MacCormack method

In computational fluid dynamics, the MacCormack method 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. The MacCormack method is elegant and easy to understand and program. In computational fluid dynamics, the MacCormack method 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. The MacCormack method is elegant and easy to understand and program. 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 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 u {displaystyle u} at time level n + 1 {displaystyle n+1} (denoted by u i n + 1 ¯ {displaystyle u_{i}^{overline {n+1}}} ) 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. Corrector step: In the corrector step, the predicted value u i n + 1 ¯ {displaystyle u_{i}^{overline {n+1}}} is corrected according to the equation Note that the corrector step uses backward finite difference approximations for spatial derivative. Note also that the time-step used in the corrector step is Δ t / 2 {displaystyle Delta t/2} in contrast to the Δ t {displaystyle Delta t} used in the predictor step. Replacing the u i n + 1 / 2 {displaystyle u_{i}^{n+1/2}} term by the temporal average