Transport: Non-diffusive, flux conservative initial value problems and ...

Transport equations 83
they are more faithful to the underlying characteristics than Eulerian schemes, and
the lack of a courant condition makes them highly attractive for high-resolution
problems. If you just need to get going though, a second order staggered leapfrog
is the simplest first approach to try. If the numerical dispersion becomes annoying
a more expensive mpdata scheme might be used but they are really superseded by
the SL schemes. In all cases beware of simple upwind differences if numerical
diffusion is not desired.
Bibliography
[1] B. P. Leonard. A survey of finite differences of opinion on numerical muddling
of the incomprehensible defective confusion equation, in: T. J. R. Hughes, ed.,
Finite element methods for convection dominated flows, vol. 34, pp. 1–17,
Amer. Soc. Mech. Engrs. (ASME), 1979.
[2] P. K. Smolarkiewicz. A simple positive definite advection scheme with small
implicit diffusion, Mon Weather Rev 111, 479–486, 1983.
[3] A. Staniforth and J. Cote. Semi-Lagrangian integration schemes for atmospheric
models—A review, Mont. Weather Rev. 119, 2206–2223, Sep 1991.
[4] W. H. Press, B. P. Flannery, S. A. Teukolsky and W. T. Vetterling. Numerical
Recipes, Cambridge Univ. Press, New York, 2nd edn., 1992.