Semi-Lagrangian Methods

  • Dale R. Durran
Part of the Texts in Applied Mathematics book series (TAM, volume 32)


Most of the fundamental equations in fluid dynamics can be derived from first principles in either a Lagrangian form or an Eulerian form. Lagrangian equations describe the evolution of the flow that would be observed following the motion of an individual parcel of fluid. Eulerian equations describe the evolution that would be observed at a fixed point in space (or at least at a fixed point in a coordinate system such as the rotating Earth whose motion is independent of the fluid). If S(x, t) represents the sources and sinks of a chemical tracer Ψ(x, t) the evolution of the tracer in a one-dimensional flow field may be alternatively expressed in Lagrangian form as
$$\frac{{d\Psi }}{{dt}} = S$$
, or in Eulerian form as
$$\frac{{\partial \Psi }}{{\partial t}} + u\frac{{\partial \Psi }}{{\partial x}} = S$$


Departure Point Advection Equation Courant Number Trajectory Calculation Back Trajectory 
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  1. 1.
    According to the discussion in Section 2.3.2, the global truncation error is of same the order as the leading-order errors in the numerical approximation to the differential form of the governing equation (6.6) and is one power of Δt lower than the truncation error in the integrated form (6.5).Google Scholar

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© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Dale R. Durran
    • 1
  1. 1.Atmospheric SciencesUniversity of WashingtonSeattleUSA

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