  The electrical current through an incompressible, viscous and resistive
liquid conductor produces an azimuthal magnetic field that becomes unstable
when the corresponding Hartmann number exceeds a critical value in the order of
20. This Tayler instability, which is not only discussed as a key ingredient of
a non-linear stellar dynamo model (Tayler-Spruit dynamo), but also as a
limiting factor for the maximum size of large liquid metal batteries, was
recently observed experimentally in a column of a liquid metal (Seilmayer et
al., Phys. Rev. Lett. 108, 244501, 2012}. On the basis of an
integro-differential equation approach, we have developed a fully
three-dimensional numerical code, and have utilized it for the simulation of
the Tayler instability at typical viscosities and resistivities of liquid
metals. The resulting growth rates are in good agreement with the experimental
data. We illustrate the capabilities of the code for the detailed simulation of
liquid metal battery problems in realistic geometries.
