13 - Lagrangian and Semi-Lagrangian Schemes

Summary

Introduction

While discussing the various methods that are employed numerically to solve the governing equations of the atmosphere, the view point considered was mainly “Eulerian.” In the Eulerian description of fluid motion, the motion is described with respect to a coordinate system that is fixed in space. These invariably result in the description of fluid properties as either a “scalar field” or a “vector field” and the evolution of fluid properties as the evolution of the resultant scalar field/vector field in space and time. In essence, in the Eulerian view point, the evolution of fluid properties were defined with respect to the time evolution of the fluid properties at fixed points in space through the use of the partial derivative with respect to time. This resulted in fluid properties being expressed as a function in space and time. In the Eulerian description of fluid motion, one is not bothered about the location and/or velocity of any particular “fluid particle,” but rather about the velocity, acceleration, etc. of whatever fluid particle happens to be at a particular location of interest at a particular time. However, as fluid is treated as a “continuum,” the Eulerian description of fluid motion is usually preferred in fluid mechanics. It is also known that the Eulerian description of fluid flow results in the appearance of nonlinear advection terms. Chapter 4.9 outlined the issues that arise because of the presence of nonlinear advection terms leading to nonlinear computational instability and the ways that one has to take recourse to overcome nonlinear computational instability.

Hence, it is not surprising that atmospheric scientists started taking serious note of the “Lagrangian” view point in terms of the Lagrangian description of fluid motion during the late 1950s and later years. In the Lagrangian description of fluid flow, individual fluid particles are “marked” and, hence, their positions, velocities, etc. can be “tracked” as a function of time and their initial position. If there were only a few fluid particles to be considered in the flow domain, the Lagrangian description would be desirable. However, fluid is treated as a continuum and, hence, it is not possible to track each and every fluid particle in a complex flow field.