With the increasing demand for global high-resolution simulation, scalability becomes an important issue and icosahedral-hexagonal grids are gaining interest in the atmospheric modeling community. Unlike the traditional lat-lon grid, the icosahedral-hexagonal grid does not have the pole-singularity problem and has better grid isotropy property. However, grid refinement strategy on the hexagonal grid remains to be an open question. In this study, we compare the wave reflection properties on the resolution interface with two different mesh refinement approaches on the planar domain: discontinuous and continuous grid refinement. For the discontinuous mesh refinement, we apply the high resolution domain nesting on the background low resolution mother domain for the regular hexagonal grid. For continuous mesh refinement, we use centroidal Voronoi tessellation (CVT) with non-constant density function assigning high density on the high resolution target area and low resolution on the background with continuous resolution transition in between. In the continuous mesh refinement case, the mesh becomes unstructured. As the spurious wave reflection develops on discontinuous resolution interface due to insufficient wave resolvability on the coarse grid, we investigate the effect of the gradual resolution change to see how much it alleviates the generation of spurious wave reflection problem. The results are presented by using both centered and upwind biased advection schemes for selected tests.