Lax-Friedrichs method
= Lax-Friedrichs method
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{wiki=Lax–Friedrichs_method}
= Lax-Friedrichs scheme
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For $u_t+cu_x=0$ 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 $|\nu|\leq1$. Stability is separate from the dissipative and dispersive approximation errors.