The upwind scheme for system of equations $$U_t + A U_x = 0$$ where $$ U =\begin{pmatrix} u_1\\ u_2 \end{pmatrix}\qquad A =\begin{pmatrix} 2 &1\\ 1 &2 \end{pmatrix}$$ is $$ \frac{U_j^{n+1} – U_j^{n}}{\Delta t} + A^+ \frac{U_{j}^n – U_{j-1}^n}{\Delta x} + A^-\frac{U_{j+1}^n – U_j^{n}}{\Delta x} =0$$ Find $A^+$ and $A^-$. For given $\Delta t$, find the minimum $\Delta x$ so that the CFL condition can be satisfied. What kind of boundary condition must be given to solve this problem?
Im trying to understand how this differs from the scalar $u_t + a u_x = 0$ case. We need to put it in the form
$$ U_j^{n+1} = U_{j}^n + \nu ( F(U) ) $$
where $\nu = \frac{A \Delta t }{\Delta x} $. So, are they asking to infer $A^+$ and $A^-$ so that the scheme is conservative? How do we do this?
Best Answer
Introduce the following diagonalization of $A = R \Lambda R^{-1}$: $$ R = \frac{1}{\sqrt 2}\begin{pmatrix} -1 & 1\\ 1 & 1 \end{pmatrix} , \qquad \Lambda = \begin{pmatrix} 1 & 0\\ 0 & 3 \end{pmatrix} , \qquad R^{-1} = \frac{1}{\sqrt 2}\begin{pmatrix} -1 & 1\\ 1 & 1 \end{pmatrix} , $$ and consider $V = R^{-1}U$. We have two independent scalar advection equation for the coordinates of $V$, since $V$ satisfies the diagonal system $V_t +\Lambda V_x = 0$. Here, both eigenvalues of $A$ are positive, so that the same upwind direction is chosen for both advection equations. Going back to the original variables $U$, we therefore have $A^+=A$ and $A^-=0$. For general linear hyperbolic systems of conservation laws, the same methodology leads to choosing $A^+$ and $A^-$ as follows: $$ A^\pm = \tfrac12 (A\pm |A|), \qquad |A| = R |\Lambda| R^{-1} . $$ The Courant number $\text{Co}$ is defined with respect to the largest absolute value of the eigenvalues of $A$. Therefore, the Courant-Friedrichs-Lewy condition writes as $$\text{Co} = 3 \frac{\Delta t}{\Delta x} \leq 1 $$ in the present case.