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| | 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> |
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| | == 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]] |
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.[1] The MacCormack method is elegant and easy to understand and program.[2]
The algorithm
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 at time level (denoted by ) is estimated as follows
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 forward differences.
Corrector step: In the corrector step, the predicted value 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 in contrast to the used in the predictor step.
Replacing the term by the temporal average
to obtain the corrector step as
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 scheme.[3]
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.
See also
References
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Template:Numerical PDE
- ↑ MacCormack, R. W., The Effect of viscosity in hypervelocity impact cratering, AIAA Paper, 69-354 (1969).
- ↑ Anderson, J. D., Jr., Computational Fluid Dynamics: The Basics with Applications, McGraw Hill (1994).
- ↑ Tannehill, J. C., Anderson, D. A., and Pletcher, R. H., Computational Fluid Dynamics and Heat Transfer, 2nd ed., Taylor & Francis (1997).