Azimuthal velocity in the radial direction of the Gresho-Chan vortex at t = 3. Here the resolution is n_x = 64 and simulation was performed with GDSPH (left three columns) and ISPH (right three columns). We compare the optimized kernel at a given N_smooth to the popular Wendland C₂ and C₄ kernels, and vary the number of neighbors in each row (N_smooth = 32,64,128,256), referred here to N32, N64, N128, N256. The solid black line show the analytical solution and the black dots show the results from the simulation. The X axis (cylindrical radius) extends from 0.0 to 0.7, and the vertical extent from the top to the bottom of the analytical solution is 1. We can see a significant improvement in the optimized kernels for the GDSPH method for N_smooth = 32,64,128, where the vortex structure becomes more pronounced. Overall less noise can be seen for all N_smooth. The ISPH method captures the overall vortex structure well for all N_smooth, due to exact linear gradients. But we can see significant improvement in the noise level of the optimized kernels, compared to the C₂ and C₄ kernels.