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We present the first astrophysical application of the ECHO code in its recentversion supplemented by a generalized Ohm law, namely a kinematic study ofdynamo effects in thick accretion disks. High-order \emph{implicit-explicit}Runge-Kutta time-stepping routines are implemented and validated within $3+1$\emph{General Relativistic MagnetoHydroDynamics} (GRMHD). The scheme is appliedto a differentially rotating torus orbiting a Kerr black hole, where themean-field dynamo process leads to strong amplification of seed magneticfields. We show that the interplay between the toroidal and poloidal componentsoccurs qualitatively in the same fashion as in the Sun, butterfly diagrams arereproduced, and a typical time-scale for the field evolution is found,depending on the dynamo and resistivity numbers, which could explainperiodicities as observed in several accreting systems.

High-order schemes for non-ideal $3+1$ GRMHD: a study of the kinematic dynamo process in accretion tori