Stephen O'Sullivan

The transport of Cosmic Rays in Self‐Excited Magnetic Turbulence

The process of diffusive shock acceleration relies on the efficacy with which hydromagnetic waves can scatter charged particles in the precursor of a shock. The growth of self-generated waves is driven by both resonant and non-resonant processes. We perform high-resolution magnetohydrodynamic simulations of the non-resonant cosmic ray driven instability, in which the unstable waves are excited beyond the linear regime. In a snapshot of the resultant field, particle transport simulations are carried out. The use of a static snapshot of the field is reasonable given that the Larmor period for particles is typically very short relative to the instability growth time. The diffusion rate is found to be close to, or below, the Bohm limit for a range of energies. This provides the first explicit demonstration that self-excited turbulence reduces the diffusion coefficient and has important implications for cosmic-ray transport and acceleration in supernova remnants.

Stochastic particle acceleration in the lobes of giant radio galaxies

We investigate the acceleration of particles by Alfven waves via the second-order Fermi process in the lobes of giant radio galaxies. Such sites are candidates for the accelerators of ultra-high-energy cosmic rays (UHECR). We focus on the nearby Fanaroff-Riley type I radio galaxy Centaurus A. This is motivated by the coincidence of its position with the arrival direction of several of the highest energy Auger events. The conditions necessary for consistency with the acceleration time-scales predicted by quasi-linear theory are reviewed. Test particle calculations are performed in fields which guarantee electric fields with no component parallel to the local magnetic field. The results of quasi-linear theory are, to an order of magnitude, found to be accurate at low turbulence levels for non-relativistic Alfven waves and at both low and high turbulence levels in the mildly relativistic case. We conclude that for pure stochastic acceleration via Alfven waves to be plausible as the generator of UHECR in Cen A, the baryon number density would need to be several orders of magnitude below currently held upper limits.

A cosmic ray current-driven instability in partially ionised media

Context. We investigate the growth of hydromagnetic waves driven by streaming cosmic rays in the precursor environment of a supernova remnant shock. Aims. It is known that transverse waves propagating parallel to the mean magnetic field are unstable to anisotropies in the cosmic ray distribution, and may provide a mechanism to substantially amplify the ambient magnetic field. We quantify the extent to which temperature and ionisation fractions modify this picture. Methods. Using a kinetic description of the plasma we derive the dispersion relation for a collisionless thermal plasma with a streaming cosmic ray current. Fluid equations are then used to discuss the effects of neutral-ion collisions. Results. We calculate the extent to which the environment into which the cosmic rays propagate influences the growth of the magnetic field, and determines the range of possible growth rates. Conclusions. If the cosmic ray acceleration is efficient, we find that very large neutral fractions are required to stabilise the growth of the non-resonant mode. For typical supernova parameters in our Galaxy, thermal effects do not significantly alter the growth rates. For weakly driven modes, ion-neutral damping can dominate over the instability at more modest ionisation fractions. In the case of a supernova shock interacting with a molecular clouds, such as in RX J1713.7-3946, with high density and low ionisation, the modes can be rapidly damped.

NONIDEAL MAGNETOHYDRODYNAMIC TURBULENT DECAY IN MOLECULAR CLOUDS

It is well known that nonideal magnetohydrodynamic (MHD) effects are important in the dynamics of molecular clouds: both ambipolar diffusion and possibly the Hall effect have been identified as significant. We present the results of a suite of simulations with a resolution of 5123 of turbulent decay in molecular clouds, incorporating a simplified form of both ambipolar diffusion and the Hall effect simultaneously. The initial velocity field in the turbulence is varied from being super-Alfvenic and hypersonic, through to trans-Alfvenic but still supersonic. We find that ambipolar diffusion increases the rate of decay of the turbulence increasing the decay from t –1.25 to t –1.4. The Hall effect has virtually no impact in this regard. The power spectra of density, velocity, and the magnetic field are all affected by the nonideal terms, being steepened significantly when compared with ideal MHD turbulence with exponents. The density power-spectra components change from ~1.4 to ~2.1 for the ideal and nonideal simulations respectively, and power spectra of the other variables all show similar modifications when nonideal effects are considered. Again, the dominant source of these changes is ambipolar diffusion rather than the Hall effect. There is also a decoupling between the velocity field and the magnetic field at short length scales. The Hall effect leads to enhanced magnetic reconnection, and hence less power, at short length scales. The dependence of the velocity dispersion on the characteristic length scale is studied and found not to be power law in nature.

