12,655 research outputs found
A High-Order Method for Stiff Boundary Value Problems with Turning Points
This paper describes some high-order collocation-like methods for the numerical solution of stiff boundary-value problems with turning points. The presentation concentrates on the implementation of these methods in conjunction with the implementation of the a priori mesh construction algorithm introduced by Kreiss, Nichols and Brown [SIAM J. Numer. Anal., 23 (1986), pp. 325â368] for such problems. Numerical examples are given showing the high accuracy which can be obtained in solving the boundary value problem for singularly perturbed ordinary differential equations with turning points
High-order adaptive time stepping for vesicle suspensions with viscosity contrast
We construct a high-order adaptive time stepping scheme for vesicle
suspensions with viscosity contrast. The high-order accuracy is achieved using
a spectral deferred correction (SDC) method, and adaptivity is achieved by
estimating the local truncation error with the numerical error of physically
constant values. Numerical examples demonstrate that our method can handle
suspensions with vesicles that are tumbling, tank-treading, or both. Moreover,
we demonstrate that a user-prescribed tolerance can be automatically achieved
for simulations with long time horizons
Studies of Thermally Unstable Accretion Disks around Black Holes with Adaptive Pseudo-Spectral Domain Decomposition Method I. Limit-Cycle Behavior in the Case of Moderate Viscosity
We present a numerical method for spatially 1.5-dimensional and
time-dependent studies of accretion disks around black holes, that is
originated from a combination of the standard pseudo-spectral method and the
adaptive domain decomposition method existing in the literature, but with a
number of improvements in both the numerical and physical senses. In
particular, we introduce a new treatment for the connection at the interfaces
of decomposed subdomains, construct an adaptive function for the mapping
between the Chebyshev-Gauss-Lobatto collocation points and the physical
collocation points in each subdomain, and modify the over-simplified
1-dimensional basic equations of accretion flows to account for the effects of
viscous stresses in both the azimuthal and radial directions. Our method is
verified by reproducing the best results obtained previously by Szuszkiewicz &
Miller on the limit-cycle behavior of thermally unstable accretion disks with
moderate viscosity. A new finding is that, according to our computations, the
Bernoulli function of the matter in such disks is always and everywhere
negative, so that outflows are unlikely to originate from these disks. We are
encouraged to study the more difficult case of thermally unstable accretion
disks with strong viscosity, and wish to report our results in a subsequent
paper.Comment: 29 pages, 8 figures, accepted by Ap
Explicit schemes for time propagating many-body wavefunctions
Accurate theoretical data on many time-dependent processes in atomic and
molecular physics and in chemistry require the direct numerical solution of the
time-dependent Schr\"odinger equation, thereby motivating the development of
very efficient time propagators. These usually involve the solution of very
large systems of first order differential equations that are characterized by a
high degree of stiffness. We analyze and compare the performance of the
explicit one-step algorithms of Fatunla and Arnoldi. Both algorithms have
exactly the same stability function, therefore sharing the same stability
properties that turn out to be optimum. Their respective accuracy however
differs significantly and depends on the physical situation involved. In order
to test this accuracy, we use a predictor-corrector scheme in which the
predictor is either Fatunla's or Arnoldi's algorithm and the corrector, a fully
implicit four-stage Radau IIA method of order 7. We consider two physical
processes. The first one is the ionization of an atomic system by a short and
intense electromagnetic pulse; the atomic systems include a one-dimensional
Gaussian model potential as well as atomic hydrogen and helium, both in full
dimensionality. The second process is the decoherence of two-electron quantum
states when a time independent perturbation is applied to a planar two-electron
quantum dot where both electrons are confined in an anharmonic potential. Even
though the Hamiltonian of this system is time independent the corresponding
differential equation shows a striking stiffness. For the one-dimensional
Gaussian potential we discuss in detail the possibility of monitoring the time
step for both explicit algorithms. In the other physical situations that are
much more demanding in term of computations, we show that the accuracy of both
algorithms depends strongly on the degree of stiffness of the problem.Comment: 24 pages, 14 Figure
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