MacCormack method

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. The MacCormack method is elegant and easy to understand and program.

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. The MacCormack method is elegant and easy to understand and program. \frac{\partial u}{\partial t} + \frac{\partial f(u)}{\partial x} = 0 To update this equation one timestep \Delta t on a grid with spacing \Delta x at grid cell i , the MacCormack method uses a "predictor step" and a "corrector step", given below \begin{align} &u_i^p = u^n_i - \frac{\Delta t}{\Delta x}\left(f^n_{i+1} - f^n_i\right) \\ &u^{n+1}_i = \frac{1}{2}(u^n_i + u^p_i) - \frac{\Delta t}{2\Delta x}(f^p_i - f^p_{i-1}) \end{align} \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 u at time level n+1 (denoted by u_i^p ) is estimated as follows u_i^p = 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 u_i^p is corrected according to the equation u_i^{n+1} = u_i^{n+1/2} - a \frac{\Delta t}{2\Delta x} \left( u_i^p - u_{i-1}^p \right) Note that the corrector step uses backward finite difference approximations for spatial derivative. The time-step used in the corrector step is \Delta t/2 in contrast to the \Delta t used in the predictor step. Replacing the u_i^{n+1/2} term by the temporal average u_i^{n+1/2} = \frac{u_i^n + u_i^p}{2} to obtain the corrector step as u_i^{n+1} = \frac{u_i^n + u_i^p}{2} - a \frac{\Delta t}{2\Delta x} \left( u_i^p - u_{i-1}^p \right)

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