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advection.pdf
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Advection: Getting Carried Away
MATH1091: ODE methods for a reaction diffusion equation
http://people.sc.fsu.edu/∼jburkardt/classes/math1091 2020/advection/advection.pdf
Wind and water will carry away any items they catch.
The Advection Equation
The wind carries along heat, humidity, dust; rivers transport quantities of silt and debris. The
advection equation models how such quantities move when they are carried along in the flowstream.
1 A model of passive transport
Consider a one-dimensional finite region 0 ≤ x ≤ xmax, such as a hose, along which some fluid is flowing. If
the fluid is water, and it enters the region on the left with a velocity c, then we expect the flow to have that
same velocity c at all times and locations in the hose. This is because water is essentially incompressible.
We will stick with this constant velocity version of the problem; the compressible flow problem, while it can
handle important problems in atmospheric dynamics, is more than we can cover in one lecture.
Now imagine that we add some secondary item to the flow, such as a small amount of dye. We can designate
by u(x, t) the local concentration of dye at location x at time t. Moreover, we can find a formula f (t) for
the concentration of dye that is being added to the water at the left endpoint for every time 0 < t < tmax.
Presumably, knowing the initial condition, that is u(x, 0) for all x, and the injection function f (t), we have
enough information to completely describe what is going to happen in this simple physical model.
But how do we mathematically describe this behavior, and how do we computationally approximate it?
We can get a handle on what is going on by thinking of a small interval, [x, x+∆x]. In a time interval of ∆t,
our substance flows into the interval in the amount c ∗ ∆t ∗ u(x), and flows out of the interval in the amount
c ∗ ∆t ∗ u(x + ∆x). Since u(x, t) represents the concentration of the substance, the amount of substance in
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