28.3. Scalar linear equation

Differential equation.

\[\begin{split}\begin{aligned} r & = \partial_t u + a \partial_x u && \text{ convective form} \\ & = \partial_t u + \partial_x ( a u ) && \text{conservative form} \end{aligned}\end{split}\]

Integral equation. On a control volume \(V\) at rest

\[0 = \dfrac{d}{dt} \int_V u + \oint_{\partial V} n_x a u \ .\]

Integral equation in an arbitrary domain \(v_t\) can be derived using Reynolds’ transport theorem.

Method of characteristics. Let \(U(t) = u(X(t), t)\). From derivative of composite functions,

\[d_t U(t) = \partial_t u|_{X(t),t} + \dot{X}(t) \partial_x u|_{X(t),t} \ ,\]

and thus the PDE can be recast as a system of two ODEs

\[\begin{split}\begin{cases} \dot{X}(t) = a \\ \dot{U}(t) = r \end{cases}\end{split}\]

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