Upwind scheme

Remarks

Example: let $a > 0$ and $0 \le \nu \le 1$. Then $u_j^{n+1} = (1 - \nu) u_j^n + \nu u_{j-1}^n$ is a convex combination, so $\min_j u_j^n \le u_j^{n+1} \le \max_j u_j^n$, and the scheme is stable in the maximum norm. The condition $\nu \le 1$ is the Courant–Friedrichs–Lewy condition: the numerical domain of dependence must contain the characteristic through the point. Non-example: for $a > 0$, the downwind scheme $u_j^{n+1} = u_j^n - \nu (u_{j+1}^n - u_j^n)$ uses the wrong side. It multiplies the Fourier mode $e^{i j \xi}$ by $g = 1 + \nu - \nu e^{i\xi}$, with $\abs{g}^2 = 1 + 2\nu(1 + \nu)(1 - \cos \xi) > 1$ for $\xi \ne 0$, so it is unstable for every $\nu > 0$. For a scalar conservation law $u_t + f(u)_x = 0$, upwinding takes the difference on the side given by the sign of the characteristic speed $f'(u)$. The scheme is first-order accurate in $\Delta x$ for fixed $\nu$.