Incomputational fluid dynamics, theMacCormack method (/məˈkɔːrmæk ˈmɛθəd/) is a widely used discretization scheme for the numerical solution ofhyperbolic partial differential equations. This second-orderfinite difference method was introduced by Robert W. MacCormack in 1969.^[1] The MacCormack method is elegant and easy to understand and program.^[2]

The MacCormack method is designed to solve hyperbolic partial differential equations of the form

${\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }t}+\frac{\mathrm{\partial }f(u)}{\mathrm{\partial }x}=0}$

To update this equation one timestep${\displaystyle \mathrm{\Delta }t}$ on a grid with spacing${\displaystyle \mathrm{\Delta }x}$ at grid cell${\displaystyle i}$, the MacCormack method uses a "predictor step" and a "corrector step", given below^[3]

To illustrate the algorithm, consider the following first order hyperbolic equation

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

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

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

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

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

Corrector step: In the corrector step, the predicted value${\displaystyle u_i^p}$ is corrected according to the equation

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

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

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

${\displaystyle u_i^{n+1/2}=\frac{u_i^n+u_i^p}{2}}$

to obtain the corrector step as

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

The MacCormack method is well suited fornonlinear equations (InviscidBurgers' equation,Euler equations, etc.) The order of differencing can be reversed for the time step (i.e., forward/backward followed by backward/forward). For nonlinear equations, this procedure provides the best results. For linear equations, the MacCormack scheme is equivalent to theLax–Wendroff method.^[4]

Unlike first-orderupwind scheme, the MacCormack does not introducediffusive errors in the solution. However, it is known to introduce dispersive errors (Gibbs phenomenon) in the region where the gradient is high.

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

• Computational fluid dynamics
• Numerical differential equations

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• Short description matches Wikidata