SLAU2-HLLD numerical flux with wiggle-sensor for stable low Mach magnetohydrodynamics simulations. SLAU2 (Simple Low-dissipation Advection-Upstream-splitting-method 2) is an approximate Riemann solver belonging to the AUSM family of schemes in gas dynamics. SLAU2 captures shock waves stably in a supersonic range and realizes high resolutions in the low-speed range (Mach number 0.01 or less) by controlling numerical dissipation. In a previous study, this scheme was successfully extended to magnetohydrodynamics (MHD) simulations, such as SLAU2-HLL (SLAU2-Harten-Lax-van Leer). However, the present study reveals that SLAU2-HLL exhibits unphysical oscillations in low-speed MHD tests. This problem has been found to be caused by insufficient numerical dissipation at the magnetic field discontinuity in a low speed range; thus, we recover the numerical dissipation only where necessary by introducing a wiggle sensor function to the pressure flux part of SLAU2-HLL. This change has enabled SLAU2-HLL to calculate low-speed MHD without unphysical oscillations. Furthermore, we constructed SLAU2-HLLD hybridized with HLLD (HLL-Discontinuity), instead of HLL, which can handle Alfvén waves. By combining this change with the wiggle-sensing stabilization method, we have succeeded in improving the stability and resolution for low Mach MHD flows from the original SLAU2-HLL. The effect of the modification is demonstrated using one- and two-dimensional numerical examples.