20,529 research outputs found
Quantum gravity: unification of principles and interactions, and promises of spectral geometry
Quantum gravity was born as that branch of modern theoretical physics that
tries to unify its guiding principles, i.e., quantum mechanics and general
relativity. Nowadays it is providing new insight into the unification of all
fundamental interactions, while giving rise to new developments in modern
mathematics. It is however unclear whether it will ever become a falsifiable
physical theory, since it deals with Planck-scale physics. Reviewing a wide
range of spectral geometry from index theory to spectral triples, we hope to
dismiss the general opinion that the mere mathematical complexity of the
unification programme will obstruct that programme.Comment: This is a contribution to the Proceedings of the 2007 Midwest
Geometry Conference in honor of Thomas P. Branson, published in SIGMA
(Symmetry, Integrability and Geometry: Methods and Applications) at
http://www.emis.de/journals/SIGMA
An almost symmetric Strang splitting scheme for nonlinear evolution equations
In this paper we consider splitting methods for the time integration of
parabolic and certain classes of hyperbolic partial differential equations,
where one partial flow can not be computed exactly. Instead, we use a numerical
approximation based on the linearization of the vector field. This is of
interest in applications as it allows us to apply splitting methods to a wider
class of problems from the sciences.
However, in the situation described the classic Strang splitting scheme,
while still a method of second order, is not longer symmetric. This, in turn,
implies that the construction of higher order methods by composition is limited
to order three only. To remedy this situation, based on previous work in the
context of ordinary differential equations, we construct a class of Strang
splitting schemes that are symmetric up to a desired order.
We show rigorously that, under suitable assumptions on the nonlinearity,
these methods are of second order and can then be used to construct higher
order methods by composition. In addition, we illustrate the theoretical
results by conducting numerical experiments for the Brusselator system and the
KdV equation
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