Linear advection: four schemes on the same square pulse

What you are seeing: the simplest PDE in numerical analysis, $u_t + c\,u_x = 0$, solved four different ways on the same square pulse. The exact solution just translates the pulse to the right at speed $c$ forever, no distortion. The four numerical schemes all try to do that. They all fail in different, instructive ways.

FTCS (forward-time, centered-space) is the obvious first guess and is unconditionally unstable; the solution blows up. Upwind is first-order accurate and dissipative; the pulse smears out but stays positive. Lax-Wendroff is second-order; it preserves the pulse shape better but produces visible oscillations near the discontinuity (Gibbs-like). MacCormack is a predictor-corrector second-order method; similar accuracy to LW. The dashed green curve is the exact solution.