A closer look at the advection equation

Most of this chapter is devoted to a discussion of the one-dimensional advection equation, (4.1) Here A is the advected quantity, and c is the advecting current. This is a linear, first-order, partial differential equation with a constant coefficient, namely c. Both space and time differencing are discussed in this chapter, but more emphasis is placed on space differencing. We have already presented the exact solution of (4.1). Before proceeding, however, it is useful to review the physical nature of advection, because the design or choice of a numerical method should always be motivated as far as possible by our understanding of the physical process at hand. In Lagrangian form, the advection equation is simply. (4.2) This means that the value of A does not change following a particle. We say that A is " conserved " following a particle. In fluid dynamics, we consider an infinite collection of fluid particles. According to (4.2), each particle maintains its value of A as it moves. If we do a survey of the values of A in our fluid system, let advection occur, and conduct a " follow-up " survey, we will find that exactly the same values of A are still in the system. The locations of the particles presumably will have changed, but the maximum value of A over the population of particles is unchanged by advection, the minimum value is unchanged, the average is unchanged, and in fact all of the statistics of the distribution of A over the mass of the fluid are completely unchanged by the advective process. This is an important characteristic of advection. Here is another way of describing this characteristic: If we worked out the probability density function (pdf) for A, by defining narrow " bins " and counting the mass associated with particles having values of A falling within each bin, we would find that the pdf was unchanged by advection. For instance, if the pdf of A at a certain time is Gaussian (or " bell shaped "), it will still be Gaussian at a later time (and with the same mean and standard deviation) if the