Burgers’ Equation#
The inviscid Burgers’ equation is the simplest nonlinear wave equation, and serves as a great stepping-stone toward doing full hydrodynamics.
This looks like the linear advection equation, except the quantity being advected is the velocity itself. This means that \(u\) is no longer a constant, but can vary in space and time.
Written in conservative form,
it appears as:
so the flux is \(F(u) = \frac{1}{2} u^2\).
In the finite volume approach, we integrate over the volume of a cell to get the update:
To second order accuracy, as we saw previously, \(\langle u \rangle_i \approx u_i\), so we’ll drop the \(\langle \rangle\) here.
Nonlinear behavior#
Our solution method is essentially the same, aside from the Riemann problem. We still want to use the idea of upwinding, but now we have a problem—the nonlinear nature of the Burgers’ equation means that information can “pile up” and we lose track of where the flow is coming from. This gives rise to a nonlinear wave called a shock.
For the linear advection equation, the solution was unchanged along the lines \(x - ut = \mbox{constant}\)—we called these the characteristic curves.
We can visualize the characteristics as show below:
The characteristic curves are the curves on which the solution is constant. For Burgers’ equation, the characteristic curves are given by \(dx/dt = u\), but now \(u\) varies in the domain. To see this, look at the change of \(u\) in a fluid element (the full, or Lagrangian time derivative):
We see that \(du/dt = 0\) since it just gives us the Burgers equation. So \(u\) is constant along the curves \(dx/dt = u\), but now \(u\) varies in the domain. So if we look at the characteristic curves in the spacetime diagram, we get:
Now we see that, for these initial conditions, at some point in the future the characteristics intersect. This means that there is not a unique curve that we can trace back along to find the value of \(u(x,t)\). The information about where the solution was coming from was lost. This is the situation of a shock. The correct solution here is to put a discontinuous jump between the left and right states where the characteristics intersect. The speed of the shock can be found from the Rankine-Hugoniot conditions.
It is also possible to get a rarefaction if the characteristics diverge: