The MacCormack Scheme

The MacCormack scheme is another multi-step scheme. It is composed of a predictor step and a corrector step. It is one of the simplest of the predictor-corrector class of numerical schemes.

The predictor step is used to obtain an estime \(\tilde{u}\) of the unknown function \(u\) at \(t^{n+1}\):

\[ \tilde{u}_m^{n+1}=u_m^n - c\frac{\Delta t}{\Delta x}\left( u_{m+1}^n - u_m^n \right) \]

Note the use of a forward formula to approximate the spatial derivative. The corrector step uses the estimates \( \tilde{u}^{n+1}\) to approximate the spatial derivative with a backward formula:

\[ u_m^{n+1}=u_m^{n+\frac{1}{2}} - c\frac{\Delta t}{2\Delta x}\left( \tilde{u}_{m}^{n+1} - \tilde{u}_{m-1}^{n+1} \right) \]

To obtain the solution at \(t^{n+\frac{1}{2}}\), we simply average the function at \(t^n\) and the estimate at \(t^{n+1}\):

\[ u_m^{n+\frac{1}{2}} = \frac{u_m^{n} + \tilde{u}_m^{n+1}}{2} \]

The final scheme is:

\[ u_m^{n+1} = \frac{u_m^{n} + \tilde{u}_m^{n+1}}{2} - c\frac{\Delta t}{2\Delta x}\left( \tilde{u}_{m}^n - \tilde{u}_{m-1}^n \right) \]

Consistency, stability and convergence

For the linear advection equation, the MacCormack scheme is equivalent to the Lax-Wendroff scheme, so its properties are the same as those of the latter.

Application: propagation of top hat function

The MacCormack scheme applied to the top hat initial condition gives the same solution as the Lax-Wendroff scheme.