# How are eigenvalues of a cell in the finite volume discretization calculated? (for euler fluid equations)

I am building a program to numerically solve the euler fluid equations based on finite volume discretization and am having trouble on a step where the eigenvalues of a cell need to be calculated according to the picture below. The eigenvalues for each cell are calculated by dotting the velocity vector with the normal of the face and adding C, the speed of sound. Dotting the velocity vector with the normal of the face of the cell make sense. Therefore, to my understanding, you must have 4 eigenvalues, one for each face, for each cell. I am confused by the next part, which says to calculate the average of the neighboring cells at each face. The notation used does not differentiate between the left or right face, making it seem like there is only one eigenvalue for each computational direction (for each cell). I am having trouble understanding the exact representation of this term and its meaning.

• What is the eigenvalue problem solved here? It should be something in the form L x = lambda x, where L is a linear operator, and x is a state vector. – Maxim Umansky May 2 at 19:26
• im not exaclty sure why they call these eigenvalues because I am not too familiar with the problem at hand as well as eigenvalues in general. To be more clear about what is being calculated. This is a term in the calculations for artificial dissipation. a more detailed description of the calculations can be seen on page 3 under 2.1 Scalar Dissipation Model for the following pdf. ntrs.nasa.gov/api/citations/20040110322/downloads/… – Frosty May 2 at 20:15
• i.gyazo.com/27198a10d6a39f655dfbc2107ad68dbf.png here is a link to the exact formulations. – Frosty May 2 at 20:16

## 1 Answer

The eigenvalues given correspond to the flux jacobians in the Euler equations. Physically, they represent the speed at which waves of information can travel in a space-time domain.
The goal is to compute the eigenvalues at each face based on the values in each neighboring cell so there is no 'left' or 'right' face in this equation, but instead a left and right cell. The face that you are calculating $$\lambda$$ is denoted $$i+\frac{1}{2}$$. The cell to the left is denoted $$i$$ and the cell to the right is $$i+1$$.

• To be clear then, λ i-1/2 would be the absolute value of the velocity of the center cell and the left cell dotted with the normal of the left face, plus C, averaged? i.e. λ i-1/2 = 1/2(|Uij.n|+C + |Ui-1j.n| + C) – Frosty May 3 at 1:59
• Based on the formulation you sent above, that is correct. – tevank May 3 at 2:16