12,310 research outputs found
Numerical Methods for the Fractional Laplacian: a Finite Difference-quadrature Approach
The fractional Laplacian is a non-local operator which
depends on the parameter and recovers the usual Laplacian as . A numerical method for the fractional Laplacian is proposed, based on
the singular integral representation for the operator. The method combines
finite difference with numerical quadrature, to obtain a discrete convolution
operator with positive weights. The accuracy of the method is shown to be
. Convergence of the method is proven. The treatment of far
field boundary conditions using an asymptotic approximation to the integral is
used to obtain an accurate method. Numerical experiments on known exact
solutions validate the predicted convergence rates. Computational examples
include exponentially and algebraically decaying solution with varying
regularity. The generalization to nonlinear equations involving the operator is
discussed: the obstacle problem for the fractional Laplacian is computed.Comment: 29 pages, 9 figure
Analytic results and weighted Monte Carlo simulations for CDO pricing
We explore the possibilities of importance sampling in the Monte Carlo
pricing of a structured credit derivative referred to as Collateralized Debt
Obligation (CDO). Modeling a CDO contract is challenging, since it depends on a
pool of (typically about 100) assets, Monte Carlo simulations are often the
only feasible approach to pricing. Variance reduction techniques are therefore
of great importance. This paper presents an exact analytic solution using
Laplace-transform and MC importance sampling results for an easily tractable
intensity-based model of the CDO, namely the compound Poissonian. Furthermore
analytic formulae are derived for the reweighting efficiency. The computational
gain is appealing, nevertheless, even in this basic scheme, a phase transition
can be found, rendering some parameter regimes out of reach. A
model-independent transform approach is also presented for CDO pricing.Comment: 12 pages, 9 figure
Resistive jet simulations extending radially self-similar magnetohydrodynamic models
Numerical simulations with self-similar initial and boundary conditions
provide a link between theoretical and numerical investigations of jet
dynamics. We perform axisymmetric resistive magnetohydrodynamic (MHD)
simulations for a generalised solution of the Blandford & Payne type, and
compare them with the corresponding analytical and numerical ideal-MHD
solutions. We disentangle the effects of the numerical and physical
diffusivity. The latter could occur in outflows above an accretion disk, being
transferred from the underlying disk into the disk corona by MHD turbulence
(anomalous turbulent diffusivity), or as a result of ambipolar diffusion in
partially ionized flows. We conclude that while the classical magnetic Reynolds
number measures the importance of resistive effects in the
induction equation, a new introduced number, \rbeta=(\beta/2)R_{\rm m} with
the plasma beta, measures the importance of the resistive effects in
the energy equation. Thus, in magnetised jets with , when \rbeta \la
1 resistive effects are non-negligible and affect mostly the energy equation.
The presented simulations indeed show that for a range of magnetic
diffusivities corresponding to \rbeta \ga 1 the flow remains close to the
ideal-MHD self-similar solution.Comment: Accepted for publication in MNRA
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