Journal of Scientific Computing

, Volume 35, Issue 2–3, pp 350–371 | Cite as

An Unconditionally Stable MacCormack Method

  • Andrew Selle
  • Ronald Fedkiw
  • ByungMoon Kim
  • Yingjie Liu
  • Jarek Rossignac
Article

Abstract

The back and forth error compensation and correction (BFECC) method advects the solution forward and then backward in time. The result is compared to the original data to estimate the error. Although inappropriate for parabolic and other non-reversible partial differential equations, it is useful for often troublesome advection terms. The error estimate is used to correct the data before advection raising the method to second order accuracy, even though each individual step is only first order accurate. In this paper, we rewrite the MacCormack method to illustrate that it estimates the error in the same exact fashion as BFECC. The difference is that the MacCormack method uses this error estimate to correct the already computed forward advected data. Thus, it does not require the third advection step in BFECC reducing the cost of the method while still obtaining second order accuracy in space and time. Recent work replaced each of the three BFECC advection steps with a simple first order accurate unconditionally stable semi-Lagrangian method yielding a second order accurate unconditionally stable BFECC scheme. We use a similar approach to create a second order accurate unconditionally stable MacCormack method.

Keywords

Advection Semi-Lagrangian CIR MacCormak Second order 

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Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Andrew Selle
    • 1
  • Ronald Fedkiw
    • 1
  • ByungMoon Kim
    • 2
  • Yingjie Liu
    • 3
  • Jarek Rossignac
    • 2
  1. 1.Computer Science DepartmentStanford UniversityStanfordUSA
  2. 2.College of ComputingGeorgia Institute of TechnologyAtlantaUSA
  3. 3.School of MathematicsGeorgia Institute of TechnologyAtlantaUSA

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