  In this study, we investigate the dispersive properties of smoothed particle
magnetohydrodynamics (SPM) in a strongly magnetized medium by using linear
analysis. In modern SPM, a correction term proportional to the divergence of
the magnetic fields is subtracted from the equation of motion to avoid a
numerical instability arising in a strongly magnetized medium. From the linear
analysis, it is found that SPM with the correction term suffer from significant
dispersive errors, especially for slow waves propagating along magnetic fields.
The phase velocity for all wave numbers is significantly larger than the exact
solution and has a peak at a finite wavenumber. These excessively large
dispersive errors occur because magnetic fields contribute an unphysical
repulsive force along magnetic fields. The dispersive errors cannot be reduced,
even with a larger smoothing length and smoother kernel functions such as the
Gaussian or quintic spline kernels. We perform the linear analysis for this
problem and find that the dispersive errors can be removed completely while
keeping SPM stable if the correction term is reduced by half. These findings
are confirmed by several simple numerical experiments.
