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Finite Volume Method
Philip Mocz
Goals
Construct a robust, 2nd order FV method for the Euler
equation (Navier-Stokes without the viscous term)
Simulate the Kelvin Helmholtz Instability!
FV formulation
State vector (conservative variables)
Equation of state:
Conservative Form
State vector
Flux
Integrate
Now a surface integral, by Gauss’ Theorem
Integrated state vector
Discretize
Flux across face
i
j
Fij
Conservative Property
Flux is anti-symmetric
Question: what are the fluid variables that are conserved?
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
Fastest
propagation
speed in the
system
Can also solve this exactly, called the Riemann Problem
Conservative <--> Primitive forms
Primitive state vector: W = (rho, vx, vy, P)’;
Euler equations in
primitive form
Question: why is this not conservative form?
Making the scheme 2nd order
Gradient estimation
Slope limiting
Detect local minima and flatten them!
Question: are there negative side effects?
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
That’s it!
Now let’s look at some code, the implementation details will
take a while to digest