Numerical diffusion

Remarks

Example: for the upwind scheme $u_j^{n+1} = u_j^n - \nu (u_j^n - u_{j-1}^n)$ with $a > 0$, Taylor expansion gives $u_t + \frac{\Delta t}{2} u_{tt} + a u_x - \frac{a \Delta x}{2} u_{xx} = O(\Delta x^2)$. Using $u_{tt} = a^2 u_{xx} + O(\Delta x)$ gives $D = \frac{a \Delta x}{2} (1 - \nu)$, which is positive for $0 < \nu < 1$. For the Lax–Friedrichs method the same computation gives $D = \frac{\Delta x^2}{2 \Delta t} (1 - \nu^2)$, which is larger than for upwind when $0 < \nu < 1$. Non-example: for the Lax–Wendroff method the coefficient of $u_{xx}$ is zero to this order, so $D = 0$. Its leading error is a third-derivative term, which is dispersive (wiggles), not diffusive (smearing). A negative $D$ means anti-diffusion, as for the downwind scheme $u_j^{n+1} = u_j^n - \nu (u_{j+1}^n - u_j^n)$ with $a > 0$, which is unstable.