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Flux computation in unstructured grids

Posted July 16, 2010 at 10:49 by praveen

Consider finite volume scheme on unstructured grids for the Euler equations.

Let n=(n_x,n_y,n_z) be normal to a cell face and whose magnitude is equal to face area. Let Q be the conserved vector. The finite volume update equation using forward Euler time discretization is

Q^{n+1}_j = Q^n_j - \Delta t \sum_{k \in N(j)} F(Q_j^n, Q_k^n, n_{jk})

Here n_{jk} 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)

Here, we have used the definition

F(U) \cdot n = \begin{bmatrix}
\rho (u \cdot n)\\
\rho u_x (u \cdot n) + p n_x\\
\rho u_y (u \cdot n) + p n_y\\
\rho u_z (u \cdot n) + p n_z\\
(E+p)(u \cdot n)

where u=(u_x,u_y,u_z) is the velocity vector, etc., and

\lambda(U,V,n) = \max\{ |u(U) \cdot n| + a(U) |n|, |u(V) \cdot n| + a(V) |n| \}

with a being speed of sound. Note that there are many other ways to define \lambda.
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