Hi:

In a celebrated paper by Beam & Warming, it was shown that a similarity transformation that diagonalizes one of these Jacobians can be developed such that it symmetrizes the remaining two jacobians occuring in the 3-D Euler equations. This holds for the particular case of a primitive variable or conservation variables( U ) formulation.

My question is this. Is it possible to determine a transformation of variables V= V(U), and the associated Euler Flux Jacobians, such that all 3 Jacobians are simultaneously diagonalized by the same similarity transformation. In other words, can the 3 Flux flux jacobians have the same eigen-vectors ??

An affirmative answer to this question will lead to new, improved methods for the solution of the Euler equations.

C.S.Venkatasubban Engineering Specialist Computational Aerodynamics Learjet Inc.

>In a celebrated paper by Beam & Warming, it was shown that a similarity
>transformation that diagonalizes one of these Jacobians can be developed
>such that it symmetrizes the remaining two jacobians occuring in the 3-D
>Euler equations. Symmetrization of the jacobian matrices as a computational tool is suggested in the context of SUPG FEM by Mallet and Hughes. See their collection of papers in Comp. Meth in Applied Mechanics and Engineering, 1986. The symmetric form of the eqns. they use is addressed by A. Harten, JCP, vol 49, 1983.
>Is it possible to determine a transformation of
>variables V= V(U), and the associated Euler Flux Jacobians, such that all 3
>Jacobians are simultaneously diagonalized by the same similarity
>transformation. The answer, at least as far I know, is no. You can do that for 1D flows, for which the equations can be written as a set of scalar eqns. using variables W = L * U where L is the matrix of left eigenvectors of A=dF/dU.

This allows to use scalar schemes on each eqn. and hence to choose optimal upwinding or optimal dissipation independently for each eqn., based on the corresponding eigenvalue. In more than one dimension, one can still use the same type of similarity transformation, with the matrix C_n = \sum A_i \n_i (n_i being real numbers, A_i being the individual jacobians: A_i = dF_i/dU) replacing A. Now the eigenvectors L_n (hence the variable W) depend upon n and the transformed set of eqns. shows coupling between the different variables, except for the entropy eqn., which fully decouples. This dependence upon (at least) one free (directional) parameter "n" prompted Deconinck et al. to look for some choices of this parameter that eventually made these coupling terms vanish: this is published in Lecture Notes in Physics, vol. 264, Springer-Verlag 1986.

More recently, P.L. Roe noticed that, with a suitable choice of a local preconditioning matrix, namely the one that is referred to as the van Leer, Lee, Roe matrix (AIAA CP 91-1552), the preconditioned equations take a block diagonal form, with entropy and now also total enthalpy decoupling as scalar eqns. The remaining block is again diagonal for 2D supersonic flows and one ends up with 4x4 diagonal system, while it shows an elliptic character for subsonic 2D flows. The 3D case is a bit more involved.

The message which seems to stem sounds like: use upwinding for all 4 scalar eqns. in supersonic flows, while in subsonic use upwinding for the entropy and total enthalpy eqns. and a "central" scheme for the remaining (non-diagonal) block.

Some references: Notes on Numerical FLuid Mechanics, vol 57, Vieweg, 1996 The Ph.D. thesis by Henry Paillere (Von Karman Institute) The Ph.D. thesis by Lisa Mesaros (Michigan) on the web page The Ph.D. thesis by W.T. Lee (Michigan) The Ph.D. thesis by D. Lee (Michigan) on the web page. A general framework for block diagonalizations is given by Turkel & Roe in VKI Lecture Series 1996-06 and other references therein. Also S. Taasan has some published paper (ICASE rep ?) on the subject.

One can NOT simultaneously diagonalize all three jacobians (unfortunately).

A very nice property of the euler equations is it's rotational invariance, though. if you compute the euler-fluxes, one can compute F*nx+G*ny+H*nz by computing F, G, H and then multiply them with the components of the normal vector. This means to evaluate 3 fluxes. Since you basically compute the flux in the direction normal to a face, one can simply rotate the velocities in to a local coordinate system and then compute ONE flux, based on a new set of variables. And then apply the inverse rotation matrix to the flux. So you only have one flux to compute, which is more elegant and probably much more effective. This is particularly nice for unstructured solvers, where you cannot simply transform your coordinates in a curvlinear coordinate system. It is just F*nx+G*ny+H*nz=T_R^{-1}*F(T_R*U) where T_R is the rotation matrix which maps the cartesian coordinate system into a local coordinate system, associated with your face, respectively Gauss-Quadrature-Point you compute the fluxes for.

Maybe this hint is valuable for you as well ?! If you are interested in more details about this, simply send me an e-mail.

Regards,

Frank Bramkamp

Hi,

You can read Hirsch's Book(Numerical Computation of Internal and External Flows, Vol2, JOHNS AND & SONS). There are quite some metrics related to multi-dimensinal euler equation.