199,398 research outputs found
A regime of linear stability for the Einstein-scalar field system with applications to nonlinear Big Bang formation
We linearize the Einstein-scalar field equations, expressed relative to
constant mean curvature (CMC)-transported spatial coordinates gauge, around
members of the well-known family of Kasner solutions on . The Kasner solutions model a spatially uniform scalar field
evolving in a (typically) spatially anisotropic spacetime that expands towards
the future and that has a "Big Bang" singularity at . We
place initial data for the linearized system along and study the linear solution's behavior in the collapsing
direction . Our first main result is the proof of an
approximate monotonicity identity for the linear solutions. Using it, we
prove a linear stability result that holds when the background Kasner solution
is sufficiently close to the Friedmann-Lema\^{\i}tre-Robertson-Walker (FLRW)
solution. In particular, we show that as , various
time-rescaled components of the linear solution converge to regular functions
defined along . In addition, we motivate the preferred
direction of the approximate monotonicity by showing that the CMC-transported
spatial coordinates gauge can be viewed as a limiting version of a family of
parabolic gauges for the lapse variable; an approximate monotonicity identity
and corresponding linear stability results also hold in the parabolic gauges,
but the corresponding parabolic PDEs are locally well-posed only in the
direction . Finally, based on the linear stability results, we
outline a proof of the following result, whose complete proof will appear
elsewhere: the FLRW solution is globally nonlinearly stable in the collapsing
direction under small perturbations of its data at .Comment: 73 page
A symmetry-adapted numerical scheme for SDEs
We propose a geometric numerical analysis of SDEs admitting Lie symmetries
which allows us to individuate a symmetry adapted coordinates system where the
given SDE has notable invariant properties. An approximation scheme preserving
the symmetry properties of the equation is introduced. Our algorithmic
procedure is applied to the family of general linear SDEs for which two
theoretical estimates of the numerical forward error are established.Comment: A numerical example adde
New collocation path-following approach for the optimal shape parameter using Kernel method
The goal of this work is to develop a numerical method combining Radial Basic Functions (RBF) kernel and a high order algorithm based on Taylor series and homotopy continuation method. The local RBF approximation applied in strong form allows us to overcome the difficulties of numerical integration and to treat problems of large deformations. Furthermore, the high order algorithm enables to transform the nonlinear problem to a set of linear problems. Determining the optimal value of the shape parameter in RBF kernel is still an outstanding research topic. This optimal value depends on density and distribution of points and the considered problem for e.g. boundary value problems, integral equations, delay-differential equations etc. These have been extensively attempts in literature which end up choosing this optimal value by tests and error or some other ad-hoc means. Our contribution in this paper is to suggest a new strategy using radial basis functions kernel with an automatic reasonable choice of the shape parameter in the nonlinear case which depends on the accuracy and stability of the results. The computational experiments tested on some examples in structural analysis are performed and the comparison with respect to the state of art algorithms from the literature is given
Exponential dichotomies of evolution operators in Banach spaces
This paper considers three dichotomy concepts (exponential dichotomy, uniform
exponential dichotomy and strong exponential dichotomy) in the general context
of non-invertible evolution operators in Banach spaces. Connections between
these concepts are illustrated. Using the notion of Green function, we give
necessary conditions and sufficient ones for strong exponential dichotomy. Some
illustrative examples are presented to prove that the converse of some
implication type theorems are not valid
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