2D Physical View
1D Eulerian Grid Density Profile
Removing the Smear: To stop the extreme diffusion of the Lax-Friedrichs scheme, we must upgrade to higher-order solvers.
Godunov's Theorem: Try the Lax-Wendroff scheme. Notice how it perfectly preserves the sharp vertical cliff of the shock front (removing the smearing). However, it introduces physical impossibilities: ripples and negative densities trailing behind the shock. This "ringing" proves that no linear second-order scheme can handle a shockwave perfectly!