Journal of Scientific Computing

, Volume 28, Issue 2–3, pp 319–335 | Cite as

The Shifted Box Scheme for Scalar Transport Problems

  • Bertil Gustafsson
  • Yaser Khalighi

When solving certain scalar transport problems, it is important to use fast methods that do not produce any parasitic oscillating solutions, but still have good energy conservation properties. The well known box scheme has these properties, but the use is restricted by severe conditions on the sign of the coefficients. In order to avoid this restriction, we introduce a modified method, that we call the shifted box scheme. It is very efficient for initial-boundary value problems, since it does not require more work per time step than an explicit scheme, while still being unconditionally stable.


Scalar transport problem box scheme initial-boundary value problem unconditional stability 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Al-Zanaidi M., Chawla M.A. (2002). A high-accuracy box scheme for first-order systems on hyperbolic conservation laws. Neural Parallel Sc. Comput. 10, 423–430MATHMathSciNetGoogle Scholar
  2. 2.
    Bradshaw P., Cebeci T., Whitelaw J.H. (1981). Engineering Calculation Methods for Turbulent Flow. Academic Press, New York, NYMATHGoogle Scholar
  3. 3.
    Cebeci K.B., Chang T. (1980). Solution of a hyperbolic system of turbulence-model by the box scheme. Comput. Methods Appl. Mech. Eng. 22, 213–227CrossRefMATHMathSciNetADSGoogle Scholar
  4. 4.
    Cebeci K.H.B., T.W. PG (1979). Seperating boundary layer flow calculations. J. Comput. Phys. 31, 363–378CrossRefMATHMathSciNetADSGoogle Scholar
  5. 5.
    Hong Y., Liu J.L. (2004). Multisimplexity of centered box scheme for a class of Hamiltonian PDEs and an application to quasi-periodically solitary waves. Math. Comput. Modell. 39, 1035–1047CrossRefMATHMathSciNetADSGoogle Scholar
  6. 6.
    Keller H.B. (1971). A new difference scheme for parabolic problems. In: Hubbard B. (eds). Numerical solution of Partial Differential Equations, vol. 2, Academic, New York, pp. 327–350Google Scholar
  7. 7.
    Keller H.B. (1978). Numerical methods in boundary-layer theory. Ann. Rev. Fluid Mech. 10, 417–433CrossRefMATHADSGoogle Scholar
  8. 8.
    Lam R.B.S., D.C.L (1976). Central differencing and the box scheme for diffusion convection problems. J. Comput. Phys. 22, 486–500CrossRefMATHGoogle Scholar
  9. 9.
    Wendroff B. (1960). On centered difference equations for hyperbolic systems. J. Soc. Ind. Appl. Math. 8, 549–555CrossRefMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  1. 1.Division of Scientific ComputingUppsala UniversityUppsalaSweden
  2. 2.Institute of Computational Mathematics in Engineering (ICME)Stanford UniversityCaliforniaUSA
  3. 3.Department of Mechanical EngineeringStanford UniversityCaliforniaUSA

Personalised recommendations