104,572 research outputs found
Time-Symmetric ADI and Causal Reconnection: Stable Numerical Techniques for Hyperbolic Systems on Moving Grids
Moving grids are of interest in the numerical solution of hydrodynamical
problems and in numerical relativity. We show that conventional integration
methods for the simple wave equation in one and more than one dimension exhibit
a number of instabilities on moving grids. We introduce two techniques, which
we call causal reconnection and time-symmetric ADI, which together allow
integration of the wave equation with absolute local stability in any number of
dimensions on grids that may move very much faster than the wave speed and that
can even accelerate. These methods allow very long time-steps, are fully
second-order accurate, and offer the computational efficiency of
operator-splitting.Comment: 45 pages, 19 figures. Published in 1994 but not previously available
in the electronic archive
General relativistic neutrino transport using spectral methods
We present a new code, Lorene's Ghost (for Lorene's gravitational handling of
spectral transport) developed to treat the problem of neutrino transport in
supernovae with the use of spectral methods. First, we derive the expression
for the nonrelativistic Liouville operator in doubly spherical coordinates (r,
theta, phi, epsilon, Theta, Phi)$, and further its general relativistic
counterpart. We use the 3 + 1 formalism with the conformally flat approximation
for the spatial metric, to express the Liouville operator in the Eulerian
frame. Our formulation does not use any approximations when dealing with the
angular arguments (theta, phi, Theta, Phi), and is fully energy-dependent. This
approach is implemented in a spherical shell, using either Chebyshev
polynomials or Fourier series as decomposition bases. It is here restricted to
simplified collision terms (isoenergetic scattering) and to the case of a
static fluid. We finish this paper by presenting test results using basic
configurations, including general relativistic ones in the Schwarzschild
metric, in order to demonstrate the convergence properties, the conservation of
particle number and correct treatment of some general-relativistic effects of
our code. The use of spectral methods enables to run our test cases in a
six-dimensional setting on a single processor.Comment: match published versio
Numerical Analysis of the Non-uniform Sampling Problem
We give an overview of recent developments in the problem of reconstructing a
band-limited signal from non-uniform sampling from a numerical analysis view
point. It is shown that the appropriate design of the finite-dimensional model
plays a key role in the numerical solution of the non-uniform sampling problem.
In the one approach (often proposed in the literature) the finite-dimensional
model leads to an ill-posed problem even in very simple situations. The other
approach that we consider leads to a well-posed problem that preserves
important structural properties of the original infinite-dimensional problem
and gives rise to efficient numerical algorithms. Furthermore a fast multilevel
algorithm is presented that can reconstruct signals of unknown bandwidth from
noisy non-uniformly spaced samples. We also discuss the design of efficient
regularization methods for ill-conditioned reconstruction problems. Numerical
examples from spectroscopy and exploration geophysics demonstrate the
performance of the proposed methods
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