Flux computation in unstructured grids
Posted July 16, 2010 at 10:49 by praveen
Tags finite volume, unstructured
Consider finite volume scheme on unstructured grids for the Euler equations.
Let
be normal to a cell face and whose magnitude is equal to face area. Let
be the conserved vector. The finite volume update equation using forward Euler time discretization is

Here
is a normal vector pointing from current cell "j" into the neighbouring cell "k". Note that the conserved variable Q is updated in the global Cartesian coordinate frame.
As an example, the Rusanov flux would be defined as
![F(U,V,n) = \frac{1}{2}[ F(U) \cdot n + F(V) \cdot n] - \frac{1}{2} \lambda(U,V,n) (V - U) F(U,V,n) = \frac{1}{2}[ F(U) \cdot n + F(V) \cdot n] - \frac{1}{2} \lambda(U,V,n) (V - U)](/Forums/vbLatex/img/afbf757e031082087c6c22031139a5e1-1.gif)
Here, we have used the definition

where
is the velocity vector, etc., and

with
being speed of sound. Note that there are many other ways to define
.
Let



Here

As an example, the Rusanov flux would be defined as
![F(U,V,n) = \frac{1}{2}[ F(U) \cdot n + F(V) \cdot n] - \frac{1}{2} \lambda(U,V,n) (V - U) F(U,V,n) = \frac{1}{2}[ F(U) \cdot n + F(V) \cdot n] - \frac{1}{2} \lambda(U,V,n) (V - U)](/Forums/vbLatex/img/afbf757e031082087c6c22031139a5e1-1.gif)
Here, we have used the definition

where


with


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