Lax-Friedrichs method

For $u_t+c{u}_x=0$ on a uniform grid, the update $U_m^{n+1}=(U_{m-1}^n+U_{m+1}^n)/2-\nu (U_{m+1}^n-U_{m-1}^n)/2$ has symbol $\cos \theta -i\nu \sin \theta$, where $\nu =c\Delta t/h$. Its squared modulus is $1+({\nu }^2-1){\sin }^2\theta$, giving contraction exactly for $|\nu |\le 1$. Stability is separate from the dissipative and dispersive approximation errors.