28.7.1. General hyperbolic problem

28.7.1.1. Integral equations

For a volume \(V\) at rest

\[ \dfrac{d}{dt} \int_{V} \mathbf{u} + \oint_{\partial V} \hat{n} \cdot \mathbf{F} = \int_{V} \mathbf{s} \]

Integral equations for a control volume \(\ V \ \) at rest

Integral equation for a control volume at rest \(V\)

\[\begin{split}\begin{aligned} & \dfrac{d}{dt} \int_{V} \mathbf{u} + \oint_{\partial V} \mathbf{u} \vec{u} \cdot \hat{n} = \int_{\partial V} \hat{n} \cdot \mathbf{f} + \int_{V} \mathbf{s} \\ & \dfrac{d}{dt} \int_{V} \mathbf{u} + \oint_{\partial V} \hat{n} \cdot \mathbf{F} = \int_{V} \mathbf{s} \\ \end{aligned}\end{split}\]

being \(\mathbf{u}\) the vector of the conservative variables, \(\mathbf{u} \in \mathbb{R}^n\), and \(\vec{u}\) the velocity field in the \(d\)-dimensional domain, and \(\mathbf{F} := - \mathbf{f} + \vec{u} \mathbf{u}\).

Integral equations for an arbitary domain \(\ v_t \ \)

Integral equation for an arbitrary domain \(v_t\), with Reynolds’ transport theorem

\[\begin{split}\begin{aligned} & \dfrac{d}{dt} \int_{v_t} \mathbf{u} + \oint_{\partial v_t} \mathbf{u} ( \vec{u} - \vec{v}_b ) \cdot \hat{n} = \int_{\partial v_t} \hat{n} \cdot \mathbf{f} + \int_{v_t} \mathbf{s} \\ & \dfrac{d}{dt} \int_{v_t} \mathbf{u} - \oint_{\partial v_t} \mathbf{u} \vec{v}_b \cdot \hat{n} + \int_{\partial v_t} \hat{n} \cdot \mathbf{F} = \int_{v_t} \mathbf{s} \\ \end{aligned}\end{split}\]

28.7.1.1.1. Jump conditions

(28.1)\[ \hat{n} \cdot \vec{v}_b \, \left[ \, \mathbf{u} \, \right] = \hat{n} \cdot \left[ \, \mathbf{F} \, \right] \]

From integral equations on \(\ v_t \ \) to jump relations

Collapsing \( v_t\) on a surface only the boundary terms on the two sides of the surface persist and the balance equations becomes

\[\begin{split}\begin{aligned} \mathbf{0} & = \oint_{\partial v_t} \left( - \hat{n} \cdot \vec{v}_b \mathbf{u} + \hat{n} \cdot \mathbf{F} \right) = \\ & = \int_{S_{-}} \left( - \hat{n}_{-} \cdot \vec{v}_b \mathbf{u}_{-} + \hat{n}_{-} \cdot \mathbf{F}_{-} \right) + \int_{S_{+}} \left( - \hat{n}_{+} \cdot \vec{v}_b \mathbf{u}_{+} + \hat{n}_{+} \cdot \mathbf{F}_{+} \right) = \\ & = \int_{S} \left( - \hat{n}_{-} \cdot \vec{v}_b ( \mathbf{u}_{-} - \mathbf{u}_{+} ) + \hat{n}_{-} \cdot ( \mathbf{F}_{-} - \mathbf{F}_{+} ) \right) \end{aligned}\end{split}\]

being \(\hat{n}_{-} = - \hat{n}_{+}\). From the arbitrariness of the surface \(S\), jump relations follow

\[\hat{n} \cdot \vec{v}_b \, \left[ \, \mathbf{u} \, \right] = \left[ \, \mathbf{F} \, \right] \ ,\]

being \([ \ \cdot \ ] = (\cdot)_{+} - (\cdot)_{-}\) the jump across the surface.

28.7.1.2. Differential equations

In regions where the fields are smooth, differential equations follow from integral equations using theorems of differential calculus.

