is solved along a given radial direction assuming that the hydrodynamic medium for the direction is spherically symmetric. To improve the accuracy of total energy conservation, we follow the prescription proposed by @mueller. For the calculations presented here, self-gravity is computed by a Newtonian monopole approximation. We use the equation of state (EOS) by @latt91 with a compressibility modulus of $K = 180$ MeV (LS180). Our fiducial 3D models are computed on a spherical polar grid with a resolution of $n_r \times n_{\theta} \times
 n_{\phi}$ = $320 \times 64 \times 128 $, in which non-equally spacial radial zones covers from the center to an outer boundary of 5000 km. The radial grid is chosen such that the resolution $\Delta
 r$ is better than 2km in the PNS interior and typically better than 5km outside the PNS. For the spectral transport, we use 20 logarithmically spaced energy bins ranging from 3 to 300 MeV and we take a ray-by-ray approximation (e.g., @buras06 [@bruenn13]), in which a ray is cast for every angular zone. In all the multi-D runs, the innermost 5 km is computed in spherical symmetry to avoid excessive time-step limitations. Seed perturbations for aspherical instabilities are imposed by hand 