  Neutron stars are usually modelled as spherical, rotating perfect conductors
with a predominant intrinsic dipolar magnetic field anchored to their stellar
crust. Due to their compactness, General Relativity corrections must be
accounted for in Maxwell's equations, leading to modified interior and exterior
electromagnetic solutions. We present analytical solutions for slowly-rotating
magnetised neutron stars taking into account the magnetic frame-dragging
correction. For typical compactness values, i.e. $R_s \sim 0.5 [R_*]$, we show
that the new terms lead to a percent order correction in the magnetic field
orientation and strength compared to the case with no magnetic frame-dragging
correction. Also, we obtain a self-consistent redistribution of the surface
azimuthal current. We verify the validity of the derived solution through
two-dimensional particle-in-cell simulations of an isolated neutron star.
Defining the azimuthal electric and magnetic field amplitudes during the
transient phase as observables, we prove that the magnetic frame-dragging
correction reduces the transient wave amplitude, as expected from the
analytical solution. We show that simulations are more accurate and stable when
we include all first-order terms. The increased accuracy at lower
spatiotemporal resolutions translates into a reduction in simulation runtimes.
