31,411 research outputs found
Space-time discontinuous Galerkin discretization of rotating shallow water equations on moving grids
A space-time discontinuous Galerkin (DG) discretization is presented for the (rotating) shallow water equations over varying topography. We formulate the space-time DG finite element discretization in an efficient and conservative discretization. The HLLC flux is used as numerical flux through the finite element boundaries. When discontinuities are present, we locally apply dissipation around these discontinuities with the help of Krivodonova's discontinuity indicator such that spurious oscillations are suppressed. The non-linear algebraic system resulting from the discretization is solved using a pseudo-time integration with a second-order five-stage Runge-Kutta method. A thorough verification of the space-time DG finite element method is undertaken by comparing numerical and exact solutions. We also carry out a discrete Fourier analysis of the one dimensional linear rotating shallow water equations to show that the method is unconditionally stable with minimal dispersion and dissipation error. The numerical scheme is validated in a novel way by considering various simulations of bore-vortex interactions in combination with a qualitative analysis of PV generation by non-uniform bores. Finally, the space-time DG method is particularly suited for problems where dynamic grid motion is required. To demonstrate this we simulate waves generated by a wave maker and verify these for low amplitude waves where linear theory is approximately valid
Governing equations of tissue modelling and remodelling: A unified generalised description of surface and bulk balance
Several biological tissues undergo changes in their geometry and in their
bulk material properties by modelling and remodelling processes. Modelling
synthesises tissue in some regions and removes tissue in others. Remodelling
overwrites old tissue material properties with newly formed, immature tissue
properties. As a result, tissues are made up of different "patches", i.e.,
adjacent tissue regions of different ages and different material properties,
within evolving boundaries. In this paper, generalised equations governing the
spatio-temporal evolution of such tissues are developed within the continuum
model. These equations take into account nonconservative, discontinuous surface
mass balance due to creation and destruction of material at moving interfaces,
and bulk balance due to tissue maturation. These equations make it possible to
model patchy tissue states and their evolution without explicitly maintaining a
record of when/where resorption and formation processes occurred. The time
evolution of spatially averaged tissue properties is derived systematically by
integration. These spatially-averaged equations cannot be written in closed
form as they retain traces that tissue destruction is localised at tissue
boundaries.
The formalism developed in this paper is applied to bone tissues, which
exhibit strong material heterogeneities due to their slow mineralisation and
remodelling processes. Evolution equations are proposed in particular for
osteocyte density and bone mineral density. Effective average equations for
bone mineral density (BMD) and tissue mineral density (TMD) are derived using a
mean-field approximation. The error made by this approximation when remodelling
patchy tissue is investigated. The specific time signatures of BMD or TMD
during remodelling events may provide a way to detect these events occurring at
lower, unseen spatial resolutions from microCT scans.Comment: 14 pages, 8 figures. V2: minor stylistic changes, more detailed
derivation of Eqs (30)-(31), additional comments on implication of BMD and
TMD signatures for microCT scan
Effects of Line-tying on Magnetohydrodynamic Instabilities and Current Sheet Formation
An overview of some recent progress on magnetohydrodynamic stability and
current sheet formation in a line-tied system is given. Key results on the
linear stability of the ideal internal kink mode and resistive tearing mode are
summarized. For nonlinear problems, a counterexample to the recent
demonstration of current sheet formation by Low \emph{et al}. [B. C. Low and
\AA. M. Janse, Astrophys. J. \textbf{696}, 821 (2009)] is presented, and the
governing equations for quasi-static evolution of a boundary driven, line-tied
magnetic field are derived. Some open questions and possible strategies to
resolve them are discussed.Comment: To appear in Phys. Plasma
Junction conditions in General Relativity with spin sources
The junction conditions for General Relativity in the presence of domain
walls with intrinsic spin are derived in three and higher dimensions. A stress
tensor and a spin current can be defined just by requiring the existence of a
well defined volume element instead of an induced metric, so as to allow for
generic torsion sources. In general, when the torsion is localized on the
domain wall, it is necessary to relax the continuity of the tangential
components of the vielbein. In fact it is found that the spin current is
proportional to the jump in the vielbein and the stress-energy tensor is
proportional to the jump in the spin connection. The consistency of the
junction conditions implies a constraint between the direction of flow of
energy and the orientation of the spin. As an application, we derive the
circularly symmetric solutions for both the rotating string with tension and
the spinning dust string in three dimensions. The rotating string with tension
generates a rotating truncated cone outside and a flat space-time with
inevitable frame dragging inside. In the case of a string made of spinning
dust, in opposition to the previous case no frame dragging is present inside,
so that in this sense, the dragging effect can be "shielded" by considering
spinning instead of rotating sources. Both solutions are consistently lifted as
cylinders in the four-dimensional case.Comment: 24 pages, no figures, CECS style. References added and misprints
corrected. Published Versio
Lagrangian perturbation theory for a superfluid immersed in an elastic neutron star crust
The inner crust of mature neutron stars, where an elastic lattice of
neutron-rich nuclei coexists with a neutron superfluid, impacts on a range of
astrophysical phenomena. The presence of the superfluid is key to our
understanding of pulsar glitches, and is expected to affect the thermal
conductivity and hence the evolution of the surface temperature. The coupling
between crust and superfluid must also be accounted for in studies of neutron
star dynamics, discussions of global oscillations and associated instabilities.
In this paper we develop Lagrangian perturbation theory for this problem,
paying attention to key issues like superfluid entrainment, potential vortex
pinning, dissipative mutual friction and the star's magnetic field. We also
discuss the nature of the core-crust interface. The results provide a
theoretical foundation for a range of interesting astrophysical applications.Comment: 13 pages, no figures, to appear in MNRA
Area Regge Calculus and Discontinuous Metrics
Taking the triangle areas as independent variables in the theory of Regge
calculus can lead to ambiguities in the edge lengths, which can be interpreted
as discontinuities in the metric. We construct solutions to area Regge calculus
using a triangulated lattice and find that on a spacelike hypersurface no such
discontinuity can arise. On a null hypersurface however, we can have such a
situation and the resulting metric can be interpreted as a so-called refractive
wave.Comment: 18 pages, 1 figur
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