1,464 research outputs found
A numerical comparison between degenerate parabolic and quasilinear hyperbolic models of cell movements under chemotaxis
We consider two models which were both designed to describe the movement of
eukaryotic cells responding to chemical signals. Besides a common standard
parabolic equation for the diffusion of a chemoattractant, like chemokines or
growth factors, the two models differ for the equations describing the movement
of cells. The first model is based on a quasilinear hyperbolic system with
damping, the other one on a degenerate parabolic equation. The two models have
the same stationary solutions, which may contain some regions with vacuum. We
first explain in details how to discretize the quasilinear hyperbolic system
through an upwinding technique, which uses an adapted reconstruction, which is
able to deal with the transitions to vacuum. Then we concentrate on the
analysis of asymptotic preserving properties of the scheme towards a
discretization of the parabolic equation, obtained in the large time and large
damping limit, in order to present a numerical comparison between the
asymptotic behavior of these two models. Finally we perform an accurate
numerical comparison of the two models in the time asymptotic regime, which
shows that the respective solutions have a quite different behavior for large
times.Comment: One sentence modified at the end of Section 4, p. 1
Strongly hyperbolic second order Einstein's evolution equations
BSSN-type evolution equations are discussed. The name refers to the
Baumgarte, Shapiro, Shibata, and Nakamura version of the Einstein evolution
equations, without introducing the conformal-traceless decomposition but
keeping the three connection functions and including a densitized lapse. It is
proved that a pseudo-differential first order reduction of these equations is
strongly hyperbolic. In the same way, densitized Arnowitt-Deser-Misner
evolution equations are found to be weakly hyperbolic. In both cases, the
positive densitized lapse function and the spacelike shift vector are arbitrary
given fields. This first order pseudodifferential reduction adds no extra
equations to the system and so no extra constraints.Comment: LaTeX, 16 pages, uses revtex4. Referee corections and new appendix
added. English grammar improved; typos correcte
A direct primitive variable recovery scheme for hyperbolic conservative equations: the case of relativistic hydrodynamics
In this article we develop a Primitive Variable Recovery Scheme (PVRS) to
solve any system of coupled differential conservative equations. This method
obtains directly the primitive variables applying the chain rule to the time
term of the conservative equations. With this, a traditional finite volume
method for the flux is applied in order avoid violation of both, the entropy
and "Rankine-Hugoniot" jump conditions. The time evolution is then computed
using a forward finite difference scheme. This numerical technique evades the
recovery of the primitive vector by solving an algebraic system of equations as
it is often used and so, it generalises standard techniques to solve these kind
of coupled systems. The article is presented bearing in mind special
relativistic hydrodynamic numerical schemes with an added pedagogical view in
the appendix section in order to easily comprehend the PVRS. We present the
convergence of the method for standard shock-tube problems of special
relativistic hydrodynamics and a graphical visualisation of the errors using
the fluctuations of the numerical values with respect to exact analytic
solutions. The PVRS circumvents the sometimes arduous computation that arises
from standard numerical methods techniques, which obtain the desired primitive
vector solution through an algebraic polynomial of the charges.Comment: 19 pages, 6 figures, 2 tables. Accepted for publication in PLOS ON
A Scheme to Numerically Evolve Data for the Conformal Einstein Equation
This is the second paper in a series describing a numerical implementation of
the conformal Einstein equation. This paper deals with the technical details of
the numerical code used to perform numerical time evolutions from a "minimal"
set of data.
We outline the numerical construction of a complete set of data for our
equations from a minimal set of data. The second and the fourth order
discretisations, which are used for the construction of the complete data set
and for the numerical integration of the time evolution equations, are
described and their efficiencies are compared. By using the fourth order scheme
we reduce our computer resource requirements --- with respect to memory as well
as computation time --- by at least two orders of magnitude as compared to the
second order scheme.Comment: 20 pages, 12 figure
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