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Observations, mainly of outbursts in dwarf novae, imply that the anomalousviscosity in highly ionized accretion discs is magnetic in origin, and requiresthat the plasma $\beta \sim 1$. Until now most simulations of the magneticdynamo in accretion discs have used a local approximation (known as theshearing box). While these simulations demonstrate the possibility of aself-sustaining dynamo, the magnetic activity generated in these modelssaturates at $\beta \gg 1$. This long-standing discrepancy has previously beenattributed to the local approximation itself. There have been recent attemptsat simulating magnetic activity in global accretion discs with parametersrelevant to the dwarf novae. These too find values of $\beta \gg 1$. Wespeculate that the tension between these models and the observations may becaused by numerical magnetic diffusivity. As a pedagogical example, we presentexact time-dependent solutions for the evolution of weak magnetic fields in anincompressible fluid subject to linear shear and magnetic diffusivity. We findthat the maximum factor by which the initial magnetic energy can be increaseddepends on the magnetic Reynolds number as ${\mathcal R}_{\rm m}^{2/3}$. Weestimate that current global numerical simulations of dwarf nova discs havenumerical magnetic Reynolds numbers around 6 orders of magnitude less than thephysical value found in dwarf nova discs of ${\mathcal R}_{\rm m} \sim10^{10}$. We suggest that, given the current limitations on computing power,expecting to be able to compute realistic dynamo action in observable accretiondiscs using numerical MHD is, for the time being, a step too far.

On the role of numerical diffusivity in MHD simulations of global accretion disc dynamos