28.4. Linear vector equation

Differential equation.

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

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

\[\int_{V} \mathbf{r} = \dfrac{d}{dt} \int_V \mathbf{u} + \oint_{\partial V} n_x \mathbf{A} \mathbf{u} \ .\]

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

Method of characteristics. Let \(\mathbf{A}\) be diagonalizable,

\[\begin{split}\begin{aligned} \mathbf{A} \mathbf{R} & = \mathbf{R} \mathbf{S} \\ \mathbf{A} & = \mathbf{R} \mathbf{S} \mathbf{L} \\ \end{aligned}\end{split}\]

Using spectral decomposition and multiplying the PDE by \(\mathbf{L}\), the PDE is transformed in a set of scalar PDE equations,

\[\begin{split}\begin{aligned} \mathbf{L} \mathbf{r} & = \mathbf{L} \left( \partial_t \mathbf{u} + \mathbf{R} \mathbf{S} \mathbf{L} \partial_x \mathbf{u} \right) = \\ & = \mathbf{L} \partial_t \mathbf{u} + \mathbf{S} \mathbf{L} \partial_x \mathbf{u} = \\ & = \partial_t \mathbf{v} + \mathbf{S} \partial_x \mathbf{v} \ , \end{aligned}\end{split}\]

or

\[\partial_t v_i + s_i \partial_x v_i = L_{ik} r_k \ ,\]

having introduced the characteristic variables \(\mathbf{v}(x,t) := \mathbf{L} \mathbf{u}(x,t)\). If the source is not function of the unknown function \(\mathbf{u}(x,t)\), this is a system of decoupled PDEs. The solution of the vector PDE follows from the solution of \(n\) scalar PDE linear equations. Using methods of characteristics, the solution of the \(i^{th}\) scalar PDE equation can be recast as the solution of pairs of ODEs, after writing \(V_i(t) = v_i(X_i(t), t)\), being \(X_i(t)\) the equation of a curve belonging to the \(i^{th}\) familiy of characteristics,

\[\begin{split}\begin{cases} \dot{X}_i = s_i \\ \dot{V}_i = L_{ik} r_k \ . \end{cases}\end{split}\]

As the spectral decomposition of a constant matrix is constant, the characteristic lines are straight lines \(X_i(t) = s_i t + c\). In general, the solution of the dynamical equations of the characteristic variables \(V_i\) is a full system of \(n\) ODEs.

Particular case: no source. As a particular case, if the source term is identically zero, \(\mathbf{r}(x,t) = \mathbf{0}\), characteristic variable \(V_i(t)\) is constant along each characteristic line \(X_i(t;\dots)\) of the \(i^{th}\) family of characteristics.

…

\[u_k(x,t) = R_{ki} v_i(x,t) = R_{ri} v_i(x_{0,i} + s_i(t-t_{0,i}), t) \ .\]