Optimal execution strategies in limit order books with general shape functions

We consider optimal execution strategies for block market orders placed in a limit order book (LOB). We build on the resilience model proposed by Obizhaeva and Wang (2005) but allow for a general shape of the LOB defined via a given density function. Thus, we can allow for empirically observed LOB shapes and obtain a nonlinear price impact of market orders. We distinguish two possibilities for modeling the resilience of the LOB after a large market order: the exponential recovery of the number of limit orders, i.e., of the volume of the LOB, or the exponential recovery of the bid-ask spread. We consider both of these resilience modes and, in each case, derive explicit optimal execution strategies in discrete time. Applying our results to a block-shaped LOB, we obtain a new closed-form representation for the optimal strategy, which explicitly solves the recursive scheme given in Obizhaeva and Wang (2005). We also provide some evidence for the robustness of optimal strategies with respect to the choice of the shape function and the resilience-type

MAESTRO: An Adaptive Low Mach Number Hydrodynamics Algorithm for Stellar Flows

Many astrophysical phenomena are highly subsonic, requiring specialized numerical methods suitable for long-time integration. In a series of earlier papers we described the development of MAESTRO, a low Mach number stellar hydrodynamics code that can be used to simulate long-time, low-speed flows that would be prohibitively expensive to model using traditional compressible codes. MAESTRO is based on an equation set derived using low Mach number asymptotics; this equation set does not explicitly track acoustic waves and thus allows a significant increase in the time step. MAESTRO is suitable for two- and three-dimensional local atmospheric flows as well as three-dimensional full-star flows. Here, we continue the development of MAESTRO by incorporating adaptive mesh refinement (AMR). The primary difference between MAESTRO and other structured grid AMR approaches for incompressible and low Mach number flows is the presence of the time-dependent base state, whose evolution is coupled to the evolution of the full solution. We also describe how to incorporate the expansion of the base state for full-star flows, which involves a novel mapping technique between the one-dimensional base state and the Cartesian grid, as well as a number of overall improvements to the algorithm. We examine the efficiency and accuracy of our adaptive code, and demonstrate that it is suitable for further study of our initial scientific application, the convective phase of Type Ia supernovae.Comment: Accepted to Astrophysical Journal Suppliment (http://iop.org). 56 pages, 15 figures

Low Mach Number Modeling of Type Ia Supernovae. IV. White Dwarf Convection

We present the first three-dimensional, full-star simulations of convection in a white dwarf preceding a Type Ia supernova, specifically the last few hours before ignition. For these long-time calculations we use our low Mach number hydrodynamics code, MAESTRO, which we have further developed to treat spherical stars centered in a three-dimensional Cartesian geometry. The main change required is a procedure to map the one-dimensional radial base state to and from the Cartesian grid. Our models recover the dipole structure of the flow seen in previous calculations, but our long-time integration shows that the orientation of the dipole changes with time. Furthermore, we show the development of gravity waves in the outer, stable portion of the star. Finally, we evolve several calculations to the point of ignition and discuss the range of ignition radii.Comment: 42 pages, some figures degraded to conserve space. Accepted to The Astrophysical Journal (http://journals.iop.org/

A Low Mach Number Model for Moist Atmospheric Flows

We introduce a low Mach number model for moist atmospheric flows that accurately incorporates reversible moist processes in flows whose features of interest occur on advective rather than acoustic time scales. Total water is used as a prognostic variable, so that water vapor and liquid water are diagnostically recovered as needed from an exact Clausius--Clapeyron formula for moist thermodynamics. Low Mach number models can be computationally more efficient than a fully compressible model, but the low Mach number formulation introduces additional mathematical and computational complexity because of the divergence constraint imposed on the velocity field. Here, latent heat release is accounted for in the source term of the constraint by estimating the rate of phase change based on the time variation of saturated water vapor subject to the thermodynamic equilibrium constraint. We numerically assess the validity of the low Mach number approximation for moist atmospheric flows by contrasting the low Mach number solution to reference solutions computed with a fully compressible formulation for a variety of test problems

