8.6. Predictor-corrector schemes

To improve the stability of numerical schemes it is possible is to incorporate a scheme with more than one step of calculations. The predictor-corrector schemes typically uses two steps, a predictor-step, where we predict a solution, and a corrector-step where we refine the final solution/estimate. These schemes also typically includes “”fictional time steps in between the regular time steps, such as a solution at time step \(n+1/2\). The method, therefore, also fit into the category of half-step schemes described in Section 8.7.

The McCormack scheme for linear advection is a good example of how this works.

8.6.1. The MacCormack Scheme

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

Step 1: The Predictor step The predictor step is used to obtain an estimate \(\tilde{u}\) of the unknown function \(u\) at timstep \(t^{n+1}\). Here, we use the FTFS scheme (Forward in time, forward in space):

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

We will also need the same FTFS step for the grid point to the left of \(u_m^{n+1}\), which is \(\tilde{u}_{m-1}^{n+1}\):

\[ \tilde{u}_{m-1}^{n+1}=u_{m-1}^n - a\frac{\Delta t}{\Delta x}\left( u_{m}^n - u_{m-1}^n \right).\]

Note the use of a forward formula to approximate the spatial derivative.

Step 2: the corrector step

The corrector step uses a FTBS (forward in time and backward in space) scheme with a half-step in time from \(u_m^{n+1/2}\) to \(u_m^{n+1}\) and the estimates \( \tilde{u}_m^{n+1}\) and \( \tilde{u}_{m-1}^{n+1}\) to approximate the backward spatial derivative :

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

To obtain the solution at the half-step in time, \(t^{n+\frac{1}{2}}\), 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:

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

For the linear advection equation, you can insert the expressions for \(\tilde{u}_m^{n+1}\), \(\tilde{u}_k^{n}\) and \(\tilde{u}_{m-1}^{n}\) into the final scheme to get a 1-step solution, which gives the same result as the Lax-Wendroff approach:

\[u_m^{n+1}=u_m^n-\frac{C}{2}(u_{m+1}^n-u_{m-1}^n)+\frac{C^2}{2}(u_{m+1}^n-2u_m^n+u_{m-1}^n),\]

where \(C=\frac{a\Delta t}{\Delta x}\) is the Courant number.

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.