In computational fluid dynamics, the MacCormack method (/məˈkɔːrmæk ˈmɛθəd/) is a widely used discretization scheme for the numerical solution of hyperbolic partial differential equations.
This second-order finite 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 To update this equation one timestep
) is estimated as follows The above equation is obtained by replacing the spatial and temporal derivatives in the previous first order hyperbolic equation using forward differences.
is corrected according to the equation Note that the corrector step uses backward finite difference approximations for spatial derivative.
The order of differencing can be reversed for the time step (i.e., forward/backward followed by backward/forward).
For linear equations, the MacCormack scheme is equivalent to the Lax–Wendroff method.
[4] Unlike first-order upwind scheme, the MacCormack does not introduce diffusive errors in the solution.
However, it is known to introduce dispersive errors (Gibbs phenomenon) in the region where the gradient is high.