Journal of Scientific Computing

, Volume 35, Issue 2–3, pp 350–371

# 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

## Preview

### References

1. 1.
Anderson, J.D.: Computational Fluid Dynamics: The Basics With Applications. McGraw-Hill, New York (1995) Google Scholar
2. 2.
Courant, R., Issacson, E., Rees, M.: On the solution of nonlinear hyperbolic differential equations by finite differences. Commun. Pure Appl. Math. 5, 243–255 (1952)
3. 3.
Dupont, T., Liu, Y.: Back and forth error compensation and correction methods for removing errors induced by uneven gradients of the level set function. J. Comput. Phys. 190(1), 311–324 (2003)
4. 4.
Dupont, T., Liu, Y.: Back and forth error compensation and correction methods for semi-Lagrangian schemes with application to level set interface computations. Math. Comp. 76, 647–668 (2007)
5. 5.
Enright, D., Fedkiw, R., Ferziger, J., Mitchell, I.: A hybrid particle level set method for improved interface capturing. J. Comput. Phys. 183, 83–116 (2002)
6. 6.
Enright, D., Losasso, F., Fedkiw, R.: A fast and accurate semi-Lagrangian particle level set method. Comput. Struct. 83, 479–490 (2005)
7. 7.
Enright, D., Marschner, S., Fedkiw, R.: Animation and rendering of complex water surfaces. ACM Trans. Graph. (SIGGRAPH Proc.) 21(3), 736–744 (2002) Google Scholar
8. 8.
Enright, D., Nguyen, D., Gibou, F., Fedkiw, R.: Using the particle level set method and a second order accurate pressure boundary condition for free surface flows. In: Proc. 4th ASME-JSME Joint Fluids Eng. Conf., no. FEDSM2003–45144. ASME (2003) Google Scholar
9. 9.
Fedkiw, R., Stam, J., Jensen, H.: Visual simulation of smoke. In: Proc. of ACM SIGGRAPH 2001, pp. 15–22 (2001) Google Scholar
10. 10.
Herrmann, M., Blanquart, G.: Flux corrected finite volume scheme for preserving scalar boundedness in reacting large-eddy simulations. AIAA J. 44(12), 2879–2886 (2006)
11. 11.
Irving, G., Guendelman, E., Losasso, F., Fedkiw, R.: Efficient simulation of large bodies of water by coupling two and three dimensional techniques. ACM Trans. Graph. (SIGGRAPH Proc.) 25(3), 805–811 (2006)
12. 12.
Jiang, G.-S., Peng, D.: Weighted ENO schemes for Hamilton-Jacobi equations. SIAM J. Sci. Comput. 21, 2126–2143 (2000)
13. 13.
Kim, B.-M., Liu, Y., Llamas, I., Rossignac, J.: Using BFECC for fluid simulation. In: Eurographics Workshop on Natural Phenomena (2005) Google Scholar
14. 14.
Kim, B.-M., Liu, Y., Llamas, I., Rossignac, J.: Advections with significantly reduced dissipation and diffusion. IEEE Trans. Vis. Comput. Graph. 13(1), 135–144 (2007)
15. 15.
Lax, P.D.: On the stability of difference approximations to solutions of hyperbolic equations with variable coefficients. Commun. Pure Appl. Math. 14, 497–520 (1961)
16. 16.
Losasso, F., Fedkiw, R., Osher, S.: Spatially adaptive techniques for level set methods and incompressible flow. Comput. Fluids 35, 995–1010 (2006)
17. 17.
Losasso, F., Gibou, F., Fedkiw, R.: Simulating water and smoke with an octree data structure. ACM Trans. Graph. (SIGGRAPH Proc.) 23, 457–462 (2004)
18. 18.
MacCormack, R.: The effect of viscosity in hypervelocity impact cratering. In: AIAA Hypervelocity Impact Conference, 1969. AIAA paper, pp. 69–354 (1969) Google Scholar
19. 19.
Min, C., Gibou, F.: A second order accurate projection method for the incompressible Navier-Stokes equation on non-graded adaptive grids. J. Comput. Phys. 219, 912–929 (2006)
20. 20.
Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Springer, New York (2002) Google Scholar
21. 21.
Osher, S., Sethian, J.: Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations. J. Comput. Phys. 79, 12–49 (1988)
22. 22.
Selle, A., Rasmussen, N., Fedkiw, R.: A vortex particle method for smoke, water and explosions. ACM Trans. Graph. (SIGGRAPH Proc.) 24(3), 910–914 (2005)
23. 23.
Shu, C.-W., Osher, S.: Efficient implementation of essentially non-oscillatory shock capturing schemes. J. Comput. Phys. 77, 439–471 (1988)
24. 24.
Stam, J.: Stable fluids. In: Proc. of SIGGRAPH 99, pp. 121–128 (1999) Google Scholar
25. 25.
Staniforth, A., Cote, J.: Semi-Lagrangian integration schemes for atmospheric models: A review. Mon. Weather Rev. 119, 2206–2223 (1991)
26. 26.
Steinhoff, J., Underhill, D.: Modification of the Euler equations for “vorticity confinement”: Application to the computation of interacting vortex rings. Phys. Fluids 6(8), 2738–2744 (1994)
27. 27.
Strain, J.: Tree methods for moving interfaces. J. Comput. Phys. 151, 616–648 (1999)
28. 28.
Sussman, M., Smereka, P., Osher, S.: A level set approach for computing solutions to incompressible two-phase flow. J. Comput. Phys. 114, 146–159 (1994)
29. 29.
Warming, R.F., Beam, R.M.: Upwind second-order difference schemes and applications in aerodynamic flows. AIAA J. 14(9), 1241–1249 (1976)

## 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