1. Finite Volume Method
Philip Mocz

2. Goals
Construct a robust, 2nd order FV method for the Euler equation (Navier-Stokes without the viscous term)
Simulate the Kelvin Helmholtz Instability!

3. FV formulation State vector (conservative variables)
Equation of state:

4. Conservative Form
State vector
Flux

5. Integrate Now a surface integral, by Gauss’ Theorem
Integrated state vector

6. Discretize i j
Flux across face
Fij

7. Conservative Property
Flux is anti-symmetric
Question: what are the fluid variables that are conserved?

8. Computing the Flux
Don’t just average the flux of 2 sides, use Upwind Flux
(i.e., add an advective term, which creates some numerical diffusion for stability)
We will use the local Rusanov Flux
Can also solve this exactly, called the Riemann Problem
Fastest propagation speed in the system

9. Conservative <--> Primitive forms
Primitive state vector: W = (rho, vx, vy, P)’;
Euler equations in primitive form
Question: why is this not conservative form?

10. Making the scheme 2nd order
Gradient estimation

11. Slope limiting Detect local minima and flatten them!
Question: are there negative side effects?

12. 2nd order flux computation
Extrapolate primitive variables in space and ½ time step before calculating the flux
Left and right states at the interface
face
cell center L R

13. That’s it!
Now let’s look at some code, the implementation details will take a while to digest