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Eigenvalues and Eigenvectors for 3-D TVD Finite Volume Solver
As some of you may remember, in October of last year I posted the following question:
"As part of my dissertation I am developing a TVD solver for 3-D inviscid flow in finite volume format. I have been extensively searching TVD papers for the eigenvector-matrices of left and right eigenvectors in terms of an arbitrary unit normal direction (Nx, Ny, Nz), consistent with finite volume formulation. So far the only papers for 3-D flow that I have seen, decompose the eigenvector matrix along the three Cartesian directions (1, 0, 0); (0, 1, 0); (0, 0, 1) and treat each matrix separately. Anyhow, I decided to derive the more general expression in terms of an arbitrary unit normal (Nx, Ny, Nz) myself with the help of Mathematica. This task was not that simple, because for repeated eigenvalues, the corresponding eigenvectors are not distinct. They form a subspace, and any vector in that subspace is also an eigenvector. I was wondering if anyone has ever seen the eigensystem (eigenvalues & eigenvectors) for 3-D inviscid flow (Euler equations) in general geometries, i.e. for any unit normal direction. If not, I am thinking about publishing my result." Although I received some interesting feedback on the subject, nobody seems to be using a general 3-D expression like the one I have derived, nor does it appear that a similar result has ever been published. For that reason, I decided to write a paper on the subject, which can be downloaded from my website. The paper has been converted to the Adobe Portable Document Format (PDF), which is platform independent and can be viewed with the latest Acrobat Reader (Version 4.0) installed. The Acrobat Reader can be downloaded for free from Adobe's website. A link to Adobe's website, an excerpt of my paper (GIF), and the paper itself (PDF) can be found at the following address, http://www.cfd4pc.com/papers.htm Please feel free to distribute this document to anyone who works in this area of research. Any comments or feedback are welcome! |

Re: Eigenvalues and Eigenvectors for 3-D TVD Finite Volume Solver
Hi Axel,
There is a series of (3-three) papers of J.S. Hestaven and David Gottlieb in SIAM journal of Scientific Computing about a method for the compressible Navier-Stokes equations in general curvilinear system of coordinates. There they treat the boundary conditions on the characteristics of the flow, and therefore they have to linearize the equations and solve the eigenvalues and eigenvectors of the system (characteristics) in a general system of coordinates (like yours). This is similar to what you are talking about. The references I have are as follows: J. S. Hesthaven and D. Gottlieb, 1996, SIAM J. Sci. Comput. vol.17, no.3, p.579 ("A stable penalty method for the compressible NS eqs: I. Open boundary conditions"). J. S. Hesthaven, (preprint) - probably 1996 or 1997 in SIAM J. Sci.Comput., ("A stable penatly...: II. One dimensional domain decomposition schemes"). J. S. Hesthaven (preprint) 1997(?), in SIAM J. Sci. Comput. , ("A stable ..: III multidimensional domain decomposition schemes"). PG |

Re: Eigenvalues and Eigenvectors for 3-D TVD Finite Volume Solver
See the papers by Peter A. Gnoffo describing Program LAURA (LANGLEY AEROTHERMODYNAMIC UPWIND RELAXATION ALGORITHM). He gives the eigenvalues/eigenvectors not only for the flow of perfect gases but chemically-reacting flows.
1. For Perfect gases, see Gnoffo, P.A " An Upwind-Biased, Point Implicit Relaxation Algorithm for Viscous, Compressible Perfect Gas Flows", NASA TP-2953, 1990. This probably has more than you need as he also gives a form for the viscous matrices. 2. In case U are interested for chemically-reacting flows see F. M Cheatwood, & Gnoffo, P.A "User's manual for the Langley Aerothermodynamic Upwind Relaxation Algorithm (LAURA) " NASA TM 4674. |

Re: Eigenvalues and Eigenvectors for 3-D TVD Finite Volume Solver
A general form of this eigensystem can be found in
Vinokur, M., "An Analysis of Finite-Difference and Finite- Volume formulation of Conservation Laws," Journal of Computational Physics, Vol. 81, pp.1-52, 1989 |

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