71,381 research outputs found
Special Second Order Non Symmetric Fitted Method for Singular Perturbation Problems
In this paper, we present a special second order non symmetric fitted difference method for solving singular perturbed two point boundary value problems having boundary layer at one end. We introduce a fitting factor in the special second order non symmetric finite difference scheme which takes care of the rapid changes occur that in the boundary layer. The value of this fitting factor is obtained from the theory of singular perturbations. The discrete invariant imbedding algorithm is used to solve the tridiagonal system obtained by the method. We discuss the existence and uniqueness of the discrete problem along with stability estimates and the convergence of the method. We present the maximum absolute errors in numerical results to illustrate the proposed method. Keywords: Singularly perturbed two-point boundary value problem, Boundary layer, Fitting factor, Maximum absolute erro
A neighboring extremal solution for an optimal switched impulsive control problem
This paper presents a neighboring extremal solution for a class of optimal switched impulsive control problems with perturbations in the initial state, terminal condition and system's parameters. The sequence of mode's switching is pre-specified, and the decision variables, i.e. the switching times and parameters of the system involved, have inequality constraints. It is assumed that the active status of these constraints is unchanged with the perturbations. We derive this solution by expanding the necessary conditions for optimality to first-order and then solving the resulting multiple-point boundary-value problem by the backward sweep technique. Numerical simulations are presented to illustrate this solution method
On Variational Data Assimilation in Continuous Time
Variational data assimilation in continuous time is revisited. The central
techniques applied in this paper are in part adopted from the theory of optimal
nonlinear control. Alternatively, the investigated approach can be considered
as a continuous time generalisation of what is known as weakly constrained four
dimensional variational assimilation (WC--4DVAR) in the geosciences. The
technique allows to assimilate trajectories in the case of partial observations
and in the presence of model error. Several mathematical aspects of the
approach are studied. Computationally, it amounts to solving a two point
boundary value problem. For imperfect models, the trade off between small
dynamical error (i.e. the trajectory obeys the model dynamics) and small
observational error (i.e. the trajectory closely follows the observations) is
investigated. For (nearly) perfect models, this trade off turns out to be
(nearly) trivial in some sense, yet allowing for some dynamical error is shown
to have positive effects even in this situation. The presented formalism is
dynamical in character; no assumptions need to be made about the presence (or
absence) of dynamical or observational noise, let alone about their statistics.Comment: 28 Pages, 12 Figure
On the stability of self-gravitating accreting flows
Analytic methods show stability of the stationary accretion of test fluids
but they are inconclusive in the case of self-gravitating stationary flows. We
investigate numerically stability of those stationary flows onto compact
objects that are transonic and rich in gas. In all studied examples solutions
appear stable. Numerical investigation suggests also that the analogy between
sonic and event horizons holds for small perturbations of compact support but
fails in the case of finite perturbations.Comment: 10 pages, accepted for publication in PR
Sensitivity And Out-Of-Sample Error in Continuous Time Data Assimilation
Data assimilation refers to the problem of finding trajectories of a
prescribed dynamical model in such a way that the output of the model (usually
some function of the model states) follows a given time series of observations.
Typically though, these two requirements cannot both be met at the same
time--tracking the observations is not possible without the trajectory
deviating from the proposed model equations, while adherence to the model
requires deviations from the observations. Thus, data assimilation faces a
trade-off. In this contribution, the sensitivity of the data assimilation with
respect to perturbations in the observations is identified as the parameter
which controls the trade-off. A relation between the sensitivity and the
out-of-sample error is established which allows to calculate the latter under
operational conditions. A minimum out-of-sample error is proposed as a
criterion to set an appropriate sensitivity and to settle the discussed
trade-off. Two approaches to data assimilation are considered, namely
variational data assimilation and Newtonian nudging, aka synchronisation.
Numerical examples demonstrate the feasibility of the approach.Comment: submitted to Quarterly Journal of the Royal Meteorological Societ
Gravitational perturbations of Schwarzschild spacetime at null infinity and the hyperboloidal initial value problem
We study gravitational perturbations of Schwarzschild spacetime by solving a
hyperboloidal initial value problem for the Bardeen-Press equation.
Compactification along hyperboloidal surfaces in a scri-fixing gauge allows us
to have access to the gravitational waveform at null infinity in a general
setup. We argue that this hyperboloidal approach leads to a more accurate and
efficient calculation of the radiation signal than the common approach where a
timelike outer boundary is introduced. The method can be generalized to study
perturbations of Kerr spacetime using the Teukolsky equation.Comment: 14 pages, 9 figure
Superheating fields of superconductors: Asymptotic analysis and numerical results
The superheated Meissner state in type-I superconductors is studied both
analytically and numerically within the framework of Ginzburg-Landau theory.
Using the method of matched asymptotic expansions we have developed a
systematic expansion for the solutions of the Ginzburg-Landau equations in the
limit of small , and have determined the maximum superheating field
for the existence of the metastable, superheated Meissner state as
an expansion in powers of . Our numerical solutions of these
equations agree quite well with the asymptotic solutions for . The
same asymptotic methods are also used to study the stability of the solutions,
as well as a modified version of the Ginzburg-Landau equations which
incorporates nonlocal electrodynamics. Finally, we compare our numerical
results for the superheating field for large- against recent asymptotic
results for large-, and again find a close agreement. Our results
demonstrate the efficacy of the method of matched asymptotic expansions for
dealing with problems in inhomogeneous superconductivity involving boundary
layers.Comment: 14 pages, 8 uuencoded figures, Revtex 3.
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