28.7.1.3. Conservative form

Conservative form of the differential equations reads

(28.2)\[ \partial_t \mathbf{u} + \nabla \cdot \mathbf{F}(\mathbf{u}) = \mathbf{s} \ . \]

From integral equations on \(\ V \ \) to conservative form of differential equations

For a volume \(V\) at rest

\[\begin{split}\begin{aligned} \dfrac{d}{dt} \int_{V} \mathbf{u} + \oint_{\partial V} \hat{n} \cdot \mathbf{F} & = \int_{V} \mathbf{s} \\ \int_{V} \partial_t \mathbf{u} + \oint_{\partial V} \hat{n} \cdot \mathbf{F} & = \int_{V} \mathbf{s} \end{aligned}\end{split}\]

or using components

\[\begin{aligned} \int_{V} \partial_t u_i + \oint_{\partial V} n_k F_{ki} & = \int_{V} s_i \ , \end{aligned}\]

and using [divergence theorem] under the assumption of sufficient regularity of the flux

\[\begin{split}\begin{aligned} 0 & = \int_{V} \partial_t u_i + \oint_{\partial V} n_k F_{ki} - \int_{V} s_i = \\ & = \int_{V} \partial_t u_i + \int_{V} \partial_k F_{ki} - \int_{V} s_i = \\ & = \int_{V} \left\{ \partial_t u_i + \partial_k F_{ki} - s_i \right\} \ , \end{aligned}\end{split}\]

and from the aribitrariness of the domain \(V\),

\[\partial_t u_i + \partial_k F_{ki} = s_i \ ,\]

or using abstract vector notation

\[\partial_t \mathbf{u} + \nabla \cdot \mathbf{F} = \mathbf{s} \ .\]

\[\partial_t \mathbf{u} + \sum_{k=1}^{d} \mathbf{A}^{(k)}(\mathbf{u}) \, \partial_k \mathbf{u} = \mathbf{s} \ ,\]

with the convection matrices

\[\left\{ \mathbf{A}^{(k)} \right\}_{i \ell} = \partial_{u_{\ell}} F_{ki} \ .\]

From conservative form to convective form of differential equations

Starting from the component expression {eq}`` of the convective differential equations

\[\begin{split}\begin{aligned} s_i & = \partial_t u_i + \partial_k F_{ki}(u_\ell) = \\ & = \partial_t u_i + \partial_{u_{\ell}} F_{ki}(u_\ell) \partial_k u_\ell = \\ & = \partial_t u_i + A_{i \ell}^{(k)}(u_\ell) \partial_k u_\ell = \\ & = \partial_t u_i + \sum_{k=1}^éd} A_{i \ell}^{(k)}(u_\ell) \partial_k u_\ell \ , \end{aligned}\end{split}\]

having defined \(A_{i \ell}^{(k)} = \partial_{u_{\ell}} F_{ki}\), or using vector notation

\[\partial_t \mathbf{u} + \sum_{k=1}^{d} \mathbf{A}^{(k)}(\mathbf{u}) \partial_k \mathbf{u} = \mathbf{s} \ .\]

28.7.1.4. Method of characteristics

28.7.1.4.1. Spectral decomposition

Summary of method of characteristics in multi-dimensional domains.

Let

\[\begin{split}\begin{aligned} \mathbf{u}(t, \vec{r}): & \ \mathbb{R} \times \mathbb{R}^d \rightarrow \mathbb{R}^n \\ \mathbf{F}( \mathbf{u}): & \ \mathbb{R}^{n} \rightarrow \mathbb{R}^{d} \times \mathbb{R}^n \\ \mathbf{s}(t, \vec{r}): & \ \mathbb{R} \times \mathbb{R}^d \rightarrow \mathbb{R}^n \\ \end{aligned}\end{split}\]

the conservative form of a hyperbolic problem reads

\[\begin{split}\begin{aligned} \partial_t \mathbf{u} + \nabla \cdot \mathbf{F}(\mathbf{u}) & = \mathbf{s} \\ \partial_t u_i + \partial_j F_{ji} & = s_i \ . \end{aligned}\end{split}\]

