### Exponential asymptotics and water waves

When an ideal fluid flows over an obstruction in the stream, the behaviour depends on a nondimensional group called the Froude number, which measures the relative strength of inertia and gravity. When the speed is low (or gravity high) an asymptotic expansion can be made in powers of the Froude number to approximate the flow. However, in 1968 the naval architect T. F. Ogilvie pointed out a peculiarity of this expansion: the approximation predicts a waveless free-surface, no matter how many terms are taken, while physically one would always expect waves to form downstream. This has come to be known as the low-speed paradox. The resolution of the apparent paradox lies in the fact that the waves generated are exponentially small in the Froude number, so that they lie "beyond-all-orders" of the power series expansion. This project is concerned with identifying and quantifying these exponentially small waves.

The power series expansion for low Froude number is divergent, and thus needs to be truncated. By truncating at the smallest term (that is, truncating optimally), the resulting remainder is exponentially small. By examining the equation for this remainder the waves can be seen to appear suddenly as certain lines (known as Stokes lines) are crossed.

The theory was developed first for steady two-dimensional flows, for which complex variable techniques can be employed. More recently it has been extended to surface piercing objects (i.e. ships, where it can be used to determine the wave drag), time-dependent flows, and three-dimensional flows. A particular three-dimensional application concerns the waves formed above a moving submarine, which are analagous to the Kelvin ship waves which form behind a swimming duck.

For more information contact Jon Chapman or Phil Trinh.

#### Key references in this area

**A. B. Olde Daalhuis, S. J. Chapman, J. R. King, J. R. Ockendon, and R. H. Tew**(1996). Stokes phenomenon and matched asymptotic expansions.*SIAM J. App. Math.***55**(6): 1469-1483.**S. J. Chapman and J.-M. Vanden-Broeck**(2006). Exponential asymptotics and gravity waves.*J. Fluid Mech.***567**: 299-326.**P. Trinh, S. J. Chapman, and J.-M. Vanden-Broeck**(2011). Do waveless ships exist? Results for single-cornered hulls.*J. Fluid Mech.***685**: 418-435.

**Quantum Lattice Boltzmann method for quantum electrodynamics**

A quantum lattice Boltzmann (QLB) scheme has been developed for the simulation of relativistic quantum systems and applied to charged particles in graphene. The QLB scheme is based on the lattice Boltzmann method for the simulation of fluid flows, and is an alternative to spectral methods of solving the Dirac equation. Unlike other finite-difference schemes, the QLB method is free from the fermion doubling problem.

Motivated by reports of irregular error behaviour in the simulation of the Schrödinger equation and a broken isotropy in the multidimensional case, a detailed error analysis of the scheme has been performed which showed it to be generally first-order accurate. An initialization condition was identified which would make the scheme second-order accurate due to cancelation of error terms, but which did not correctly correspond to the non-relativistic limit required to simulate the systems governed by the Schrödinger equation. By rederiving the multidimensional version we found the cause of the reported broken symmetry and recovered isotropy of the scheme.

The QLB scheme is even more useful in simulating relativistic quantum systems; an immediate application is the simulation of charge carriers in graphene, governed by the Dirac equation. The QLB scheme was improved to global second-order accuracy, and verified in systems of relativistic quantum tunnelling, simulating the Klein paradox in one and two dimensions. It was also expanded to include non-zero time-dependant vector potentials. Future goals include using the QLB scheme to investigate the ability of a configuration of multiple potential barriers to confine charge carriers in graphene, achieving a high on/off current ratio, and deriving QLB schemes for bilayer graphene and Bose-Einstein condensates in hexagonal optical lattices governed by a non-linear Dirac equation.

For more information please contact Paul Dellar or Denis Lapitski.

### Key references in this area

**S. Succi and R. Benzi**(1993).*Physica D***69**: 327-332.**S. Palpacelli and S. Succi**(2008).*Commun. Comput. Phys***4**: 980-1007.**D. Lapitski & P. J. Dellar**(2011).*Phil. Trans. R. Soc. Lond.*A**369**: 2155-2163.**P. J. Dellar, D. Lapitski, S. Palpacelli & S. Succi**(2011).*Phys. Rev. E***83**: 46706.