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We perform an extensive study of numerical convergence for hot-Jupiter atmospheric flow solutions in simulations employing a setup commonly-used in extrasolar planet studies, a resting state thermally forced to a prescribed temperature distribution on a short time-scale at high altitudes. Convergence is assessed rigorously with: (i) a highly-accurate pseudospectral model which has been explicitly verified to perform well under hot-Jupiter flow conditions and (ii) comparisons of the kinetic energy spectra, instantaneous (unaveraged) vorticity fields and temporal evolutions of the vorticity field from simulations which are numerically equatable. In the simulations, the (horizontal and vertical) resolutions, dissipation operator order and viscosity coefficient are varied with identical physical and initial setups. All of the simulations are compared against a fiducial, reference simulation at high horizontal resolution and dissipation order (T682 and $\nabla^{16}$, respectively) -- as well as against each other. Broadly, the reference solution features a dynamic, zonally (east-west) asymmetric jet with a copious amount of small-scale vortices and gravity waves. Here we show that simulations converge to the reference simulation only at T341 resolution and with $\nabla^{16}$ dissipation order. Below this resolution and order, simulations either do not converge or converge to unphysical solutions. The general convergence behaviour is independent of the vertical range of the atmosphere modelled, from $\sim\! 2\!\times\! 10^{-3}$ MPa to $\sim\! 2\!\times\! 10^1$ MPa. Ramifications for current extrasolar planet atmosphere modelling and observations are discussed.