Expanding the divergence of the flux, the convective (quasi-linear? It makes little to no sense to me…) form follows

\[\begin{split}\begin{aligned} s_i & = \partial_t u_i + \sum_{j=1}^{d} \partial_j F_{ji} = \\ & = \partial_t u_i + \sum_{j=1}^{d} \sum_{k=1}^{n} \partial_j u_k \partial_{u_k} F_{ji} = \\ & = \partial_t u_i + \sum_{j=1}^{d} \sum_{k=1}^{n} A^{j}_{ik}(\mathbf{u}) \, \partial_j u_k \ , \end{aligned}\end{split}\]

for every component \(i = 1:n\), or

\[\mathbf{s} = \partial_t \mathbf{u} + \sum_{j=1}^{d} \mathbf{A}^j \partial_j \mathbf{u} \ .\]

This problem can be made a little more general with by defining \(\mathbf{x} = (t, \mathbf{r}) = ( x_0, \vec{r} )\), as

\[\sum_{j=0}^{d} \mathbf{A}^j \, \partial_j \mathbf{u} = \mathbf{s} \ .\]

The former expression of the hyperbolic problem immediately follows if \(\mathbf{A}_0 = \mathbf{I}_n\), \(A^0_{ik} = \delta_{ik}\), and

\[\widetilde{\nabla} \cdot \widetilde{\mathbf{F}} = \partial_0 \mathbf{F}_0 + \sum_{j=1}^{d} \partial_j \mathbf{F}_j = \widetilde{\nabla} \cdot \begin{bmatrix} \ \mathbf{u} \ | \ \mathbf{F} \ \end{bmatrix} \ .\]

as, for \(i = 1:n\),

\[\delta_{ik} = \partial_{u_k} F_{0i} \quad \rightarrow \quad F_{0i} = u_k \delta_{ik} = u_i \ .\]

Taylor expansion of a solution. Starting from the solution on a manifold determined by the equation \(S: f(\mathbf{x}) = 0\), \(\mathbf{x}_0 \in S\), if the solution is differentiable, the solution in a point \(\mathbf{x} = \mathbf{x}_0 + \Delta \mathbf{x}\) reads

\[\begin{split}\begin{aligned} \mathbf{u}(\mathbf{x}) & \sim \mathbf{u}(\mathbf{x}_0) + \Delta \mathbf{x} \cdot \nabla \mathbf{u}(\mathbf{x}_0) \\ u_i(\mathbf{x}) & \sim u_i(\mathbf{x}_0) + \Delta x_j \frac{\partial u_i}{\partial x_j}(\mathbf{x}_0) \ . \end{aligned}\end{split}\]

Now, with a change of coordinates from \(\mathbf{x}\) to \(\boldsymbol\xi = (n, \mathbf{t})\), i.e. to local normal and tangential directions,

\[\begin{aligned} \nabla \mathbf{u} = \hat{\mathbf{x}}_i \frac{\partial}{\partial x_i} \mathbf{u} = \hat{\mathbf{x}}_i \frac{\partial \mathbf{u}}{\partial x_i} = \hat{\mathbf{x}}_i \frac{\partial \mathbf{u}}{\partial \xi_k} \frac{\partial \xi_k}{\partial x_i} = \hat{\boldsymbol\xi}_k \frac{\partial \mathbf{u}}{\partial \xi_k} \end{aligned}\]

