MacCormack method

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

:$$

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

To update this equation one timestep $\backslash Delta\; t$ on a grid with spacing $\backslash Delta\; x$ at grid cell $i$, the MacCormack method uses a "predictor step" and a "corrector step", given below{{Cite web |last=Cameron |first=Maria |date=2011 |title=Notes on Burger's Equation |url=https://www.math.umd.edu/~mariakc/burgers.pdf#page=13 |website=University of Maryland}}

:$\backslash begin\{align\}$

&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}

== Linear Example ==

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

:$$

\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 $\backslash Delta\; t/2$ in contrast to the $\backslash Delta\; t$ 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)

The MacCormack method is well suited for nonlinear equations (Inviscid Burgers 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 the Lax–Wendroff method.Tannehill, J. C., Anderson, D. A., and Pletcher, R. H., Computational Fluid Mechanics and Heat Transfer, 2nd ed., Taylor & Francis (1997).

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.

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

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Category:Computational fluid dynamics

Category:Numerical differential equations