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For the equation:

$\partial_t u + \partial_x(\frac{u^2} 2)$ = 0

with the initial condtions: $$ u(x,0) = g(x) = \begin{cases} 2, \quad \text{si} \quad x<1 \\ 1,\quad \text{si} \quad 1<x<2\\ 2, \quad \text{si} \quad x>2 \\ \end{cases} $$

The characteristic equation is $$ x(t) = \xi + q'(g(\xi))t = \xi + g(\xi)t = \begin{cases} \xi + 2t, \quad \text{si} \quad \xi \leq1 \\ \xi + t, \quad \text{si} \quad 1<\xi<2 \\ \xi + 2t, \quad \text{si} \quad \xi>2 \end{cases} $$

There should one shock wave going from (1;0) and a refaction one on (2;0)

a) Shock wave (1,0)

$$ \begin{cases} s'(t) = \frac{q(u_R) - q(u_L)}{u_R-u_L} = \frac{(u_R+u_L)}{2} = \frac{(2+1)}{2} = \frac{3}{2} \\ s(0) = 1 \end{cases} \implies x = s(t) = \frac{3}{2}t + 1 $$

b) Rarefaction wave $(2,0)$ between $x(t) = 1+t$ and $x(t) = 1+2t$

$$ u(x,t) = (q')^{-1}\left(\frac{x-1}{t}\right) = \frac{x-2}{t} \\ $$ Can't find the intersection (x*,t*)

Rasule
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1 Answers1

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Note that the rarefaction is continuous at its boundaries. Thus, one boundary of the rarefaction is incorrect. You actually need to find when the shock trajectory $x=s(t)=1+\frac32 t$ intersects the boundary $x={\color{red}2}+t$ of the rarefaction. A plot of the characteristic lines might help (see this related post).

EditPiAf
  • 20,898