49,438 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
Implicit and Implicit-Explicit Strong Stability Preserving Runge-Kutta Methods with High Linear Order
When evolving in time the solution of a hyperbolic partial differential
equation, it is often desirable to use high order strong stability preserving
(SSP) time discretizations. These time discretizations preserve the
monotonicity properties satisfied by the spatial discretization when coupled
with the first order forward Euler, under a certain time-step restriction.
While the allowable time-step depends on both the spatial and temporal
discretizations, the contribution of the temporal discretization can be
isolated by taking the ratio of the allowable time-step of the high order
method to the forward Euler time-step. This ratio is called the strong
stability coefficient. The search for high order strong stability time-stepping
methods with high order and large allowable time-step had been an active area
of research. It is known that implicit SSP Runge-Kutta methods exist only up to
sixth order. However, if we restrict ourselves to solving only linear
autonomous problems, the order conditions simplify and we can find implicit SSP
Runge-Kutta methods of any linear order. In the current work we aim to find
very high linear order implicit SSP Runge-Kutta methods that are optimal in
terms of allowable time-step. Next, we formulate an optimization problem for
implicit-explicit (IMEX) SSP Runge-Kutta methods and find implicit methods with
large linear stability regions that pair with known explicit SSP Runge-Kutta
methods of orders plin=3,4,6 as well as optimized IMEX SSP Runge-Kutta pairs
that have high linear order and nonlinear orders p=2,3,4. These methods are
then tested on sample problems to verify order of convergence and to
demonstrate the sharpness of the SSP coefficient and the typical behavior of
these methods on test problems
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