Advection Scheme Shootout

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.

Figure 1. 1D linear advection

u_t + c u_x = 0

on a periodic domain, square pulse initial condition. Four schemes side-by-side: FTCS, upwind, Lax-Wendroff, MacCormack.

c1.00

CFL0.80

speed3

WHAT TO TRY

Watch the same square pulse advected four ways: FTCS blows up (unconditionally unstable), first-order upwind smears the pulse into a diffusive lump, and Lax-Wendroff and MacCormack ring with dispersive wiggles near the edges.
Raise the CFL number toward 1: the schemes sharpen, but push past the stability limit and even the stable schemes blow up. The total-variation readout flags the instability.
Compare the L2 errors: no second-order linear scheme escapes either diffusion or dispersion at a discontinuity, the Godunov barrier that motivates flux limiters.