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Dwarf novae (DNe) and low mass X-ray binaries (LMXBs) show eruptions that are thought to be due to a thermal-viscous instability in their accretion disk. These eruptions provide constraints on angular momentum transport mechanisms. We explore the idea that angular momentum transport could be controlled by the dynamical evolution of the large scale magnetic field. We study the impact of different prescriptions for the magnetic field evolution on the dynamics of the disk. This is a first step in confronting the theory of magnetic field transport with observations. We develop a version of the disk instability model that evolves the density, the temperature and the large scale vertical magnetic flux together. We take into account the accretion driven by turbulence or by a magnetized outflow. To evolve the magnetic flux, we use a toy model with physically motivated prescriptions depending mainly on the local magnetization. We find that allowing magnetic flux to be advected inwards provides the best agreement with DNe lightcurves. This leads to a hybrid configuration with an inner magnetized disk, driven by angular momentum losses to an MHD outflow, sharply transiting to an outer weakly-magnetized turbulent disk, where the eruptions are triggered. The dynamical impact is equivalent to truncating a viscous disk so that it does not extend down to the compact object, with the truncation radius dependent on the magnetic flux and evolving as $\dot{M}^{-2/3}$. Models of DNe and LMXBs lightcurves typically require the outer, viscous disk to be truncated in order to match observations. There is no generic explanation for this truncation. We propose that it is a natural outcome of the presence of large-scale magnetic fields in both DNe and LMXBs, the magnetic flux accumulating towards the center to produce a magnetized disk with a fast accretion timescale.