# Finite Volume II Philip Mocz. Goals Construct a robust, 2nd order FV method for the Euler equation (Navier-Stokes without the viscous term, compressible)

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Finite Volume II Philip Mocz

Goals Construct a robust, 2nd order FV method for the Euler equation (Navier-Stokes without the viscous term, compressible) Simulate some shocks!

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

Conservative Form State vector Flux

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

Discretize Flux across face ij F ij

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

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 Can also solve this exactly, called the Riemann Problem (see Mathematica demo) Fastest propagation speed in the system

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

Making the scheme 2 nd order Gradient estimation

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

2 nd order flux computation Extrapolate primitive variables in space and ½ time step before calculating the flux facecell center LR Left and right states at the interface

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

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