Timeline for Interpretation of Eigenvalues and Eigenvectors of an hyperbolic conservation law $\partial_t W + A \partial_x W = 0$
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 8, 2016 at 22:43 | vote | accept | Amine HANINI | ||
| Feb 8, 2016 at 21:51 | comment | added | Amine HANINI | Let us continue this discussion in chat. | |
| Feb 8, 2016 at 20:20 | comment | added | tpg2114 | @AmineHANINI It looks like you skipped a step -- you found the eigenvectors and eigenvalues of $\mathcal{A}$, not of $J^{-1} \mathcal{A} J$. In other words, you didn't symmetrize $\mathcal{A}$ first. | |
| Feb 8, 2016 at 17:17 | comment | added | Amine HANINI | I don't have any problem with the mathematical analysis of the hyperbolic pde, I am just a little confused with the physical interpretation. With my specific problem, namely Shallow Water Equation : $W=(h,hu)$, $ \mathcal A = \begin{pmatrix} 0 & 1 \\ - u^2 + c^2 & 2 u \\ \end{pmatrix}$ we have : $ \mathcal \Lambda = \begin{pmatrix} u-c & 0 \\ 0 & u+c \\ \end{pmatrix}$ and $ \mathcal T = \begin{pmatrix} 1 & 1 \\ u-c & u+c \\ \end{pmatrix}$ | |
| Feb 8, 2016 at 16:21 | comment | added | tpg2114 | And it works in any dimension. But, in higher dimension you cannot (at least for the Euler equations) diagonalize both dimensions at the same time. Read through the linked PDF, it explains pretty well what happens and it provides all of the equations needed to work through and verify the answers for 1D and 2D in both conservative and non-conservative form of the Euler equations. | |
| Feb 8, 2016 at 16:19 | comment | added | tpg2114 | @AmineHANINI Doing things with non-linear equations throws it entirely out the window. What we do in the Euler equations is assume the Jacobian is fixed and this makes the equation "linear", or sometimes called quasi-linear. It's only true in a small neighborhood around a particular operating point, but that's enough to do the analysis needed. Regarding your question about the Shallow Water Equation -- I'm guessing that is the equation you need to do this for? I'll happily answer specific questions about particular steps, but I won't do the full derivation -- that is an exercise for you to do. | |
| Feb 8, 2016 at 15:54 | comment | added | Amine HANINI | if it's possible can you illustrate me your point on the Sallow Water Equation example in the conservative form ? Thanks | |
| Feb 8, 2016 at 15:32 | comment | added | Amine HANINI | thank you for your answer. If i understand well your explanation works only in the linear case, what about the non-linear one? And what is it the same with the 2D case ? | |
| Feb 5, 2016 at 0:38 | history | edited | tpg2114 | CC BY-SA 3.0 |
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| Feb 5, 2016 at 0:30 | history | answered | tpg2114 | CC BY-SA 3.0 |