Ngô Quốc Anh

March 24, 2010

Wave equations: Jumps conditions

Filed under: PDEs — Ngô Quốc Anh @ 23:46

To obtain a restriction about how a solution across a discontinuity propagates we consider the integral conservation law. At first, the conservation law tells us that

\displaystyle u_t(x,t)+\phi_x(x,t)=0

where u is called density and \phi is called flux. The integral form is

\displaystyle \frac{d}{dt}\int_a^b u(x,t)dx=\phi(a,t)-\phi (b,t).

The above equation states that the time rate of change of the total amount of u inside the interval [a,b] must equal the rate that u flows into [a,b] minus the rate that u flows out of [a,b]. Recall that \phi may depend on x and t through dependence on u, then the conservation law can be rewritten as

\displaystyle u_t(x,t)+c(u)u_x(x,t)=0

with c(u)=\phi'(u).

We now assume that x=s(t) is a smooth curve in spacetime along which u suffers a simple discontiuity; that is, assume that u is continuously differentiable for x>s(t) and x<s(t), and that u and its derivatives have finite one-sided limits as x \to s(t)^- and x\to s(t)^+. Then choosing a<s(t) and b>s(t), the integral conservation law may be written

\displaystyle \frac{d}{{dt}}\int_a^{s(t)} u (x,t)dx + \frac{d}{{dt}}\int_{s(t)}^b u (x,t)dx =\phi (a,t) - \phi (b,t).

Leibniz’s rule for differentiating an integral whose integrand and limits depend on a parameter can be applied on the left side

\displaystyle\int_a^{s(t)} {{u_t}} (x,t)dx + \int_{s(t)}^b {{u_t}} (x,t)dx + u({s^ - },t)s' - u({s^ + },t)s' = \phi (a,t) - \phi (b,t)

where

\displaystyle\mathop {\lim }\limits_{x \to s{{(t)}^ - }} u(x,t) = u({s^ - },t), \quad \mathop {\lim }\limits_{x \to s{{(t)}^ + }} u(x,t) = u({s^ + },t).

Now we take a \to s(t)^- and b \to s(t)^+. The first two terms go to zero because the integrand is bounded and the interval shrinks to zero. Therefore we obtain

\displaystyle -s'[u]+[\phi(u)]=0,

where the brackets denote the jump of the quantity inside across the discontinuity (the value of the left minus the value on the right). This is called the jump condition. It relates conditions both ahead of and behind the discontinuity to the speed of discontinuity itself. In this context, the discontinuity in u that propagates along the curve x=s(t) is called a shock wave and the curve x=x(t) is called the shock path, or just shock, s' is called the shock speed and the magnitude of the jump in u is called the shock strength.

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

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