A Stability Study of a New Explicit Numerical Scheme for a System of Differential Equations with a Large Skew-Symmetric Component

Explicit numerical methods for the solution of a system of stiff differential equations suffer from a time step size that approaches zero in order to satisfy stability conditions. Implicit schemes allow a larger time step, but require more computations. When the differential equations are dominated by a skew-symmetric component, the problem is not stiffness in the sense that the size of the eigenvalues are unequal, rather that the real eigenvalues are dominated by imaginary eigenvalues. A skew-dominated system of this type may be seen in magnetohydrodynamics and in control problems. We present and compare analytical results for stable time step limits for several explicit methods including the super-time-stepping method of Alexiades, Amiez, and Gremaud which is an explicit Runge-Kutta method for parabolic partial differential equations and a new method modeled on a predictor-corrector scheme with multiplicative operator splitting. This new explicit method, presented in regular and super-time-stepping form, increases stability without forcing the step size to zero.

ENVIRONMENTAL LIMITS ON THE NONRESONANT COSMIC-RAY CURRENT-DRIVEN INSTABILITY

We investigate the so-called nonresonant cosmic-ray streaming instability, first discussed by Bell (2004). The extent to which thermal damping and ion-neutral collisions reduce the growth of this instability is calculated. Limits on the growth of the nonresonant mode in SN1006 and RX J1713.7-3946 are presented.

Pricing European And American Options In The Heston Model With Accelerated Explicit Finite Differencing Methods

We present an acceleration technique, effective for explicit finite difference schemes describing diffusive processes with nearly symmetric operators, called Super-Time-Stepping (STS). The technique is applied to the two-factor problem of option pricing under stochastic volatility. It is shown to significantly reduce the severity of the stability constraint known as the Courant-Friedrichs-Lewy condition whilst retaining the simplicity of the chosen underlying explicit method.For European and American put options under Hestons stochastic volatility model we demonstrate degrees of acceleration over standard explicit methods sufficient to achieve comparable, or superior, efficiency to benchmark implicit schemes. We conclude that STS accelerated methods are powerful numerical tools for the pricing of options which inherit the simplicity of explicit methods whilst achieving high accuracy at low computational cost and offer a compelling alternative to conventional implicit techniques.

An Explicit Super‐Time‐Stepping Scheme for Non‐Symmetric Parabolic Problems

Explicit numerical methods for the solution of a system of differential equations may suffer from a time step size that approaches zero in order to satisfy stability conditions. When the differential equations are dominated by a skew‐symmetric component, the problem is that the real eigenvalues are dominated by imaginary eigenvalues. We compare results for stable time step limits for the super‐time‐stepping method of Alexiades, Amiez, and Gremaud (super‐time‐stepping methods belong to the Runge‐Kutta‐Chebyshev class) and a new method modeled on a predictor‐corrector scheme with multiplicative operator splitting. This new explicit method increases stability of the original super‐time‐stepping whenever the skew‐symmetric term is nonzero, which occurs in particular convection‐diffusion problems and more generally when the iteration matrix represents a nonlinear operator. The new method is stable for skew symmetric dominated systems where the regular super‐time‐stepping scheme fails. This method is second ord...

Accelerated Trinomial Trees Applied to American Basket Options and American Options Under the Bates Model

The accelerated trinomial tree (ATT) is a derivatives pricing lattice method that circumvents the restrictive time step condition inherent in standard trinomial trees and explicit finite difference methods (FDMs), in which the time step must scale with the square of the spatial step. ATTs consist of L uniform supersteps, each of which contains an inner lattice/trinomial tree with N nonuniform subtime steps. Similarly to implicit FDMs, the size of the superstep in ATTs, a function of N, is constrained primarily by accuracy demands. ATTs can price options up to N times faster than standard trinomial trees (explicit FDMs). ATTs can be interpreted as using risk-neutral extended probabilities: extended in the sense that values can lie outside the range [0, 1] on the substep scale but aggregate to probabilities within the range [0,1] on the superstep scale. Hence, it is only strictly at the end of each superstep that a practically meaningful solution may be extracted from the tree. We demonstrate that ATTs with L supersteps are more efficient or have comparable efficiency to competing implicit methods that use L time steps in pricing Black–Scholes American put options and two-dimensional American basket options. Crucially, this performance is achieved using an algorithm that requires only a modest modification of a standard trinomial tree. This is in contrast to implicit FDMs, which may be relatively complex in their implementation. We also extend ATTs to the pricing of American options under the Heston model and the Bates model in order to demonstrate the general applicability of the approach.