Lane–Emden test case, preservation of initially static configuration. Top panel: using the standard scheme, a convection like velocity field develops. Shown is absolute velocity (from purple to white; sound speed ranging from 0.76 to 1.39) with velocity arrows (rainbow colored according to magnitude) for the axi-symmetric 256² mesh after 300 τ_s. Bottom panel: using the well-balanced scheme, an initially static polytrope on a 3D Cartesian mesh (128³, star in a box) is preserved to machine precision. Shown are iso-surfaces in density after 300 τ_s. They remain perfectly spherical despite the Cartesian grid, because velocities remain zero to machine precision, thanks to the well-balanced scheme.