An embedded boundary approach for efficient simulations of viscoplastic fluids in three dimensions

We present a methodology for simulating three-dimensional flow of incompressible viscoplastic fluids modelled by generalised Newtonian rheological equations. It is implemented in a highly efficient framework for massively parallelisable computations on block-structured grids. In this context, geometric features are handled by the embedded boundary approach, which requires specialised treatment only in cells intersecting or adjacent to the boundary. This constitutes the first published implementation of an embedded boundary algorithm for simulating flow of viscoplastic fluids on structured grids. The underlying algorithm employs a two-stage Runge-Kutta method for temporal discretisation, in which viscous terms are treated semi-implicitly and projection methods are utilised to enforce the incompressibility constraint. We augment the embedded boundary algorithm to deal with the variable apparent viscosity of the fluids. Since the viscosity depends strongly on the strain rate tensor, special care has been taken to approximate the components of the velocity gradients robustly near boundary cells, both for viscous wall fluxes in cut cells and for updates of apparent viscosity in cells adjacent to them. After performing convergence analysis and validating the code against standard test cases, we present the first ever fully three-dimensional simulations of creeping flow of Bingham plastics around translating objects. Our results shed new light on the flow fields around these objects

Evaluation of solar cell materials for a Solar Power Satellite

Alternative solar cell materials being considered for the solar power satellite are described and price, production, and availability projections through the year 2000 are presented. The chief materials considered are silicon and gallium arsenide

Multi-Dimensional Simulations of Pair-Instability Supernovae

We present preliminary results from multidimensional numerical studies of pair instability supernova (PSN), studying the fluid instabilities that occur in multiple spatial dimensions. We use the new radiation-hydrodynamics code, CASTRO, and introduce a new mapping procedure that defines the initial conditions for the multidimensional runs in such a way that conservation of physical quantities is guaranteed at any level of resolution.Comment: Accepted for publication in Computer Physics Communications. 3 pages. 2 fig

Evaluation of solar cells and arrays for potential solar power satellite applications

Proposed solar array designs and manufacturing methods are evaluated to identify options which show the greatest promise of leading up to the develpment of a cost-effective SPS solar cell array design. The key program elements which have to be accomplished as part of an SPS solar cell array development program are defined. The issues focussed on are: (1) definition of one or more designs of a candidate SPS solar array module, using results from current system studies; (2) development of the necessary manufacturing requirements for the candidate SPS solar cell arrays and an assessment of the market size, timing, and industry infrastructure needed to produce the arrays for the SPS program; (3) evaluation of current DOE, NASA and DOD photovoltaic programs to determine the impacts of recent advances in solar cell materials, array designs and manufacturing technology on the candidate SPS solar cell arrays; and (4) definition of key program elements for the development of the most promising solar cell arrays for the SPS program

Pair Instability Supernovae of Very Massive Population III Stars

Numerical studies of primordial star formation suggest that the first stars in the universe may have been very massive. Stellar models indicate that non-rotating Population III stars with initial masses of 140-260 Msun die as highly energetic pair-instability supernovae. We present new two-dimensional simulations of primordial pair-instability supernovae done with the CASTRO code. Our simulations begin at earlier times than previous multidimensional models, at the onset of core collapse, to capture any dynamical instabilities that may be seeded by collapse and explosive burning. Such instabilities could enhance explosive yields by mixing hot ash with fuel, thereby accelerating nuclear burning, and affect the spectra of the supernova by dredging up heavy elements from greater depths in the star at early times. Our grid of models includes both blue supergiants and red supergiants over the range in progenitor mass expected for these events. We find that fluid instabilities driven by oxygen and helium burning arise at the upper and lower boundaries of the oxygen shell $\sim$ 20 - 100 seconds after core bounce. Instabilities driven by burning freeze out after the SN shock exits the helium core. As the shock later propagates through the hydrogen envelope, a strong reverse shock forms that drives the growth of Rayleigh--Taylor instabilities. In red supergiant progenitors, the amplitudes of these instabilities are sufficient to mix the supernova ejecta.Comment: 42 pages, 15 figures (accepted to ApJ