It is now widely accepted that most ultraluminous X-ray sources (ULXs) are binary systems whose large (above $10^{39}$ erg s$^{-1}$) apparent luminosities are explained by super-Eddington accretion onto a stellar-mass compact object. Many of the ULXs, especially those containing magnetized neutron stars, are highly variable; some exhibit transient behaviour. Large luminosities might imply large accretion discs that could be therefore prone to the thermal-viscous instability known to drive outbursts of dwarf novae and low-mass X-ray binary transient sources. The aim of this paper is to extend and generalize the X-ray transient disc-instability model to the case of large (outer radius larger than $10^{12}$ cm) accretion discs and apply it to the description of systems with super-Eddington accretion rates at outburst and, in some cases, super-Eddington mass transfer rates. We have used our disc-instability-model code to calculate the time evolution of the accretion disc and the outburst properties. We show that, provided that self-irradiation of the accretion disc is efficient even when the accretion rate exceeds the Eddington value, possibly due to scattering back of the X-ray flux emitted by the central parts of the disc on the outer portions of the disc, heating fronts can reach the discs outer edge generating high accretion rates. We also provide analytical approximations for the observable properties of the outbursts. We have reproduced successfully the observed properties of galactic transients with large discs, such as V404 Cyg, as well as some ULXs such as M51 XT-1. Our model can reproduce the peak luminosity and decay time of ESO 243-39 HLX-1 outbursts if the accretor is a neutron star. Observational tests of our predicted relations between the outburst duration and decay time with peak luminosity would be most welcome.
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