The discontinuous Galerkin method for fractional degenerate convection-diffusion equations

We propose and study discontinuous Galerkin methods for strongly degenerate convection-diffusion equations perturbed by a fractional diffusion (L\'evy) operator. We prove various stability estimates along with convergence results toward properly defined (entropy) solutions of linear and nonlinear equations. Finally, the qualitative behavior of solutions of such equations are illustrated through numerical experiments

Bisexual erasure: Perceived attraction patterns of bisexual women and men

This is the author accepted manuscript. The final version is available from Wiley via the DOI in this recordBisexual individuals face identity denial and erasure and qualitative analyses suggest that it may be gendered, such that people stereotype bisexual women as truly heterosexual and bisexual men as truly gay. Across three studies (total N = 787), we examined perceptions of bisexual targets’ attraction patterns. Participants rated the attraction of either a female or male bisexual target to both the same gender/sex and opposite gender/sex. An internal meta‐analysis revealed that heterosexual, lesbian, and gay participants all perceived bisexual men as more attracted to men than to women. No such pattern emerged for bisexual women. These differences between the perception of bisexual women and bisexual men were also reflected in the endorsement of an explicit measure of bisexual erasure. Our findings add to the understanding of the unique bias bisexual people face by showing that perceived attraction patterns may underlie the labelling of bisexual men as “actually gay”.Economic and Social Research Council (ESRC

Coronal heating distribution due to low-frequency wave-driven turbulence

The heating of the lower solar corona is examined using numerical simulations and theoretical models of magnetohydrodynamic turbulence in open magnetic regions. A turbulent energy cascade to small length scales perpendicular to the mean magnetic field can be sustained by driving with low-frequency Alfven waves reflected from mean density and magnetic field gradients. This mechanism deposits energy efficiently in the lower corona, and we show that the spatial distribution of the heating is determined by the mean density through the Alfven speed profile. This provides a robust heating mechanism that can explain observed high coronal temperatures and accounts for the significant heating (per unit volume) distribution below two solar radius needed in models of the origin of the solar wind. The obtained heating per unit mass on the other hand is much more extended indicating that the heating on a per particle basis persists throughout all the lower coronal region considered here.Comment: 19 pages, 5 figures. Accepted for publication in Ap

Fully adaptive multiresolution schemes for strongly degenerate parabolic equations with discontinuous flux

A fully adaptive finite volume multiresolution scheme for one-dimensional strongly degenerate parabolic equations with discontinuous flux is presented. The numerical scheme is based on a finite volume discretization using the Engquist--Osher approximation for the flux and explicit time--stepping. An adaptivemultiresolution scheme with cell averages is then used to speed up CPU time and meet memory requirements. A particular feature of our scheme is the storage of the multiresolution representation of the solution in a dynamic graded tree, for the sake of data compression and to facilitate navigation. Applications to traffic flow with driver reaction and a clarifier--thickener model illustrate the efficiency of this method

On the acceleration of explicit finite difference methods for option pricing

Implicit finite difference methods are conventionally preferred over their explicit counterparts for the numerical valuation of options. In large part the reason for this is a severe stability constraint known as the Courant-Friedrichs-Lewy (CFL) condition which limits the latter class's efficiency. Implicit methods, however, are difficult to implement for all but the most simple of pricing models, whereas explicit techniques are easily adapted to complex problems. For the first time in a financial context, we present an acceleration technique, applicable to explicit finite difference schemes describing diffusive processes with symmetric evolution operators, called Super-Time-Stepping. We show that this method can be implemented as part of a more general approach for non-symmetric operators. Formal stability is thereby deduced for the exemplar cases of European and American put options priced under the Black-Scholes equation. Furthermore, we introduce a novel approach to describing the efficiencies of finite difference schemes as semi-empirical power laws relating the minimal real time required to carry out the numerical integration to a solution with a specified accuracy. Tests are described in which the method is shown to significantly ameliorate the severity of the CFL constraint whilst retaining the simplicity of the underlying explicit method. Degrees of acceleration are achieved yielding comparable, or superior, efficiencies to a set of benchmark implicit schemes. We infer that the described method is a powerful tool, the explicit nature of which makes it ideally suited to the treatment of symmetric and non-symmetric diffusion operators describing complex financial instruments including multi-dimensional systems requiring representation on decomposed and/or adaptive meshes.Numerical methods for option pricing, Black-Scholes model, Computational finance, Equity options, American options, Exotic options,