3,713 research outputs found
Cauchy-perturbative matching revisited: tests in spherical symmetry
During the last few years progress has been made on several fronts making it
possible to revisit Cauchy-perturbative matching (CPM) in numerical relativity
in a more robust and accurate way. This paper is the first in a series where we
plan to analyze CPM in the light of these new results.
Here we start by testing high-order summation-by-parts operators, penalty
boundaries and contraint-preserving boundary conditions applied to CPM in a
setting that is simple enough to study all the ingredients in great detail:
Einstein's equations in spherical symmetry, describing a black hole coupled to
a massless scalar field. We show that with the techniques described above, the
errors introduced by Cauchy-perturbative matching are very small, and that very
long term and accurate CPM evolutions can be achieved. Our tests include the
accretion and ring-down phase of a Schwarzschild black hole with CPM, where we
find that the discrete evolution introduces, with a low spatial resolution of
\Delta r = M/10, an error of 0.3% after an evolution time of 1,000,000 M. For a
black hole of solar mass, this corresponds to approximately 5 s, and is
therefore at the lower end of timescales discussed e.g. in the collapsar model
of gamma-ray burst engines.
(abridged)Comment: 14 pages, 20 figure
Adaptive Mesh Refinement for Hyperbolic Systems based on Third-Order Compact WENO Reconstruction
In this paper we generalize to non-uniform grids of quad-tree type the
Compact WENO reconstruction of Levy, Puppo and Russo (SIAM J. Sci. Comput.,
2001), thus obtaining a truly two-dimensional non-oscillatory third order
reconstruction with a very compact stencil and that does not involve
mesh-dependent coefficients. This latter characteristic is quite valuable for
its use in h-adaptive numerical schemes, since in such schemes the coefficients
that depend on the disposition and sizes of the neighboring cells (and that are
present in many existing WENO-like reconstructions) would need to be recomputed
after every mesh adaption.
In the second part of the paper we propose a third order h-adaptive scheme
with the above-mentioned reconstruction, an explicit third order TVD
Runge-Kutta scheme and the entropy production error indicator proposed by Puppo
and Semplice (Commun. Comput. Phys., 2011). After devising some heuristics on
the choice of the parameters controlling the mesh adaption, we demonstrate with
many numerical tests that the scheme can compute numerical solution whose error
decays as , where is the average
number of cells used during the computation, even in the presence of shock
waves, by making a very effective use of h-adaptivity and the proposed third
order reconstruction.Comment: many updates to text and figure
Convex-Arc Drawings of Pseudolines
A weak pseudoline arrangement is a topological generalization of a line
arrangement, consisting of curves topologically equivalent to lines that cross
each other at most once. We consider arrangements that are outerplanar---each
crossing is incident to an unbounded face---and simple---each crossing point is
the crossing of only two curves. We show that these arrangements can be
represented by chords of a circle, by convex polygonal chains with only two
bends, or by hyperbolic lines. Simple but non-outerplanar arrangements
(non-weak) can be represented by convex polygonal chains or convex smooth
curves of linear complexity.Comment: 11 pages, 8 figures. A preliminary announcement of these results was
made as a poster at the 21st International Symposium on Graph Drawing,
Bordeaux, France, September 2013, and published in Lecture Notes in Computer
Science 8242, Springer, 2013, pp. 522--52
- âŠ