Semi-Lagrangian Methods

Abstract

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

$$\frac{{d\Psi }}{{dt}} = S$$

((6.1))

, or in Eulerian form as

$$\frac{{\partial \Psi }}{{\partial t}} + u\frac{{\partial \Psi }}{{\partial x}} = S$$

.