Astrophysical dynamos are usually characterised by huge values of the magnetic Reynolds number (Rm). This reflects the short turn-over time compared to the resistive time. The extreme values of Rm relevant to astrophysical objects cannot be tackled with today's numerical resources and this number is always under-estimated by several orders of magnitudes in numerical models. Here we chose to focus on an extremely simplified problem (dynamo action from a periodic steady flow) and take advantage

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... this simplicity to numerically investigate the limit of very large magnetic Reynolds number. We present results recently published in physics journals, which highlight the difficulty of approaching the limit in which dynamo action is independent of the value of the ohmic resistivity (measured by 1/Rm), known as the "fast dynamo" limit. Using state of the art high performance computing, we present high resolution simulations (up to (40963) and extend the value of (Rm) up to (5⋅105).