Inserting in the hyperbolic equation

\[\begin{split}\begin{aligned} s_i & = \sum_{j=0}^{d} A^{j}_{ik} \frac{\partial u_k}{\partial x_j} = \\ & = \sum_{j=0}^{d} A^{j}_{ik} \sum_{\ell=0}^{d} \frac{\partial u_k}{\partial \xi_\ell} \frac{\partial \xi_\ell}{\partial x_j} = \\ & = \sum_{\ell=0}^{d} \sum_{j=0}^{d} A^{j}_{ik} \frac{\partial \xi_\ell}{\partial x_j} \frac{\partial u_k}{\partial \xi_\ell} = \\ & = \sum_{\ell=0}^{d} \sum_{j=0}^{d} A^{j}_{ik} \xi^\ell_j \frac{\partial u_k}{\partial \xi_\ell} \ , \end{aligned}\end{split}\]

and separating the normal \(\ell = 0\) from the tangential \(\ell=1:d\) coordinates,

\[\begin{split}\begin{aligned} s_i & = \sum_{j=0}^{d} A^j_{ik} \xi^0_j \frac{\partial u_k}{\partial \xi_0} + \sum_{\ell=1}^{d} \sum_{j=0}^{d} A^j_{ik} \xi^t_j \frac{\partial u_k}{\partial \xi_t} = \\ & = \sum_{j=0}^{d} A^j_{ik} n_j \frac{\partial u_k}{\partial n} + \sum_{\ell=1}^{d} \sum_{j=0}^{d} A^j_{ik} t^\ell_j \frac{\partial u_k}{\partial t_\ell} = \\ & = \sum_{j=0}^{d} n_j \mathbf{A}^j \partial_n \mathbf{u} + \sum_{\ell=1}^{d} \sum_{j=0}^{d} t^\ell_j \mathbf{A}^j \partial_{t_\ell} \mathbf{u} \ . \end{aligned}\end{split}\]

On a smooth surface where the solution is known, all the tangential derivatives of the solution are known as well. In order to evaluate the normal derivative \(\partial_n \mathbf{u}\) from the PDE, the (formal) inversion of the matrix

\[\mathbf{A}_{\hat{\mathbf{n}}} := \sum_{j=0}^{d} n_j \, \mathbf{A}^j \ ,\]

must be feasible, and thus \(\mathbf{A}_{\hat{\mathbf{n}}}\) must be invertible, to formally get

\[\partial_n \mathbf{u} = \mathbf{A}_{\hat{\mathbf{n}}}^{-1} \left( \mathbf{s} - \sum_{\ell=1}^{d} \mathbf{A}_{\hat{\mathbf{t}}_\ell} \partial_{t_\ell} \mathbf{u} \right) \ .\]

This expression of the normal derivative of the solution - whenever it exists - can be used to find the approximation of the solution in normal direction w.r.t. the surface \(S\), i.e. with \(\Delta \mathbf{x} = \Delta \ell \hat{\mathbf{n}}\) as

\[\begin{split}\begin{aligned} \mathbf{u}(\mathbf{x}) & \sim \mathbf{u}(\mathbf{x}_0) + \Delta \ell \, \hat{\mathbf{n}} \cdot \nabla \mathbf{u}(\mathbf{x}_0) = \\ & = \mathbf{u}(\mathbf{x}_0) + \Delta \ell \, \partial_{n} \mathbf{u}(\mathbf{x}_0) \ . \end{aligned}\end{split}\]

If the matrix \(\mathbf{A}_{\hat{\mathbf{n}}}\) is diagonalizable, it’s invertible if it has no zero eigenvalue. Let \(\hat{\mathbf{n}}_i\) be a unit vector so that the matrix \(\mathbf{A}_{\hat{\mathbf{n}}}\) has an eigenvalue equal to zero \(s_0 = 0\), with right and left eigenvectors \(\mathbf{r}_0\), \(\mathbf{l}_0\). Recalling the expression of the PDE in normal and tangential components

\[\mathbf{s} = \mathbf{A}_{\hat{\mathbf{n}}} \partial_n \mathbf{u} + \sum_{\ell=1}^{d} \mathbf{A}_{\hat{\mathbf{d}}_\ell} \partial_{t_\ell} \mathbf{u} \ ,\]

