MacCormack method

Remarks

Example: for $f(u) = a u$ and $\nu = a \lambda$, the predictor gives $u_j^* - u_{j-1}^* = (u_j^n - u_{j-1}^n) - \nu (u_{j+1}^n - 2u_j^n + u_{j-1}^n)$. Putting this into the corrector gives $$u_j^{n+1} = u_j^n - \frac{\nu}{2} (u_{j+1}^n - u_{j-1}^n) + \frac{\nu^2}{2} (u_{j+1}^n - 2u_j^n + u_{j-1}^n),$$ which is the Lax–Wendroff method; so the method is second-order accurate and stable for $\abs{\nu} \le 1$ in this case. Non-example: using a forward difference in the corrector as well gives, for $f(u) = a u$, the scheme $u_j^{n+1} = u_j^n - \nu (u_{j+1}^n - u_j^n) + \frac{\nu^2}{2}(u_{j+2}^n - 2u_{j+1}^n + u_j^n)$, whose Taylor expansion leaves the error term $-\frac{a \Delta x}{2} u_{xx}$, so it is only first order. The order of the one-sided differences may be reversed (backward in the predictor, forward in the corrector); alternating the two orders from step to step is common.