|
|
| Line 1: |
Line 1: |
| In [[computational fluid dynamics]], the '''MacCormack method''' is a widely used discretization scheme for the numerical solution of [[hyperbolic partial differential equation]]s. This second-order [[finite difference method]] was introduced by Robert W. MacCormack in 1969.<ref>MacCormack, R. W., The Effect of viscosity in hypervelocity impact cratering, AIAA Paper, 69-354 (1969).</ref> The MacCormack method is elegant and easy to understand and program.<ref>[[John D. Anderson|Anderson, J. D., Jr.]], Computational Fluid Dynamics: The Basics with Applications, McGraw Hill (1994).</ref>
| | Hi, everybody! <br>I'm Portuguese female :). <br>I really love Sand castle building!<br><br>Feel free to surf to my blog ... [http://www.supertankgames.com/profile/fe60m backup plugin] |
| | |
| == The algorithm ==
| |
| The MacCormack method is a variation of the [[Lax–Wendroff method|two-step Lax–Wendroff scheme]] but is much simpler in application. To illustrate the algorithm, consider the following first order hyperbolic equation
| |
| :<math>
| |
| \qquad \frac{\partial u}{\partial t} + a \frac{\partial u}{\partial x} = 0 .
| |
| </math> | |
| 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 <math>u</math> at time level <math>n+1</math> (denoted by <math>u_i^{\overline{n+1}}</math>) is estimated as follows
| |
| :<math>
| |
| u_i^{\overline{n+1}} = u_i^n - a \frac{\Delta t}{\Delta x} \left( u_{i+1}^n - u_i^n \right)
| |
| </math>
| |
| It may be noted that the above equation is obtained by replacing the spatial and temporal derivatives in the previous first order hyperbolic equation using [[Finite difference#Forward, backward and central differences|forward difference]]s.
| |
| | |
| '''Corrector step:''' In the corrector step, the predicted value <math>u_i^{\overline{n+1}}</math> is corrected according to the equation
| |
| :<math>
| |
| u_i^{n+1} = u_i^{n+1/2} - a \frac{\Delta t}{2\Delta x} \left( u_i^{\overline{n+1}} - u_{i-1}^{\overline{n+1}} \right)
| |
| </math>
| |
| Note that the corrector step uses [[Finite difference#Forward, backward and central differences|backward finite difference]] approximations for spatial derivative. Note also that the time-step used in the corrector step is <math>\Delta t/2</math> in contrast to the <math>\Delta t</math> used in the predictor step.
| |
| | |
| Replacing the <math>u_i^{n+1/2}</math> term by the temporal average
| |
| :<math> | |
| u_i^{n+1/2} = \frac{u_i^n + u_i^{\overline{n+1}}}{2}
| |
| </math>
| |
| to obtain the corrector step as
| |
| :<math>
| |
| u_i^{n+1} = \frac{u_i^n + u_i^{\overline{n+1}}}{2} - a \frac{\Delta t}{2\Delta x} \left( u_i^{\overline{n+1}} - u_{i-1}^{\overline{n+1}} \right)
| |
| </math>
| |
| | |
| === Some remarks ===
| |
| The MacCormack method is well suited for [[Nonlinear system#Nonlinear differential equations|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 scheme.<ref>Tannehill, J. C., [[Dale A. Anderson|Anderson, D. A.]], and Pletcher, R. H., Computational Fluid Dynamics and Heat Transfer, 2nd ed., Taylor & Francis (1997).</ref>
| |
| | |
| Unlike first-order [[upwind scheme]], the MacCormack does not introduce [[Numerical diffusion|diffusive errors]] in the solution. However, it is known to introduce dispersive errors ([[Gibbs phenomenon]]) in the region where the gradient is high.
| |
| | |
| == See also ==
| |
| *[[Lax–Wendroff method]]
| |
| *[[Upwind scheme]]
| |
| *[[Hyperbolic partial differential equation]]s
| |
| | |
| ==References==
| |
| {{reflist|colwidth=30em}}
| |
| | |
| {{Numerical PDE}}
| |
| | |
| {{DEFAULTSORT:Maccormack Method}}
| |
| [[Category:Computational fluid dynamics]]
| |
| [[Category:Numerical differential equations]]
| |
Hi, everybody!
I'm Portuguese female :).
I really love Sand castle building!
Feel free to surf to my blog ... backup plugin