left-multiplying by \(\mathbf{l}\) gives

\[\begin{split}\begin{aligned} \mathbf{l}^T_0 \mathbf{s} & = \mathbf{l}_0^T \left( \mathbf{A}_{\hat{\mathbf{n}}} \partial_{\hat{n}} \mathbf{u} + \sum_{\ell=1}^{d} \mathbf{A}_{\hat{\mathbf{d}}_\ell} \partial_{t_\ell} \mathbf{u} \right) = \\ & = \mathbf{l}_0^T \sum_{\ell=1}^{d} \mathbf{A}_{\hat{\mathbf{d}}_\ell} \partial_{t_\ell} \mathbf{u} = \\ & = \mathbf{l}_0^T \sum_{k=0}^{d} \mathbf{A}^k \partial_k \mathbf{u} \ , \end{aligned}\end{split}\]

i.e. the compatibility condition on the surface with unit normal \(\hat{\mathbf{n}}\) that makes \(s_0 = 0\),

\[\mathbf{l}_0^T \mathbf{s} = \mathbf{l}_0^T \sum_{k=0}^{d} \mathbf{A}^k \partial_k \mathbf{u} = \mathbf{l}_0^T \widetilde{\nabla} \cdot \widetilde{\mathbf{F}} \ .\]

Surfaces with unit normal vectors \(\hat{\mathbf{n}}\) making eigenvalues of the matrix \(\mathbf{A}_{\mathbf{n}}\) equal to zero are defined as characteristic surfaces. On each characteristic surface, the corresponding compatibility condition holds.

28.7.1.4.2. Change of variables

Let \(\mathbf{u}\) the set of conservative variables1 and let \(\mathbf{v}\) another set of variables. Let the transfomration of variables \(\mathbf{v}(\mathbf{u})\) be invertible and let \(\mathbf{u}(\mathbf{v})\) be the inverse transformation. Let \(\mathbf{U}_{\mathbf{v}}\) the gradient of the transformation \(\mathbf{u}(\mathbf{v})\) w.r.t. variables \(\mathbf{v}\) and \(\mathbf{V}_{\mathbf{u}}\) the gradient of the inverse transformation \(\mathbf{v}(\mathbf{u})\), so that the differentials of the variables read

\[d \mathbf{u} = \mathbf{U}_{\mathbf{v}} d \mathbf{v} \quad , \quad d \mathbf{v} = \mathbf{V}_{\mathbf{u}} d \mathbf{u} \ .\]

These gradients are mutually inverse, \(\mathbf{U}_{\mathbf{v}} = \mathbf{V}^{-1}_{\mathbf{u}}\). The convective form of the equations using \(\mathbf{u}\) or \(\mathbf{v}\) as primary variables are respectively

\[\begin{split}\begin{aligned} & \partial_t \mathbf{u} + \sum_{k=1}^{d} \mathbf{A}^{\mathbf{u}, k} \partial_k \mathbf{u} = \mathbf{s}^{\mathbf{u}} \\ & \partial_t \mathbf{v} + \sum_{k=1}^{d} \mathbf{A}^{\mathbf{v}, k} \partial_k \mathbf{v} = \mathbf{s}^{\mathbf{v}} \\ \end{aligned}\end{split}\]

with

\[\begin{split}\begin{aligned} \mathbf{A}^{\mathbf{v},k} & = \mathbf{V}_{\mathbf{u}} \mathbf{A}^{\mathbf{u},k} \mathbf{U}_{\mathbf{v}} \\ \mathbf{s}^{\mathbf{v}} & = \mathbf{V}_{\mathbf{u}} \mathbf{s}^{\mathbf{u}} \ . \end{aligned}\end{split}\]

The equations using \(\mathbf{v}\) immediately follows after introducing the relation between the differentials and pre-multiplication of the equations by \(\mathbf{V}_{\mathbf{u}}\),

Change of variables and spectral decomposition of the matrix \(\ \mathbf{A}_{\hat{n}} \ \).

