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.

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