The second reason for using a hyperbolic formulation for
numerical analysis is that well posedness of the system gives
bounds on the growth of the solution and its derivatives, as long
as the solution is smooth
. This property, when used in conjuction with stable algorithms,
implies that one can bound the errors made on the simulation.
That is, one knows not only that the error goes to zero as some
power of the step size, but also the proportionality factor of
that power law. In simulating phenomena whose observation would
require hundreds of millions of dollars, a tight control of the
accuracy reached should be required. Nevertheless it should be
noted that raw hyperbolicity estimates alone usually give
exponential bounds with very large growth coefficients, and that
they are not of much value for numerical work.
Many of the systems we shall analyze can be cast as flux conservative equations with sources. This is a direct consequence of the facts that the principal part of the equations depends only on the metric variable, and that the equation for the time derivative of it does not contain derivatives of any of the dynamical variables. This property is important when using codes with variable grid spacing, even more if one considers that there are many standard codes for fluids -which are truly flux conserving- with adaptive grid schema.
It has to be said that flux conservation is important when
dealing with systems that develop shock waves, that is in
convective or more precisely in genuinely non-linear systems (for
a definition of this term and many of the results, see [28,
29]). One should be cautious about any expectation of improvement
by using flux conservative properties in general relativity,
since here the shocks would probably not develop -in particular
the systems are not genuinely non-linear. Rather, when
singularities appear, they would be much worse than mere
discontinuities of some of the dynamical fields.
Due to bad gauge choices, discontinuities resembling shocks have
been observed in numerical simulations, see [33
]. Perhaps, instead of trying to devise an algorithm which allows
one to go through these discontinuities, one should concentrate
on finding better gauges, where it could even happen that the
system cannot be put in flux conserving form. Thus, it is not
clear whether flux conservative forms are relevant for vacuum
general relativity.
Going Further
For the reader wishing to delve further into this precious theory
of hyperbolic systems, while keeping a physicist's approach, I
recommend [20]. For those wishing to see more of the machinery at work, I
recommend the book of Kreiss and Lorentz [21]. Finally, for those who really want to get the latest on the
technical aspects and the modern approach to the problem, I
recommend Taylor's book, [23]. Considerations about numerical analysis and algorithms can be
found in [22]. In particular, that book contains general stable algorithms
for strongly and symmetric hyperbolic systems and numerical error
bounds in terms of analytic bounds of the exact solutions
applicable to non-linear systems.
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Hyperbolic methods for Einstein's Equations
Oscar A. Reula http://www.livingreviews.org/lrr-1998-3 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |