Eigenvalues & Eigenvectors in General Geometries for 3-D TVD Solver
 

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), i.e. 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.

Re: Eigenvalues & Eigenvectors in General Geometries for 3-D TVD Solver
 

Sometimes, people solve riemann problems in the x,y,z direction (or i,j,k) and then multiply it by the normal vectors. E.g. F*n_x+G*n_y+H*N_z. So one solves 3 riemann problems. This is (of course) not required. A more general and better approach is (that is what I do) to use the rotational invariance of the euler equations, which reduces everything to a solely 1D problem.

It is simply,

F*n_x+G*n_y+H*N_z = T_R^{-1}*F(T_R*U)

where U is the vector of conserved variables and T_R a rotational matrix which rotates the equations for the momentum equations into a local coordinate system, alligned with each face (in fact for each gauss quadrature point). So one computes ONE Riemann problem and maps it back into the x,y,z coordinate system. The euler equations have this nice property.

In my opinion, that makes live much easier and one can use the standard eigenvalues and vectors. In this case e.g. u denotes the velocity, perpendicular to the face and v,w the appropriate tangential velocities.

The nice thing is: The difference between a 3D and 2D code gets in fact very small and can be treated via ifdefs in a nice way.

I already did this stuff for a 3D unstructured finite volume scheme, that works on arbitrary grids of any topology using 2D/3D reconstruction schemes, Vankatakrishnan limiter, Barth etc...works fine so far.

Hope this helps a bit.

Frank

Re: Eigenvalues & Eigenvectors in General Geometries for 3-D TVD Solver
 

I think the group in Brown University has been working in curved geometry with spectral methods. Since spectral methods require the treatment of the boundary conditions on the characteristics of the flow, they probably solved for the eigenvector (characteristics of the flow) and eigenvalue in curved geometry using a Jacobian, etc..

You might want to try to contact them:

Jan Hesthaven: jansh@cfm.brown.edu

David Gottlieb: dig@cfm.brown.edu

David Gottlieb is probably pretty busy, so try Jan.

I have myself solved for the egienvalues and eigenvectors in spherical and cylindrical coordinates for the Normal vector in the radial direction. The matrices were similar to the one in cartesian coordinates (after linearization of the euler equations).

Patrick

Re: Eigenvalues & Eigenvectors in General Geometries for 3-D TVD Solver
 

You should consult papers on "truly multidimensional" numerical methods, such as those by Hirsch, Roe, and van Leer. Most ordinary "TVD" papers will not have what you want - it is very difficult to usefully exploit truly-multidimensional characteristics in a numerical method and thus few attempt it.

The last chapter of my book, "Computational Gasdynamics," includes some expressions for the multidimensional characteristics of the Euler equations that you might find helpful. Unfortunately, there was not room to include all of the derivations such as those involving the Jacobian matrices. I have quite lengthy formal notes with the full details, originally planned for inclusion in the book, which I may someday see fit to distribute.

Bascially, in multidimensions, instead of a Jacobian matrix (two-dimensional), you have a Jacobian tensor (three-dimensional, two deep for two-dimensions, and three deep for three-dimensions). Multiplying by unit vectors, such as coordinate unit axis vectors, brings the Jacobian tensor from three- to two-dimensions, where it is much easier to express and work with. On the other hand, the three-dimensional tensor is invariant to coordinate transformations, which is not true of the Jacobian matrices derived from the coordinate unit axis vectors.