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.
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