A class of high-order Runge-Kutta-Chebyshev stability polynomials

The analytic form of a new class of factorized Runge-Kutta-Chebyshev (FRKC) stability polynomials of arbitrary order $N$ is presented. Roots of FRKC stability polynomials of degree $L=MN$ are used to construct explicit schemes comprising $L$ forward Euler stages with internal stability ensured through a sequencing algorithm which limits the internal amplification factors to $\sim L^2$. The associated stability domain scales as $M^2$ along the real axis. Marginally stable real-valued points on the interior of the stability domain are removed via a prescribed damping procedure. By construction, FRKC schemes meet all linear order conditions; for nonlinear problems at orders above 2, complex splitting or Butcher series composition methods are required. Linear order conditions of the FRKC stability polynomials are verified at orders 2, 4, and 6 in numerical experiments. Comparative studies with existing methods show the second-order unsplit FRKC2 scheme and higher order (4 and 6) split FRKCs schemes are efficient for large moderately stiff problems.Comment: 24 pages, 5 figures. Accepted for publication in Journal of Computational Physics, 22 Jul 2015. Revise

Runge-Kutta-Gegenbauer explicit methods for advection-diffusion problems

In this paper, Runge-Kutta-Gegenbauer (RKG) stability polynomials of arbitrarily high order of accuracy are introduced in closed form. The stability domain of RKG polynomials extends in the the real direction with the square of polynomial degree, and in the imaginary direction as an increasing function of Gegenbauer parameter. Consequently, the polynomials are naturally suited to the construction of high order stabilized Runge-Kutta (SRK) explicit methods for systems of PDEs of mixed hyperbolic-parabolic type. We present SRK methods composed of $L$ ordered forward Euler stages, with complex-valued stepsizes derived from the roots of RKG stability polynomials of degree $L$. Internal stability is maintained at large stage number through an ordering algorithm which limits internal amplification factors to $10 L^2$. Test results for mildly stiff nonlinear advection-diffusion-reaction problems with moderate ($\lesssim 1$) mesh P\'eclet numbers are provided at second, fourth, and sixth orders, with nonlinear reaction terms treated by complex splitting techniques above second order.Comment: 20 pages, 7 figures, 3 table

Pricing Options under Heston’s Stochastic Volatility Model via 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 Heston’s stochastic volatility model we demonstrate degrees of acceleration over standard explicit methods sufficient to achieve comparable, or superior, efficiencies to a benchmark implicit scheme. We conclude that STS is a powerful tool for the numerical pricing of options and propose them as the method-of-choice for exotic financial instruments in two and multi-factor models.

Predatory lending: attempts to plug the money drain

Abusive lending practices have made headlines around the nation, and the issue remains hotly debated. Stephen O'Sullivan gives an overview and an update on the issue. He focuses on how various regulators and legislators are trying to curb abuses without limiting access to the subprime market.Predatory lending ; Loans

Strong-field tidal distortions of rotating black holes: II. Horizon dynamics from eccentric and inclined orbits

In a previous paper, we developed tools for studying the horizon geometry of a Kerr black hole that is tidally distorted by a binary companion using techniques that require large mass ratios but can be applied to any bound orbit and allow for arbitrary black hole spin. We now apply these tools to generic Kerr black hole orbits. This allows us to investigate horizon dynamics: the tidal field perturbing the horizon's geometry varies over a generic orbit, with significant variations for eccentric orbits. Many of the features of the horizon's behavior found previously carry over to the dynamical case in a natural way. In particular, we find significant offsets between the applied tide and the horizon's response. This leads to bulging in the horizon's geometry which can lag or lead the orbit, depending upon the hole's rotation and the orbit's geometry. An interesting and apparently new feature we find are small-amplitude, high-frequency oscillations in the horizon's response. We have not been able to identify a mechanism for producing these oscillations, but find that they appear most clearly when rapidly rotating black holes are distorted by very strong-field orbits.Comment: 24 pages, 15 figures. Final accepted version, to appear in Phys. Rev.