Starting from a set of variables \(\mathbf{u}\) and the convective form of the equations

\[\mathbf{s} = \partial_t \mathbf{u} + \sum_{k} \mathbf{A}^k \partial_k \mathbf{u} \ ,\]

using another set of variables \(\mathbf{v}\), the equations become

\[\begin{split}\begin{aligned} \mathbf{s} & = \partial_t \mathbf{u} + \sum_{k} \mathbf{A}^k \partial_k \mathbf{u} \\ \mathbf{s} & = \mathbf{U}_{\mathbf{v}}(\mathbf{v}) \partial_t \mathbf{v} + \sum_{k} \mathbf{A}^k \mathbf{U}_{\mathbf{v}}(\mathbf{v}) \partial_k \mathbf{v} \ , \end{aligned}\end{split}\]

being \(d \mathbf{u} = \mathbf{U}_{\mathbf{v}} d \mathbf{v}\). If the transformation is non-singular, and \(\mathbf{U}_{\mathbf{v}}^{-1} = \mathbf{V}_{\mathbf{u}}(\mathbf{v})\), then the differential equations can be recast as

\[\partial_t \mathbf{v} + \sum_{k} \mathbf{V}_{\mathbf{u}} \mathbf{A}^k \mathbf{U}_{\mathbf{v}} \partial_k \mathbf{v} = \mathbf{V}_{\mathbf{u}} \mathbf{s}\]

With a coordinate transformation, the normal and tangential matrices become

\[\begin{split}\begin{aligned} \mathbf{A}^{\mathbf{v}}_{\mathbf{n}} & = \sum_k n_k \mathbf{V}_{\mathbf{u}} \mathbf{A}^k \mathbf{U}_{\mathbf{v}} = \\ & = \mathbf{V}_{\mathbf{u}} \underbrace{\sum_k n_k \mathbf{A}^k}_{ \mathbf{A}^{\mathbf{u}}_{\mathbf{n}} } \mathbf{U}_{\mathbf{v}} = \mathbf{V}_{\mathbf{u}} \mathbf{A}^{\mathbf{u}}_{\mathbf{n}} \mathbf{U}_{\mathbf{v}} \ , \end{aligned}\end{split}\]

\[\mathbf{A}^{\mathbf{v}}_{\mathbf{t}_i} = \dots = \mathbf{V}_{\mathbf{u}} \mathbf{A}^{\mathbf{u}}_{\mathbf{t}_i} \mathbf{U}_{\mathbf{v}} \ .\]

If the change of variables is non-singular, the Jacobian matrices \(\mathbf{V}_\mathbf{u}\), \(\mathbf{U}_\mathbf{v}\) are non singular and reciprocally inverse. Thus, the determinant of the matrix \(\mathbf{A}_{\mathbf{n}}\) is independent from the set of chosen variables, as

\[\begin{split}\begin{aligned} \text{det}\left( \mathbf{A}_{\mathbf{n}}^{\mathbf{v}} \right) & = \text{det}\left( \mathbf{V}_{\mathbf{u}} \right) \text{det}\left( \mathbf{A}_{\mathbf{n}}^{\mathbf{u}} \right) \text{det}\left( \mathbf{U}_{\mathbf{v}} \right) = \\ & = \text{det}\left( \mathbf{V}_{\mathbf{u}} \right) \text{det}\left( \mathbf{A}_{\mathbf{n}}^{\mathbf{u}} \right) \text{det}\left( \mathbf{V}_{\mathbf{u}} \right)^{-1} = \\ & = \text{det}\left( \mathbf{A}_{\mathbf{n}}^{\mathbf{u}} \right) \ . \end{aligned}\end{split}\]

Thus, the eigenvalues are independent from the set of variables as well

\[\begin{split}\begin{aligned} 0 & = \text{det}\left( s \mathbf{I} - \mathbf{A}^{\mathbf{v}}_{\mathbf{n}} \right) = \\ & = \text{det} \left( \mathbf{V}_{\mathbf{u}} \right) \text{det}\left( s \mathbf{I} - \mathbf{A}^{\mathbf{u}}_{\mathbf{n}} \right) \text{det} \left( \mathbf{U}_{\mathbf{v}} \right) = \\ & = \text{det}\left( s \mathbf{I} - \mathbf{A}^{\mathbf{u}}_{\mathbf{n}} \right) \ . \end{aligned}\end{split}\]

