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2.1.5 Solution along characteristic curves

For a quasi-linear PDE

\frac{\partial \rho}{\partial t} + v_{} \! \left( \rho, x, t \right)   \frac{\partial \rho}{\partial x} = 0
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a curve \bgroup\color{DEFcolor}$c(t)$\egroup with
\frac{\mathrm{d} c}{\mathrm{d} t} = v_{}
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or sometimes the corresponding map \bgroup\color{DEFcolor}$\mathbb{R} \rightarrow \mathbb{R}^2\!\!: t \rightarrow \left( c \left( t \right) , t \right)$\egroup is called characteristic curve or characteristic. For the linear advection equation (63) these curves have the general form
c(t) = v_{}   t + x_0
\enspace .
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Along \bgroup\color{DEFcolor}$c(t)$\egroup we get for a solution \bgroup\color{DEFcolor}$\tilde{\rho}$\egroup of Eq. (63)
\frac{\mathrm{d} }{\mathrm{d} t} \tilde{\rho} \left( c \lef...
...{} \frac{\partial \tilde{\rho}}{\partial x}
\enspace .
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The solution \bgroup\color{DEFcolor}$\tilde{\rho}$\egroup is constant along the characteristic \bgroup\color{DEFcolor}$c(t)$\egroup.