5,211 research outputs found
A discrete least squares collocation method for two-dimensional nonlinear time-dependent partial differential equations
In this paper, we develop regularized discrete least squares collocation and
finite volume methods for solving two-dimensional nonlinear time-dependent
partial differential equations on irregular domains. The solution is
approximated using tensor product cubic spline basis functions defined on a
background rectangular (interpolation) mesh, which leads to high spatial
accuracy and straightforward implementation, and establishes a solid base for
extending the computational framework to three-dimensional problems. A
semi-implicit time-stepping method is employed to transform the nonlinear
partial differential equation into a linear boundary value problem. A key
finding of our study is that the newly proposed mesh-free finite volume method
based on circular control volumes reduces to the collocation method as the
radius limits to zero. Both methods produce a large constrained least-squares
problem that must be solved at each time step in the advancement of the
solution. We have found that regularization yields a relatively
well-conditioned system that can be solved accurately using QR factorization.
An extensive numerical investigation is performed to illustrate the
effectiveness of the present methods, including the application of the new
method to a coupled system of time-fractional partial differential equations
having different fractional indices in different (irregularly shaped) regions
of the solution domain
Optimal symmetric flight studies
Several topics in optimal symmetric flight of airbreathing vehicles are examined. In one study, an approximation scheme designed for onboard real-time energy management of climb-dash is developed and calculations for a high-performance aircraft presented. In another, a vehicle model intermediate in complexity between energy and point-mass models is explored and some quirks in optimal flight characteristics peculiar to the model uncovered. In yet another study, energy-modelling procedures are re-examined with a view to stretching the range of validity of zeroth-order approximation by special choice of state variables. In a final study, time-fuel tradeoffs in cruise-dash are examined for the consequences of nonconvexities appearing in the classical steady cruise-dash model. Two appendices provide retrospective looks at two early publications on energy modelling and related optimal control theory
VADER: A Flexible, Robust, Open-Source Code for Simulating Viscous Thin Accretion Disks
The evolution of thin axisymmetric viscous accretion disks is a classic
problem in astrophysics. While models based on this simplified geometry provide
only approximations to the true processes of instability-driven mass and
angular momentum transport, their simplicity makes them invaluable tools for
both semi-analytic modeling and simulations of long-term evolution where two-
or three-dimensional calculations are too computationally costly. Despite the
utility of these models, the only publicly-available frameworks for simulating
them are rather specialized and non-general. Here we describe a highly
flexible, general numerical method for simulating viscous thin disks with
arbitrary rotation curves, viscosities, boundary conditions, grid spacings,
equations of state, and rates of gain or loss of mass (e.g., through winds) and
energy (e.g., through radiation). Our method is based on a conservative,
finite-volume, second-order accurate discretization of the equations, which we
solve using an unconditionally-stable implicit scheme. We implement Anderson
acceleration to speed convergence of the scheme, and show that this leads to
factor of speed gains over non-accelerated methods in realistic
problems, though the amount of speedup is highly problem-dependent. We have
implemented our method in the new code Viscous Accretion Disk Evolution
Resource (VADER), which is freely available for download from
https://bitbucket.org/krumholz/vader/ under the terms of the GNU General Public
License.Comment: 58 pages, 13 figures, accepted to Astronomy & Computing; this version
includes more discussion, but no other changes; code is available for
download from https://bitbucket.org/krumholz/vader
Einstein equations in the null quasi-spherical gauge III: numerical algorithms
We describe numerical techniques used in the construction of our 4th order
evolution for the full Einstein equations, and assess the accuracy of
representative solutions. The code is based on a null gauge with a
quasi-spherical radial coordinate, and simulates the interaction of a single
black hole with gravitational radiation. Techniques used include spherical
harmonic representations, convolution spline interpolation and filtering, and
an RK4 "method of lines" evolution. For sample initial data of "intermediate"
size (gravitational field with 19% of the black hole mass), the code is
accurate to 1 part in 10^5, until null time z=55 when the coordinate condition
breaks down.Comment: Latex, 38 pages, 29 figures (360Kb compressed
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