  The Biermann Battery effect is frequently invoked in cosmic magnetogenesis
and studied in High-Energy Density laboratory physics experiments. Generation
of magnetic fields by the Biermann effect due to mis-aligned density and
temperature gradients in smooth flow <i>behind</i> shocks is well known. We
show that a Biermann-effect magnetic field is also generated <i>within</i>
shocks. Direct implementation of the Biermann effect in MHD codes does not
capture this physical process, and worse, produces unphysical magnetic fields
at shocks whose value does not converge with resolution. We show that this
convergence breakdown is due to naive discretization, which fails to account
for the fact that discretized irrotational vector fields have spurious
solenoidal components that grow without bound near a discontinuity. We show
that careful consideration of the kinetics of ion viscous shocks leads to a
formulation of the Biermann effect that gives rise to a convergent algorithm.
We note two novel physical effects: a <i>resistive magnetic precursor</i> in
which Biermann-generated field in the shock "leaks" resistively upstream; and a
<i>thermal magnetic precursor</i>, in which field is generated by the Biermann
effect ahead of the shock front due to gradients created by the shock's
electron thermal conduction precursor. Both effects appear to be potentially
observable in experiments at laser facilities. We re-examine published studies
of magnetogenesis in galaxy cluster formation, and conclude that the
simulations in question had inadequate resolution to reliably estimate the
field generation rate. Corrected estimates suggest primordial field values in
the range $B\sim 10^{-22}$G --- $10^{-19}$G by $z=3$.
