6 - Linear Advection Equation

Summary

Introduction

The advection equation is a very important equation to investigate as this equation conserves the quantity that gets advected following a motion. In this chapter, the various time and space differencing schemes as applied to the advection equation is taken up for detailed discussion.

Centered Time and Space Differencing Schemes for Linear Advection Equation

Consider a linear one-dimensional advection equation

where c is a constant and u = u(x; t), and its general solution is given by u(x; t) = f (x-ct), where f is an arbitrary function. If the space derivative in Equation (6.1) is approximated by a central finite difference, one obtains

Applying the leapfrog scheme to Equation (6.2) gives

Substituting in Equation (6.2), a tentative solution of the form

one obtains an equation of the form

Equation (6.5) is of the form of oscillation equation with

Figure 6.1 depicts the schematic of the solution of the one-dimensional linear advection equation moving along the characteristic. Here, the solution u(x; t) = u(x-ct;0). i.e., the initial shape of u translates without change of shape with constant speed c. Equation (6.1) is the linear one-dimensional advection equation as expressed in the Eulerian description of fluid motion in which u is the advected quantity and c is the advection speed. It can be written in terms of the Lagrangian description of fluid motion as

The meaning of this Lagrangian description of fluid motion is that the value of the advected quantity (u in this case) does not change following the motion of a fluid element, i.e., the advected quantity u is “conserved” following the motion of the fluid element. The definition of a fluid element follows the continuum hypothesis where a fluid is assumed to be continuous; the fluid element is assumed to be made up of a very large number of molecules such that one can ascribe macroscopic properties such as density, temperature, etc. to the fluid element. In fluid dynamics, one considers an infinite collection of fluid elements. Equation (6.7) indicates that each fluid element maintains its value of the advected quantity u as it moves in time. Assuming that advection is the only process occurring in a fluid system, exactly the same values of the advected quantity u are still in the fluid system at any two different times.