  Supermassive stars are Population III stars with masses exceeding
$10^4\,M_{\odot}$ that could be the progenitors of the first supermassive black
holes. Their interiors are in a regime where radiation pressure dominates the
equation of state. In this work, we use the explicit gas dynamics code PPMstar
to simulate the hydrogen-burning core of a $10^4\,M_{\odot}$ supermassive
main-sequence star. These are the first 3D hydrodynamics simulations of core
convection in supermassive stars. We perform a series of ten simulations at
different heating rates and on Cartesian grids with resolutions of $768^3$,
$1152^3$ and $1728^3$. We examine different properties of the convective flow,
including its large-scale morphology, its velocity spectrum and its mixing
properties. We conclude that the radiation pressure-dominated nature of the
interior does not noticeably affect the behaviour of convection compared to the
case of core convection in a massive main-sequence star where gas pressure
dominates. Our simulations also offer support for the use of mixing-length
theory in 1D models of supermassive stars.
