MacCormack method

The MacCormack method is a variation of the two-step Lax–Wendroff scheme but is much simpler in application. To illustrate the algorithm, consider the following first order hyperbolic equation

${\displaystyle \qquad {\frac {\partial u}{\partial t}}+a{\frac {\partial u}{\partial x}}=0.}$

The application of MacCormack method to the above equation proceeds in two steps; a predictor step which is followed by a corrector step.

Predictor step: In the predictor step, a "provisional" value of ${\displaystyle u}$ at time level ${\displaystyle n+1}$ (denoted by ${\displaystyle u_{i}^{\overline {n+1}}}$) is estimated as follows

${\displaystyle u_{i}^{\overline {n+1}}=u_{i}^{n}-a{\frac {\Delta t}{\Delta x}}\left(u_{i+1}^{n}-u_{i}^{n}\right)}$

The above equation is obtained by replacing the spatial and temporal derivatives in the previous first order hyperbolic equation using forward differences.

Corrector step: In the corrector step, the predicted value ${\displaystyle u_{i}^{\overline {n+1}}}$ is corrected according to the equation

${\displaystyle u_{i}^{n+1}=u_{i}^{n+1/2}-a{\frac {\Delta t}{2\Delta x}}\left(u_{i}^{\overline {n+1}}-u_{i-1}^{\overline {n+1}}\right)}$

Note that the corrector step uses backward finite difference approximations for spatial derivative. The time-step used in the corrector step is ${\displaystyle \Delta t/2}$ in contrast to the ${\displaystyle \Delta t}$ used in the predictor step.

Replacing the ${\displaystyle u_{i}^{n+1/2}}$ term by the temporal average

${\displaystyle u_{i}^{n+1/2}={\frac {u_{i}^{n}+u_{i}^{\overline {n+1}}}{2}}}$

to obtain the corrector step as

${\displaystyle u_{i}^{n+1}={\frac {u_{i}^{n}+u_{i}^{\overline {n+1}}}{2}}-a{\frac {\Delta t}{2\Delta x}}\left(u_{i}^{\overline {n+1}}-u_{i-1}^{\overline {n+1}}\right)}$

• Lax–Wendroff method
• Upwind scheme
• Hyperbolic partial differential equations