We proceed to investigate numerically the validity of the proposed scheme, by employing a solver based on a posteriori error control for the Schr ̈odinger equation. Thus, for the problems we study, we generate rigorous upper bounds for the error in our asymptotic approximation. For 1-dimensional problems without interference, we obtain compelling agreement between the regularized asymptotics and the full solution. In problems with interference, there is a quantum effect that seems to survive in the classical limit. We discuss the scope of applicability of the proposed regularization approach, and formulate a precise conjecture
An axisymmetric generalized harmonic evolution code
We describe the first axisymmetric numerical code based on the generalized
harmonic formulation of the Einstein equations which is regular at the axis. We
test the code by investigating gravitational collapse of distributions of
complex scalar field in a Kaluza-Klein spacetime. One of the key issues of the
harmonic formulation is the choice of the gauge source functions, and we
conclude that a damped wave gauge is remarkably robust in this case. Our
preliminary study indicates that evolution of regular initial data leads to
formation both of black holes with spherical and cylindrical horizon
topologies. Intriguingly, we find evidence that near threshold for black hole
formation the number of outcomes proliferates. Specifically, the collapsing
matter splits into individual pulses, two of which travel in the opposite
directions along the compact dimension and one which is ejected radially from
the axis. Depending on the initial conditions, a curvature singularity develops
inside the pulses.Comment: 21 page, 18 figures. v2: minor corrections, added references, new
Fig. 9; journal version
Plasmonic nanoparticle monomers and dimers: From nano-antennas to chiral metamaterials
We review the basic physics behind light interaction with plasmonic
nanoparticles. The theoretical foundations of light scattering on one metallic
particle (a plasmonic monomer) and two interacting particles (a plasmonic
dimer) are systematically investigated. Expressions for effective particle
susceptibility (polarizability) are derived, and applications of these results
to plasmonic nanoantennas are outlined. In the long-wavelength limit, the
effective macroscopic parameters of an array of plasmonic dimers are
calculated. These parameters are attributable to an effective medium
corresponding to a dilute arrangement of nanoparticles, i.e., a metamaterial
where plasmonic monomers or dimers have the function of "meta-atoms". It is
shown that planar dimers consisting of rod-like particles generally possess
elliptical dichroism and function as atoms for planar chiral metamaterials. The
fabricational simplicity of the proposed rod-dimer geometry can be used in the
design of more cost-effective chiral metamaterials in the optical domain.Comment: submitted to Appl. Phys.
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