Flux-Corrected Transport pp 23-65 | Cite as

# The Design of Flux-Corrected Transport (FCT) Algorithms for Structured Grids

## Abstract

A given flux-corrected transport (FCT) algorithm consists of three components: (1) a high order algorithm to which it reduces in smooth parts of the flow; (2) a low order algorithm to which it reduces in parts of the flow devoid of smoothness; and (3) a flux limiter which calculates the weights assigned to the high and low order fluxes in various regions of the flow field. One way of optimizing an FCT algorithm is to optimize each of these three components individually. We present some of the ideas that have been developed over the past 30 years toward this end. These include the use of very high order spatial operators in the design of the high order fluxes, non-clipping flux limiters, the appropriate choice of constraint variables in the critical flux-limiting step, and the implementation of a “failsafe” flux-limiting strategy. This chapter confines itself to the design of FCT algorithms for structured grids, using a finite volume formalism, for this is the area with which the present author is most familiar. The reader will find excellent material on the design of FCT algorithms for unstructured grids, using both finite volume and finite element formalisms, in the chapters by Professors Löhner, Baum, Kuzmin, Turek, and Möller in the present volume.

## Keywords

Grid Point Advection Equation Courant Number Numerical Flux Flux Limiter## References

- 1.Boris, J.P.: A fluid transport algorithm that works. In: Computing as a Language of Physics, International Atomic Energy Agency, pp. 171–189 (1971) Google Scholar
- 2.Boris, J.P., Book, D.L.: Flux-Corrected Transport I: SHASTA, a fluid-transport algorithm that works. J. Comput. Phys.
**11**, 38–69 (1973) ADSMATHCrossRefGoogle Scholar - 3.Chorin, A.J.: Random choice solution of hyperbolic systems. J. Comput. Phys.
**22**, 517–536 (1976) MathSciNetADSMATHCrossRefGoogle Scholar - 4.Chorin, A.J.: Random choice methods with application to reacting gas flow. J. Comput. Phys.
**25**, 252–272 (1977) MathSciNetADSCrossRefGoogle Scholar - 5.Colella, P., Woodward, P.R.: The Piecewise-Parabolic Method (PPM) for gas-dynamical simulations. J. Comput. Phys.
**54**, 174–201 (1984) MathSciNetADSMATHCrossRefGoogle Scholar - 6.DeVore, C.R.: An improved limiter for multidimensional flux-corrected transport. NRL Memorandum Report 6440-98-8330, Naval Research Laboratory, Washington, DC (1998) Google Scholar
- 7.Forester, C.K.: Higher order monotonic convective difference schemes. J. Comput. Phys.
**23**, 1–22 (1977) MathSciNetADSMATHCrossRefGoogle Scholar - 8.Glimm, J.: Solution in the large for nonlinear hyperbolic systems of equations. Commun. Pure Appl. Math.
**18**, 697–715 (1955) MathSciNetCrossRefGoogle Scholar - 9.Kreiss, H.-O., Oliger, J.: Comparison of accurate methods for the integration of hyperbolic equations. Tellus
**24**, 199 (1972) MathSciNetADSCrossRefGoogle Scholar - 10.McDonald, B.E.: Flux-corrected pseudospectral method for scalar hyperbolic conservation laws. J. Comput. Phys.
**82**, 413 (1989) MathSciNetADSMATHCrossRefGoogle Scholar - 11.Sod, G.A.: A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws. J. Comput. Phys.
**27**, 1–31 (1978) MathSciNetADSMATHCrossRefGoogle Scholar - 12.Woodward, P.R., Colella, P.: The numerical simulation of two-dimensional flow with strong shocks. J. Comput. Phys.
**54**, 115–173 (1984) MathSciNetADSMATHCrossRefGoogle Scholar - 13.Zalesak, S.T.: Fully multidimensional Flux-Corrected Transport algorithms for fluids. J. Comput. Phys.
**31**, 335–362 (1979) MathSciNetADSMATHCrossRefGoogle Scholar - 14.Zalesak, S.T.: Very high order and pseudospectral Flux-Corrected Transport (FCT) algorithms for conservation laws. In: Vichnevetsky, R., Stepleman, R.S. (eds.) Advances in Computer Methods for Partial Differential Equations IV, IMACS, Rutgers University, pp. 126–134 (1981) Google Scholar
- 15.Zalesak, S.T.: A preliminary comparison of modern shock-capturing schemes: Linear advection. In: Vichnevetsky, R., Stepleman, R.S. (eds.) Advances in Computer Methods for Partial Differential Equations VI, IMACS, Rutgers University, pp. 15–22 (1987) Google Scholar