We use a flux transport model to simulate the evolution of the Sun’s total and open magnetic flux over the last 26 solar cycles (1713–1996). Polar field reversals are maintained by varying the meridional flow speed between 11 and 20 m s-1, with the poleward-directed surface flow being slower during low-amplitude cycles. If the strengths of the active regions are fixed but their numbers are taken to be proportional to the cycle amplitude, the open flux is found to scale approximately as the square root of the cycle amplitude. However, the scaling becomes linear if the number of active regions per cycle is fixed but their average strength is taken to be proportional to the cycle amplitude. Even with the inclusion of a secularly varying ephemeral region background, the increase in the total photospheric flux between the Maunder minimum and the end of solar cycle 21 is at most $one-third of its minimum-to-maximum variation during the latter cycle. The simulations are compared with geomagnetic activity and cosmogenic isotope records and are used to derive a new reconstruction of total solar irradiance (TSI). The increase in cycle-averaged TSI since the Maunder minimum is estimated to be $1 W m-2. Because the diffusive decay rate accelerates as the average spacing between active regions decreases, the photospheric magnetic flux and facular brightness grow more slowly than the sunspot number and TSI saturates during the highest amplitude cycles. Subject heading gs: interplanetary medium — solar-terrestrial relations — Sun: activity —
Sun: magnetic fields — Sun: photosphere