Computational fluid dynamics

Remarks

Example: for the advection equation $\partial_t u + a \, \partial_x u = 0$ with $a > 0$, the upwind scheme on a grid $x_j = j h$, $t_n = n \Delta t$ is $$u_j^{n+1} = u_j^n - \frac{a \Delta t}{h} (u_j^n - u_{j-1}^n).$$ When $0 \le a \Delta t / h \le 1$, the new value is a convex combination of $u_j^n$ and $u_{j-1}^n$, so $\max_j \abs{u_j^n}$ does not grow; this is the Courant–Friedrichs–Lewy condition, and the scheme converges. If $a \Delta t / h > 1$, the coefficient of $u_j^n$ is negative and errors grow, so the scheme is unstable. Non-example: the exact parabolic profile of Poiseuille flow, found by hand, is analytic fluid dynamics, not CFD, since nothing is discretised. By the Lax equivalence theorem, a consistent linear scheme for a well-posed linear problem converges if and only if it is stable.