Ngô Quốc Anh

March 26, 2010

Wave equations: Rarefaction waves

Filed under: PDEs — Ngô Quốc Anh @ 21:15

Another difficulty can occur with nonlinear equations having discontinuous initial or boundary data.

Example. Consider the equation

$\displaystyle u_t+uu_x=0, \quad x\in \mathbb R, t>0$,

subject to the initial condition

$\displaystyle {u_0}(x) = \left\{ \begin{gathered} 0, \quad x < 0, \hfill \\ 1, \quad x > 0. \hfill \\ \end{gathered} \right.$

The characteristic curves are the following: $u=0$ on $x={\rm const.}$ and $u=1$ on $x=t$. Because $u$ is constant along characteristics. the data $u = 1$ are carried into the region $x > t$ along characteristics with speed $1$, and the data $u = 0$ are carried into the region $x < 0$ along vertical (speed $0$) characteristics.

There is a region $0 < x < t$ void of the characteristics. In this case there is a continuous solution that connects the solution $u = 1$ ahead to the solution $u = 0$ behind. We simply insert characteristics (straight lines in this case) passing through the origin into the void in such a way that $u$ is constant on the characteristics and $u$ varies continuously from $1$ to $0$ along these characteristics. In other words, along the characteristic $x = ct$, $0 < c < 1$, take $u = c$. Consequently, the solution to the Riemann problem is

$\displaystyle u(x,t) = \left\{ \begin{gathered} 0,\quad x < 0, \hfill \\ \frac{x}{t},\quad 0 < \frac{x}{t} < 1, \hfill \\ 1,\quad x > t. \hfill \\ \end{gathered} \right.$

A solution of this type is called a centered expansion wave, or a fan; other terms are release wave or rarefaction wave. The idea is that the wave spreads as time increases.

Source: J.D. Logan, An introduction to nonlinear partial differential equations, 2nd, 2008; Section 3.1.