### Convergence of a three-dimensional
quantum lattice Boltzmann scheme towards solutions of the Dirac equation

D. Lapitski & P. J. Dellar (2011) Convergence
of a three-dimensional quantum lattice Boltzmann scheme towards
solutions of the Dirac equation (in press) Phil. Trans. R. Soc. Lond. A

Preprint available ( DSFD2010_Dirac.pdf
330K)

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Abstract

We investigate the convergence properties of a three-dimensional
quantum lattice Boltzmann scheme for the Dirac equation. These
schemes were constructed as discretisations of the Dirac equation based
on operator splitting, but their output has previously only been
compared against solutions of the Schrödinger equation. The
Schrödinger equation arises as the non-relativistic limit of the
Dirac equation, describing solutions that vary slowly compared to the
Compton frequency. We demonstrate first-order convergence towards
solutions of the Dirac equation obtained by an independent numerical
method based on fast Fourier transforms and matrix

exponentiation.