An explicit scheme for multifluid magnetohydrodynamics

When modeling astrophysical fluid flows, it is often appropriate to discard the canonical magnetohydrodynamic approximation thereby freeing the magnetic field to diffuse with respect to the bulk velocity field. As a consequence, however, the induction equation can become problematic to solve via standard explicit techniques. In particular, the Hall diffusion term admits fast-moving whistler waves which can impose a vanishing timestep limit. Within an explicit differencing framework, a multifluid scheme for weakly ionised plasmas is presented which relies upon a new approach to integrating the induction equation efficiently. The first component of this approach is a relatively unknown method of accelerating the integration of parabolic systems by enforcing stability over large compound timesteps rather than over each of the constituent substeps. This method, Super Time Stepping, proves to be very effective in applying a part of the Hall term up to a known critical value. The excess of the Hall term above this critical value is then included via a new scheme for pure Hall diffusion.Comment: 8 pages; 4 figures; accepted by MNRAS; minor corrections to equations; addition of appendi

Modelling and forecasting UK public finances

In this paper, we present a new model of UK public finances which aims to shed light on recent problems of forecasting the PSBR. The main elements of public spending are treated as endogenous variables which rise in line with GDP over the medium term. Also, the cyclical response of public borrowing to rises in the level of economic activity is more muted when growth is export-led than when it is consumer-led. These two features go a long way towards explaining the rapid deterioration of public finances in the early 1990s and the slow pace of improvement since 1993.

Legal protection of investors, corporate governance, and investable premia in emerging markets

We examine the interaction between the legal protection of investors, corporate governance within firms, institutional development between countries, and investable premia in emerging markets. In a multi country setting and using a novel dataset we find that better-governed firms experience significantly greater stock price increases upon equity market liberalization. We look to see whether well-governed firms in poorly governed countries enjoy an investability premium as measured by Tobin’s q. We find they do. Investors look beyond the seemingly weak country-level governance structures, and focus on corporate governance.Investability, Corporate Governance, Tobin's q, Emerging Markets

Strong-field tidal distortions of rotating black holes: III. Embeddings in hyperbolic 3-space

In previous work, we developed tools for quantifying the tidal distortion of a black hole's event horizon due to an orbiting companion. These tools use techniques which require large mass ratios (companion mass $\mu$ much smaller than black hole mass $M$), but can be used for arbitrary bound orbits, and for any black hole spin. We also showed how to visualize these distorted black holes by embedding their horizons in a global Euclidean 3-space, ${\mathbb{E}}^3$. Such visualizations illustrate interesting and important information about horizon dynamics. Unfortunately, we could not visualize black holes with spin parameter $a_* > \sqrt{3}/2 \approx 0.866$: such holes cannot be globally embedded into ${\mathbb{E}}^3$. In this paper, we overcome this difficulty by showing how to embed the horizons of tidally distorted Kerr black holes in a hyperbolic 3-space, ${\mathbb{H}}^3$. We use black hole perturbation theory to compute the Gaussian curvatures of tidally distorted event horizons, from which we build a two-dimensional metric of their distorted horizons. We develop a numerical method for embedding the tidally distorted horizons in ${\mathbb{H}}^3$. As an application, we give a sequence of embeddings into ${\mathbb{H}}^3$ of a tidally interacting black hole with spin $a_*=0.9999$. A small amplitude, high frequency oscillation seen in previous work shows up particularly clearly in these embeddings.Comment: 10 pages, 6 figure

Cormac McCarthy’s cold pastoral: the overturning of a national allegory

This dissertation will argue that the novels of Cormac McCarthy represent a sustained attack on American literature’s abiding fixation with pastoral. It further argues that such a fixation is very much a national allegory, one that, paradoxically, cannot help but produce a sense of doubt lurking beneath the numerous assertions of individual and national confidence. Cormac McCarthy very much engages with the antinomies of this national allegory. His use of pastoral allegory comes in the form of a broken allegory: a strategy that is very much in keeping with Walter Benjamin’s vision of allegorical fragmentation resulting from permanent historical crisis. This crisis, as McCarthy shows, reaches tipping-point in the modern era: the pastoral’s dream of ‘pure-utility’ is shown to be completely incompatible with the predominance of exchange value and commoditized social relations. The study is in four parts. The first section divides the first four novels in order to explore how they shatter the South’s notion of uniqueness through a depiction of a desecrated pastoral. The second section considers the novel Blood Meridian on its own in order to demonstrate how the novel’s absurdist renunciation of pastoral and the western mythos helps set up the late novels themes of generic and cultural termination. The third looks at the Border Trilogy, and discusses how recourse to the more open wildernesses of the south-west curiously introduces a countervailing theme of disenchantment and pastoral attenuation. The fourth and final section groups together No Country for Old Men and The Road, in order to argue that these late novels elicit a final rejection of pastoral as it collides headlong with the imaginary of late-capitalism