5,761 research outputs found
Symplectic integrators with adaptive time steps
In recent decades, there have been many attempts to construct symplectic
integrators with variable time steps, with rather disappointing results. In
this paper we identify the causes for this lack of performance, and find that
they fall into two categories. In the first, the time step is considered a
function of time alone, \Delta=\Delta(t). In this case, backwards error
analysis shows that while the algorithms remain symplectic, parametric
instabilities arise because of resonance between oscillations of \Delta(t) and
the orbital motion. In the second category the time step is a function of phase
space variables \Delta=\Delta(q,p). In this case, the system of equations to be
solved is analyzed by introducing a new time variable \tau with dt=\Delta(q,p)
d\tau. The transformed equations are no longer in Hamiltonian form, and thus
are not guaranteed to be stable even when integrated using a method which is
symplectic for constant \Delta. We analyze two methods for integrating the
transformed equations which do, however, preserve the structure of the original
equations. The first is an extended phase space method, which has been
successfully used in previous studies of adaptive time step symplectic
integrators. The second, novel, method is based on a non-canonical
mixed-variable generating function. Numerical trials for both of these methods
show good results, without parametric instabilities or spurious growth or
damping. It is then shown how to adapt the time step to an error estimate found
by backward error analysis, in order to optimize the time-stepping scheme.
Numerical results are obtained using this formulation and compared with other
time-stepping schemes for the extended phase space symplectic method.Comment: 23 pages, 9 figures, submitted to Plasma Phys. Control. Fusio
Modelling ripples in Orion with coupled dust dynamics and radiative transfer
In light of the recent detection of direct evidence for the formation of
Kelvin-Helmholtz instabilities in the Orion nebula, we expand upon previous
modelling efforts by numerically simulating the shear-flow driven gas and dust
dynamics in locations where the H region and the molecular cloud
interact. We aim to directly confront the simulation results with the infrared
observations. Methods: To numerically model the onset and full nonlinear
development of the Kelvin-Helmholtz instability we take the setup proposed to
interpret the observations, and adjust it to a full 3D hydrodynamical
simulation that includes the dynamics of gas as well as dust. A dust grain
distribution with sizes between 5-250 nm is used, exploiting the gas+dust
module of the MPI-AMRVAC code, in which the dust species are represented by
several pressureless dust fluids. The evolution of the model is followed well
into the nonlinear phase. The output of these simulations is then used as input
for the SKIRT dust radiative transfer code to obtain infrared images at several
stages of the evolution, which can be compared to the observations. Results: We
confirm that a 3D Kelvin-Helmholtz instability is able to develop in the
proposed setup, and that the formation of the instability is not inhibited by
the addition of dust. Kelvin-Helmholtz billows form at the end of the linear
phase, and synthetic observations of the billows show striking similarities to
the infrared observations. It is pointed out that the high density dust regions
preferentially collect on the flanks of the billows. To get agreement with the
observed Kelvin-Helmholtz ripples, the assumed geometry between the background
radiation, the billows and the observer is seen to be of critical importance.Comment: 8 pages, 10 figure
Time integration and steady-state continuation for 2d lubrication equations
Lubrication equations allow to describe many structurin processes of thin
liquid films. We develop and apply numerical tools suitable for their analysis
employing a dynamical systems approach. In particular, we present a time
integration algorithm based on exponential propagation and an algorithm for
steady-state continuation. In both algorithms a Cayley transform is employed to
overcome numerical problems resulting from scale separation in space and time.
An adaptive time-step allows to study the dynamics close to hetero- or
homoclinic connections. The developed framework is employed on the one hand to
analyse different phases of the dewetting of a liquid film on a horizontal
homogeneous substrate. On the other hand, we consider the depinning of drops
pinned by a wettability defect. Time-stepping and path-following are used in
both cases to analyse steady-state solutions and their bifurcations as well as
dynamic processes on short and long time-scales. Both examples are treated for
two- and three-dimensional physical settings and prove that the developed
algorithms are reliable and efficient for 1d and 2d lubrication equations,
respectively.Comment: 33 pages, 16 figure
Coarsening dynamics in one dimension: The phase diffusion equation and its numerical implementation
Many nonlinear partial differential equations (PDEs) display a coarsening
dynamics, i.e., an emerging pattern whose typical length scale increases
with time. The so-called coarsening exponent characterizes the time
dependence of the scale of the pattern, , and coarsening
dynamics can be described by a diffusion equation for the phase of the pattern.
By means of a multiscale analysis we are able to find the analytical expression
of such diffusion equations. Here, we propose a recipe to implement numerically
the determination of , the phase diffusion coefficient, as a
function of the wavelength of the base steady state .
carries all information about coarsening dynamics and, through the relation
, it allows us to determine the coarsening exponent. The
main conceptual message is that the coarsening exponent is determined without
solving a time-dependent equation, but only by inspecting the periodic
steady-state solutions. This provides a much faster strategy than a forward
time-dependent calculation. We discuss our method for several different PDEs,
both conserved and not conserved
A mollified Ensemble Kalman filter
It is well recognized that discontinuous analysis increments of sequential
data assimilation systems, such as ensemble Kalman filters, might lead to
spurious high frequency adjustment processes in the model dynamics. Various
methods have been devised to continuously spread out the analysis increments
over a fixed time interval centered about analysis time. Among these techniques
are nudging and incremental analysis updates (IAU). Here we propose another
alternative, which may be viewed as a hybrid of nudging and IAU and which
arises naturally from a recently proposed continuous formulation of the
ensemble Kalman analysis step. A new slow-fast extension of the popular
Lorenz-96 model is introduced to demonstrate the properties of the proposed
mollified ensemble Kalman filter.Comment: 16 pages, 6 figures. Minor revisions, added algorithmic summary and
extended appendi
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