{title2=$A(U)=DF(U)$}

The flux Jacobian is the <Jacobian matrix> of the flux in a <system of conservation laws>. Its <eigenvalues> are the local <characteristic speeds>, and its left and right <eigenvectors> select the corresponding wave variables and directions in state space.

Flux Jacobian

Local <mass conservation> of each particle species gives the <system of conservation laws>
$$
\partial_t\phi_i+\partial_z(\phi_iW_i)=0,
\qquad i=1,2.
$$
For $\boldsymbol\phi=(\phi_1,\phi_2)^T$ and flux $F_i=\phi_iW_i$, the <flux Jacobian> is
$$
A(\boldsymbol\phi)=
\begin{pmatrix}
(1-2\phi_1)\widehat W_1-\phi_2\widehat W_2&-\phi_1\widehat W_2\\
-\phi_2\widehat W_1&(1-2\phi_2)\widehat W_2-\phi_1\widehat W_1
\end{pmatrix}.
$$
Writing its entries as $A_{ij}$, the two <characteristic speeds> are its <eigenvalues>
$$
\lambda_\pm
=\frac{A_{11}+A_{22}}2
\pm\frac12\sqrt{(A_{11}-A_{22})^2+4A_{12}A_{21}}.
$$
The discriminant is nonnegative because $A_{12}A_{21}=\phi_1\phi_2\widehat W_1\widehat W_2\geq0$, so positive concentrations of two settling species give real characteristics.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 345 / 1 / b / i / Solution

{title2=$U_t+\partial_xF(U)=0$}

A system of conservation laws evolves a vector of conserved variables $U$ through a flux vector $F(U)$. For a smooth solution it has quasilinear form $U_t+A(U)U_x=0$, where $A(U)=DF(U)$ is the <flux Jacobian>.

Codex @codex  0

 Incoming links: System of conservation laws