The relation between right eigenvectors immediately follows

\[\begin{split}\begin{aligned} \mathbf{A}^{\mathbf{v}}_{\mathbf{n}} \mathbf{R}^\mathbf{v} & = \mathbf{R}^\mathbf{v} \mathbf{S} \\ \mathbf{V}_{\mathbf{u}} \mathbf{A}^{\mathbf{u}}_{\mathbf{n}} \mathbf{U}_{\mathbf{v}} \mathbf{R}^\mathbf{v} & = \mathbf{R}^\mathbf{v} \mathbf{S} \\ \mathbf{A}^{\mathbf{u}}_{\mathbf{n}} \underbrace{\mathbf{U}_{\mathbf{v}} \mathbf{R}^\mathbf{v}}_{\mathbf{R}^\mathbf{u}} & = \underbrace{\mathbf{U}_{\mathbf{v}} \mathbf{R}^\mathbf{v}}_{\mathbf{R}^\mathbf{u}} \mathbf{S} \\ \mathbf{A}^{\mathbf{u}}_{\mathbf{n}} \mathbf{R}^\mathbf{u} & = \mathbf{R}^\mathbf{u} \mathbf{S} \ , \end{aligned}\end{split}\]

with \(\mathbf{R}^{\mathbf{u}} = \mathbf{U}_{\mathbf{v}} \mathbf{R}^\mathbf{v}\), or for any individual eigenvector \(\mathbf{r}^\mathbf{u}_i = \mathbf{U}_{\mathbf{v}} \mathbf{r}^{\mathbf{v}}_i\).

The relation between left eigenvectors analogously follows

\[\begin{split}\begin{aligned} \mathbf{L}^\mathbf{v} \mathbf{A}^{\mathbf{v}}_{\mathbf{n}} & = \mathbf{S} \mathbf{L}^\mathbf{v} \\ \mathbf{L}^\mathbf{v} \mathbf{V}_{\mathbf{u}} \mathbf{A}^{\mathbf{u}}_{\mathbf{n}} \mathbf{U}_{\mathbf{v}} & = \mathbf{S} \mathbf{L}^\mathbf{v} \\ \underbrace{\mathbf{L}^\mathbf{v} \mathbf{V}_{\mathbf{u}}}_{\mathbf{L}^\mathbf{u}} \mathbf{A}^{\mathbf{u}}_{\mathbf{n}} & = \mathbf{S} \underbrace{\mathbf{L}^\mathbf{v} \mathbf{V}_{\mathbf{u}}}_{\mathbf{L}^\mathbf{u}} \\ \mathbf{L}^\mathbf{u} \mathbf{A}^{\mathbf{u}}_{\mathbf{n}} & = \mathbf{S} \mathbf{L}^\mathbf{u} \ , \end{aligned}\end{split}\]

with \(\mathbf{L}^{\mathbf{u}} = \mathbf{L}^\mathbf{v} \mathbf{U}_{\mathbf{v}}\), or for any individual eigenvector \(\left( \mathbf{l}^\mathbf{u}_i \right)^T = \left( \mathbf{l}^\mathbf{v}_i \right)^T \mathbf{U}_\mathbf{v}\), or \(\mathbf{l}^\mathbf{u}_i = \mathbf{U}_\mathbf{v}^T \mathbf{l}^\mathbf{v}_i\).

• Eigenvalues, right and left eigenvectors

• Characteristic lines/surfaces, and compatibility conditions

28.7.1.5. Riemann problem

• Useful in numerical schemes in finite volume methods, using Godunov flux

28.7.1.5.1. Linearization - Roe intermediate state

• Local linearization of the problem, to reduce the computational cost of solving non-linear Riemann problems at all the interfaces in a grid in FVM

28.7.1.6. Boundary conditions

• characteristic-based

• wall

• …

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1

Once the problem is written in convective form, this set of variable could be an arbitrary set of variables and not only the conservative variables.