42 research outputs found
Bouncing droplets on a billiard table
In a set of experiments, Couder et. al. demonstrate that an oscillating fluid
bed may propagate a bouncing droplet through the guidance of the surface waves.
We present a dynamical systems model, in the form of an iterative map, for a
droplet on an oscillating bath. We examine the droplet bifurcation from
bouncing to walking, and prescribe general requirements for the surface wave to
support stable walking states. We show that in addition to walking, there is a
region of large forcing that may support the chaotic bouncing of the droplet.
Using the map, we then investigate the droplet trajectories for two different
wave responses in a square (billiard ball) domain. We show that for waves which
are quickly damped in space, the long time trajectories in a square domain are
either non-periodic dense curves, or approach a quasiperiodic orbit. In
contrast, for waves which extend over many wavelengths, at low forcing,
trajectories tend to approach an array of circular attracting sets. As the
forcing increases, the attracting sets break down and the droplet travels
throughout space
Spatial Manifestations of Order Reduction in Runge-Kutta Methods for Initial Boundary Value Problems
This paper studies the spatial manifestations of order reduction that occur
when time-stepping initial-boundary-value problems (IBVPs) with high-order
Runge-Kutta methods. For such IBVPs, geometric structures arise that do not
have an analog in ODE IVPs: boundary layers appear, induced by a mismatch
between the approximation error in the interior and at the boundaries. To
understand those boundary layers, an analysis of the modes of the numerical
scheme is conducted, which explains under which circumstances boundary layers
persist over many time steps. Based on this, two remedies to order reduction
are studied: first, a new condition on the Butcher tableau, called weak stage
order, that is compatible with diagonally implicit Runge-Kutta schemes; and
second, the impact of modified boundary conditions on the boundary layer theory
is analyzed.Comment: 41 pages, 9 figure
Unconditional Stability for Multistep ImEx Schemes: Theory
This paper presents a new class of high order linear ImEx multistep schemes
with large regions of unconditional stability. Unconditional stability is a
desirable property of a time stepping scheme, as it allows the choice of time
step solely based on accuracy considerations. Of particular interest are
problems for which both the implicit and explicit parts of the ImEx splitting
are stiff. Such splittings can arise, for example, in variable-coefficient
problems, or the incompressible Navier-Stokes equations. To characterize the
new ImEx schemes, an unconditional stability region is introduced, which plays
a role analogous to that of the stability region in conventional multistep
methods. Moreover, computable quantities (such as a numerical range) are
provided that guarantee an unconditionally stable scheme for a proposed
implicit-explicit matrix splitting. The new approach is illustrated with
several examples. Coefficients of the new schemes up to fifth order are
provided.Comment: 33 pages, 7 figure