When is it safe to ignore the diffusion term in an advection-diffusion equation?

Question

Given the one dimensional equation:

$\epsilon\frac{\partial^2u}{\partial x^2} +\frac{\partial u}{\partial x} = 0 $

with $0\le\epsilon \ll1$ with boundary conditions $u(0) = 0$ and $u(1) = 2$, we can't neglect the diffusive term because of the boundary conditions. In which situations (with very small $\epsilon$ could we? What's the mathematical theory behind that could explain it?

Also, if we had a time-dependent equation:

$\frac{\partial u}{\partial t} = \epsilon\frac{\partial^2u}{\partial x^2} +\frac{\partial u}{\partial x}$

how would the situation change? what if the convective term were nonlinear, such as in the Burgers equation? I haven't been able to find references on this topic, I would appreciate any.

Answer 1

The stationary equation you show transports information from the right to the left via the advection term; it also diffuses slightly. If you switch off the diffusion term altogether, then you only have transport from the right to the left, and you need to also drop the boundary condition at the left: because information is from the right to the left, nothing that happens at the left end of the domain has any effect on the solution.

A similar argument can be made for the time dependent equation.

In general, these equations are examples of "singularly perturbed problems". You will be able to find a lot of literature on the subject.

Answer 2

In the time-dependent nonlinear case, if you drop the diffusive term then you have a nonlinear hyperbolic problem. Solutions will naturally generate singularities (discontinuities) in finite time. To extend the solution beyond that time, one must consider weak solutions, and uniqueness is lost. To specify a unique, physically relevant solution one typically invokes an entropy condition, which is equivalent to enforcing that the weak solution is the "vanishing-viscosity limit" of a diffusive equation.

The canonical reference in this area is Whitham. I am partial to the two texts by LeVeque, although they focus on numerics more than theory.

Answer 3

In addition to AJK's recommendation, here are a couple more recommendations:

1. Perturbation methods in fluid mechanics by Milton Van Dyke
2. Introduction to Perturbation Methods by M.H. Holmes, (2013).

I recommend the second to get started on perturbation methods. The first one can be read after going through the first 2-3 chapters of the second.

To answer your question about when the diffusion term can be ignored (in the first equation), it can be ignored when $u(0) \approx u(1)$. In this case, $u(x)$ will be more or less constant. This can of course be quantified. In your case with $u(0)=0$ and $u(1) = 2$, the diffusion term cannot be ignored. There is a boundary layer near $x=0$. See a plot of the exact solution below for $\epsilon = 0.01$: