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Perhaps I am being pedantic but I saw the word 'propagate' used in the method of characteristics multiple times but I am not sure what it means. Here are a few examples I encountered.

$1)$ The boundary conditions and initial conditions propagates along the characteristics.

$2)$ Consider the ODE $U_x+U_y=0$ subject to $U(0,y)=1_{\{y\geq 0\}}.$ Then the discontinuity of the initial condition propagates along the characteristics.

I am not sure what the meaning of these sentences are trying to convey, could someone please kindly explain to me?

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This is the idea behind the method of characteristics. The dynamic of the PDE reduces to ODE systems along the characteristic curves. The initial conditions of the ODE are given by the boundary conditions of the PDE, so that these get "propagated" along the characteristic curves to the area where these are regular, do not intersect.

In the example the characteristic curves are described by $y-x=c$. Along these the value of $U$ stays constant, this constant is thus propagated to the whole line, in total $U(x,y)=U(0,y-x)={\bf 1}_{\{y\ge x\}}$.

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